TUBULENT FLUXES OF HEAT, MOISTURE AND MOMENTUM: MEASUREMENTS AND PARAMETERIZATION

General definitions and dimensions:

Flux of property c: Fc = wc = wc + w’c’ (1)

Momentum flux

=w’v’ [kg/(ms2)=N/m2] (3)

Sensible heat flux:

Qh=Cpw’’ [J/(m2s)=W/m2] (4)

Latent heat flux:

Qe=Lw’q’ [J/(m2s)=W/m2] (5)

Thus, we need to measure w’v’, w’’, w’q’

wherew’ is vertical velocity fluctuationv’ is horizontal velocity fluctuation is air density’ is potential temperature fluctuationCp is specific heat capacity at

constant pressureq’ is specific humidity fluctuationL is specific heat of evaporation

If the area considered is small and horizontally uniform and during a given time atmospheric conditions are steady, than

w = 0 (2)

The eddy correlation method

Statistical meaning of fluxes:

w’v’, w’’, w’q’ can be considered as the second mixed moments, i.e. co-variances of variables

Requirements for the direct measurements Covariances should be measured, and not variances Time resolution should be high (10-20 Hz) Time of record should be relatively long (more than 20 min)

Instrumentation: Sonic anemometer Fast-response thermometer Fast-response infrared hydrometer

ship

Stationary platform

Problems: Technically difficult Expensive

Description of a typical eddy correlation package:The 3-D sonic anemometer uses three pairs of orthogonally oriented, ultrasonic transmit/receive transducers to measure the transit time of sound signals traveling between the transducer pairs. The wind speed along each transducer axis is determined from the difference in transit times. The sonic temperature is computed from the speed of sound which is determined from the average transit time along the vertical axis. A pair of measurements are made along each axis normally 100 times per second. 10 measurements are averaged to produce 10 wind measurements along each axis and 10 temperatures each second. The infrared hygrometer measures the water vapor density by detecting the absorption of infrared radiation by water vapor in the light path. Two infrared wavelength bands are used, one centered on a band strongly absorbed by water vapor, and one centered on a band (the reference band) which is not aborbed. By normalizing the absorption band by the reference band, instrument drifts caused by light source and photodetector changes are eliminated. Measurements are made 40 times per second. 4 measurements are averaged to produce 10 water vapor density measurements each second.

Sonic anemometer is based on the estimation of speed of sound

V = [(t1-t2)C2] / 2d

We need to measure variances of wind velocity in 3 directions

w’v’, w’’, w’q’

SST, Ta, q, V, SLPmassively measured

Parameterization of surface turbulent fluxes

parameterization

z

uKmzx

From K-theory the relationship between stress and vertical distribution of wind under stationary conditions:

Km(u/z) ~ [U2]

What should be a combination of massively measured parameters to be compared with direct flux measurements?

What is this velocity? This is the velocity related to the mean speed of turbulent eddies u*. Now for the stress:

2*uzx

Km is eddy diffusivity ~ [m2s-1] ~ [UL]. u/z is velocity gradient ~ [s-1] ~ [U/L]

(10)i.e. eddy velocities in neutral conditionsare independent on height.

Velocity u* is called friction velocity and serves as indicator of velocity scale in the exchange theory. It is proportional to root-mean square vertical velocity:

u* ~ wWe need also the size of the largest eddies. It is related to the height, because the upper limit of eddy is determined by the distance from the sea surface. Thus, for Km:

Km:= u*zWhere is the von Karman constant of proportionality (0.4)(11)

z

uKmzx

2*uzx non-dimensional wind shear

z

u

uz

u

u

zm ln**

(12)

Measurements of mean velocity atseveral levels, plotted on a log-scale

give a straight live with a slope of

/u*

u/u*

ln z

z0 is the integration constant. It

represents the height, on which the mean wind speed calculated by (13) goes to zero. This is the so-called roughness length.

We can calculate friction velocity and wind speed at any level step by step

0

* lnz

zuu

Land data:

Integration of (12) gives equation for the vertical distribution of mean velocity:

(13)

Von Karman constant

Sea surface:

the shape of the surface and the roughness lengths vary with wind speed; marine roughness length may depend on the characteristics of short capillary waves, but does not depend on longer waves (sea);

wind sea is not what theroughness length is about!!!

marine roughness elements are not stationary, they move together with significant waves.

Dimensional analysis under light wind speed when the ocean surface

is smooth shows that: z0 ~ /u* however, no observational

evidence has been found for that, except for Roll (1961).

Henry Charnock (1955) assumed that roughness lenght depends on surface stress and gravity, and derived experimental formula:

z0 = 0.0123u*2 / g (14)

where 0.0123 is the Charnock «constant», which varies in different studies from 0.011 to 0.018.

0

*2* ln,

z

zuuuzx

under neutral conditions:

neutral drag coefficient:

On aerodynamically smooth surfaces he stress is exerted by viscosity and within the laminar sublayer the stress is proportional to wind speed

z0 ~ /u*

On rough surfaces, the effects of viscosity are negligible and the transfer of stress to the surface is done by the pressure differences between the upwind and downwind sides of the obstacles

z0 = 0.0123u*2 / g

0

2

2*

ln,

zz

Cu

uC dndn

2uCdnzx

Stress: (15)

z

qKqw

zKw qt

'',''

Passive mean properties under neutral conditions (,q)

For the potential temperature and humidity with the use of K-theory:

(16)

We introduce now (with the analogy to u*) two new scales: q* and *

**** '','' uqwuw For non-dimensional temperature and humidity gradients:

q

mq

t

mt K

K

z

q

qz

q

q

z

K

K

zz

z

ln,

ln ****

Ratio of exchange coefficients for passive property and momentum is a fundamental question of boundary layer turbulence. In general, it depends on stability, but for the neutral conditions can be taken as constant.

(17)

Eddy diffusivity for temperature Eddy diffusivity for humidity

(18)

Measurements of mean velocity atseveral levels, plotted on a log-scale

give a straight live with a slope of

/u*/q*

lnz

q

aa

aqa

z

z

z

zqqq

ln

ln

*

*

Unlike the momentum, the mean values of θ, q do not approach zero at the surface, but come to the values which depend on th eprocesses at

the surface. Let’s define them as θa, qa and integrate (18):

Integration constants zaθ and zaq are the heights at which <θ> and <q> are equal to the surface values θa and qa. Although these are analogues of the roughness length, the mechanisms producing zaθ and zaq are completely different from those for zaq.

aq

at

qquCqw

uCw

''

''under neutral conditions:

neutral coefficientsfor heat and moisturetransfer:

Bulk formulas for the transfer of sensible heat and moisture(still under the neutral conditions)

qa

qn

ta

tn

zz

zz

C

zz

zz

C

lnln

lnln

0

2

0

2

Now, if the conditions are not neutral, we can:

1. Measure the mean variables and derive the already known product u(SST-)

2. Measure the flux directly using eddy correlation method3. Try to statistically compare these two

w’’

(SST-Ta) V

Ct=?

But!!!! what to do when the conditions are not neutral???

In this case we have to account for the modification of profiles of wind and passive properties due to surface layer instability.

TKEEuuu

M

iii 222

2'22

We have to consider the balance of turbulent kinetic energy

Turbulent kinetic energy per unit mass:

TKE = ½ u’i2 = ½ (u’2+ v’2+ w’2) (21)

Reynolds averaging of TKE:

(22)

Equation of TKE transformation can be derived theoretically:

see, e.g. Blackadar, A. “Turbulence an diffusion in the atmosphere”. We will consider it as a generally assumed conservation equation of energy.

The energy of mean motion The eddy energy

ikik

T

i

upxdt

dpgw

dt

ud

12

2

Work doneby stress atboundary

Transformationthrough the

potential energy

Transformation to andfrom mechanical energy

Transformation to and from

internal energy

External sources

Conservation of the total kinetic energy:

(23)

2/1 2ikiikikiiki

k

MM

uuuuuupupx

Dt

Dpw

gwg

Dt

TKEED

Steps skipped (Blackadar 1997, Appendix A):

2) Multiplication of the Reynolds averaged Navier-Stockes equation term-by-term by ui and summing over i:

iikkikk

iikM

M uuupxx

uuu

Dt

Dpwg

Dt

DE

1

3) Subtraction (25) from (24) term by term:

21 2ikiki

kk

iik uuup

xx

uuuw

g

Dt

TKED

(24)

1) Reynolds averaging of the total kinetic energy of conservation:

(25)

(26)

21 2ikiki

kk

iik uuup

xx

uuuw

g

Dt

TKED

EXMB

Dt

TKED (27)

Production of TKE by

buoyancy (+/-)(reversible)

TKE transform into internal energy (+)

irreversible

Mechanicalproduction of

TKE(normally positive)

External sources

(with either sign)

z

u

z

uK

z

uM m

3*

2

Ri – gradient

Richardson number

The mechanical production rate (from K-theory):

The buoyant production rate (in the dry atmosphere):

zK

TCQ

TC

gQw

gw

gB tph

p

h

,

The flux Richardson number:

2

z

u

zK

Kg

z

uTC

gQ

M

BR

m

t

p

hf

defines the stability: Rf>0 (stable), Rf<0 (unstable, turbulence

is maintained by convection).

(28)

(29)

(30)

h

p

h

pL gQ

uTCL

gQ

uTCh

3

*3* ,

Universal functionto be estimated

Height on which the two rates of TKE production are equal gives the length scale, known as Monin-Obukhov length:

independent on height has the same sign as the Richardson number, defining the stability falls to infinity under neutral conditions

Now we can derive a non-dimensional height:

)(*

mm z

u

u

z

(31)

z L , (32)which gives the length scale and implies similarity of wind profiles (Monin-Obukhov 1954):

(33)

SellersPanofsky

Yamamoto Ellison Kazansky

116 34 mm

Solution exists but it is not convenient to use

Estimation of ():

1. KEYPS – equation:

2. Bussinger et al. (1971) from the experiments in Kansas:

)(0,151

)(0,7.41

41

unstable

stable

m

m

3. Dyer (1974):

)(0,161

)(0,51

41

unstable

stable

m

m

4. Large and Pond (1981):

)(0,161

)(0,71

41

unstable

stable

m

m

(34)

ln zneutral

U

stable

unstable

General formulation:

)(0,1

)(0,1

unstable

stable

m

m

Similarity functions for temperature and humidity:

)(0,

)(0,2

2

unstable

stable

mqt

mqt

(35)

Now we can finally derive the bulk formulae!!!

Since we need to estimate flux at a given height z, the equations:

)(),(),(***

qtm z

q

q

z

z

z

z

u

u

z

should be integrated from the surface to this height, that gives the values of mean variables at height z:

qqq

qz

ttt

tz

mmu

z

z

zqqq

z

z

z

zuuu

0

*0

0

*0

0

*0

ln

,ln

,lnStability correctionfunctions, which arethe integrals of non-dimensional profiles(Paulson 1970):

(36)

)(0,1),(0,2

1ln2

)(0,1),(0,2

tan22

1ln

2

1ln2

1

1111

stableunstable

stableunstable

tttt

tt

mmmmm

m

(37)

)(

),(

,

0

0

2

qqCLQ

CCQ

uC

zqq

ztph

zd

Bulk formulae:

mu

qq

qq

mu

tt

tt

mu

d

z

z

z

zC

z

z

z

zC

z

zC

0

1

0

2

1

0

1

0

2

2

0

2

lnln

,lnln

ln

(38)

The problem of transfer coefficients:

1

0

1

0

2

1

0

1

0

2

2

0

2

10ln

10ln

,10

ln10

ln

,10

ln

zzC

zzC

zC

qqn

ttn

dn

qqdnqn

dnd

tn

t

ttdntn

dnd

tn

t

umudndn

d

zCC

CC

C

C

zCC

CC

C

C

LzzCC

C

10ln1

,10ln1

,10ln1

1

2/1

2/1

Neutral coefficients at a given reference height (h=10m):

Thus, we need to know either roughness length, or neutral transfer coefficients to determine the fluxes!

PU XXMBDt

TKEDXE

Inertial dissipation method:

TKE – equation:

for the steady horizontally homogeneous flow:

pwz

uw

zz

uuw

g 1

20

22*

Scaling parameter: z u*3

PUmL

z

Vertical divergenceof turbulent transport

Vertical divergenceof pressure transport

PU XXpwz

uw

z

1

2

2

McBean and Elliott (1975):

is independent on L

Large and Pond (1981): PU

Very important: this does not mean that both U, P are small (a

very mistake of many), simply they balance each other. We do not neglect terms XU and XP, but we neglect the sum of these!

Now we can assume: B + P = (39)

or

L

zm

Surprise!!!!: if we know , we can find u* !!!!!!!!!!!

(40)

Turbulence is dissipative by nature

Only from the lager scale eddies

How to know ?

Molecular destruction of turbulent motions is largest for the smallest size eddies. Thus, if we have only “very small eddies”, they will be destructed by molecular processes and turbulence will decay. But from where “the smallest eddies” come from?

Thus, there should be a dissipation which allows larger scale eddies to become smaller eddies. And the more intense small scale turbulence is, the greater rate of dissipation occurs.Thus, dissipation, as well as the other terms in the TKE equation should depend on the size of eddies. This allows us to consider the TKE equation in a spectral form, where the contributions of terms will depend on the wave length or the eddy size

Attribute of turbulence No.5 from the last lecture:

0 MB

Dt

TKED

Let’s assume that we observed that this works for a particular environment during a given time:

large eddies small eddies

TKE

Let’s assume that we are able to estimate all terms for the eddies of different sizes. Now, let’s plot TKE vs eddy size (wave number)

?

Traditional TKE equation, assuming that the vertical divergence of turbulent transport and pressure transport are neglected and the flow is steady:

Is it also valid for the particular range of the eddy sizes?

ktSkz

ktTR

z

uktkt

g

t

ktS,2

,,,

, 2

Spectral representation of TKE equation (Batchelor 1953, Stull 1988):

?MBDt

TKED

The local timetendency of

the k-th component

of TKE

Buoyantproduction

associated withthe k-th component

of <w’’>

Mechanicalproduction

associated withthe k-th

componentof <w’u’>

Viscousdissipation of the k-th

componentof TKE

Convergence ofTKE transport

across the spectrum

Mid-size eddies feel neither the effect of viscosity nor the generation. How do they get their energy? How do they loss it?

S

Large eddies Small eddies

Generation of TKE

Viscosity

Here eddies get the energy inertially from the larger size eddies and loss it in the same way to smaller size eddies!

Kolmogorov (1941), Obukhov (1941):

The cascade rate of energy down the spectrum must balance the dissipationrate at the smallest eddies sizes.

3/53/2)( kKkS K is the Kolmogorov constant (0.5510%)

k = f / U, f = /2

links time and space for turbulent motion

Spatial structure of turbulence: frozen turbulence (Taylor 1938):

Now we can re-write the Kolmogorov’s hypothesis:

3/23/53/2 2/)( UfKfS

But now we can measure the time series of, e.g. wind speed at high frequency and derive the friction velocity from (40), (41):

3/12/1

*

2)(

f

Lz

z

UK

ffSu

m

If the mean wind U is directed in x-direction, spatial statistics of U can be considered, assuming that the U(x) is frozen in time, i.e. these statistics move along the x-axis with the mean speed U.

(41)

(42)

For the fluxes of passive scalars you need to determine the spectral level of fluctuations in the inertial sub-range and solve the budget equation for this scalar:

3/23/53/2 2/)( UfKfS

2*

2/1

)(

)(u

fS

ffSKqw

u

q

q

Kolmogorov constant pertaining to q

dissipation function for q

Spectral similarity

3/23/53/2 2/)( UfKfS

Summary of inertial dissipation method:

So, you do not need to measure <w’x’> anymore, you now need only:

1. Make fast response measurements of velocity to get the time series

2. Calculate the spectrum of the time series3. Plot the spectra on log-log graph4. Find inertial subrange, i.e. the portion of the spectrum that

exhibits a –5/3 slope5. Fit a strait line to this part of your graph6. Pick any point on this line and determine the values of S and k7. Compute mean wind speed U during your measurements 8. Solve the equation for :

9. Find wind sress:

3/12/1

*

2)(

f

Lz

z

UK

ffSu

m

Find sensible and latent fluxes:2*

2/1

)(

)(u

fS

ffSKqw

u

q

q