Evaluation of the Inertial Dissipation Methodover Land

Examensarbete vid Institutionen för geovetenskaper ISSN 1650-6553 Nr 51

Evaluation of the Inertial Dissipation Method over Land

Björn Carlsson

Evaluation of the Inertial Dissipation Method over Land

Abstract

The focus was to evaluate the Inertial Dissipation Method (IDM) over land during unstable conditions. This was done by comparing the friction velocity, u , from the

_*

eddy-correlation method (ECM) with u from IDM. The result can be used to see if

_*

IDM can rely on its assumptions, since the surface-layer theory is more fulfilled over land, where we for example do not have wave influence. The measurements were taken from the flat agricultural site Marsta, 8 km north of Uppsala, Sweden.

The result shows that IDM works well over land (relative standard deviation of about 10 %). For weakly unstable stratification, it is enough to use an assumption of neutral conditions in the IDM calculations. If it is more unstable, one should include the influence of stability and also include an imbalance term. The imbalance term is introduced implicitly by varying the effective Kolmogorov’s constant with stability.

The effective Kolmogorov’s constant used here, varied from 0.50 up to above 0.80.

To calculate u

_*

using IDM, a first estimation of u

_*

was calculated from a

parameterised drag coefficient C

D

. Also, to imitate the measuring setup on a moving platform on the sea, the stability parameter, z/L, was calculated using a bulk-

estimated heat flux. The large scatter showed that it is important that the parameterisations of C

D

and the heat flux are good.

One can conclude that the IDM as a method to determine turbulent fluxes over land

works satisfactory. The larger scatter over sea is probably an effect of sea wave

influence, even though the sea surface is considered more homogeneous and the

conditions more stationary.

Sammanfattning av “Utvärdering av inertial- dissipationsmetoden över land”

Syftet var att utvärdera inertial-dissipationsmetoden (IDM) över land under instabila förhållanden. Detta gjordes genom att jämföra friktionshastigheten, u , från eddy-

_*

correlation-metoden (ECM) med u

_*

från IDM. Resultaten kan användas till att se huruvida man kan använda de antaganden som IDM vilar på. Ytskiktsteorin är troligen mer uppfylld över land, eftersom det t.ex. inte finns något våginflytande.

Mätningarna är gjorda i Marsta, som ligger i ett flackt jordbrukslandskap, 8 km norr om Uppsala.

Resultatet visar att IDM fungerar tillfredställande över land (relativ standardavvikelse ca 10 %). För svagt instabila förhållanden, räcker det med att anta neutral skiktning i IDM-beräkningarna. Om det är mer instabilt, bör man ta hänsyn till stabiliteten och även inkludera en obalansterm. Obalanstermen introduceras implicit genom att variera den effektiva Kolmogorovkonstanten med stabiliteten. Den effektiva

Kolmogorovkonstanten som användes här, varierade från 0.5 till över 0.8.

För att beräkna u

_*

med hjälp av IDM, beräknades en första uppskattning av u

_*

genom att använda en parametriserad ”drag-coefficient” C

D

. För att efterlikna de medel man har att tillgå på skepp och bojar i rörelse ute på havet, beräknades stabilitetsparametern, z/L, med hjälp av ett bulkbestämt värmeflöde. Den stora spridningen av data visade att det är viktigt att parameteriseringen av C

D

samt värmeflödet är bra.

Man kan dra slutsatsen att dissipationsmetoden fungerar tillfredsställande, som metod

att bestämma turbulenta flöden över land. Den större spridningen som man fått över

hav är förmodligen en effekt av t.ex. vågor, trots att havsytan anses vara mer

homogen och förhållandena mer stationära.

Table of contents

1. INTRODUCTION ...7

2. THEORY ...8

2.1 S

URFACE

-

LAYER

T

HEORY AND THE

E

DDY

-C

ORRELATION

M

ETHOD

...8

2.2 T

HE

B

ULK

A

ERODYNAMIC

M

ETHOD

...10

2.3 T

^HE

T

^URBULENT

K

^INETIC

E

^NERGY

B

^UDGET

...11

2.4 T

^HE

I

^NERTIAL

D

ISSIPATION

M

^ETHOD

...12

2.4.1 The Inertial Subrange...13

3. SITE AND MEASUREMENTS...14

3.1 T

^{HE SITE}

M

^ARSTA

...14

3.2 M

EASUREMENTS

...14

4. DATA ...15

5. RESULT...18

5.1 T

HE

D

RAG

C

OEFFICIENT

...18

5.2 T

^HE

R

^OUGHNESS

P

^ARAMETER

...20

5.3 T

^HE

H

^EAT

F

^LUX

...22

5.4 T

HE

S

TABILITY

P

ARAMETER

...23

5.5 T

HE

F

RICTION

V

ELOCITY

...25

5.6 T

HE

I

MBALANCE

T

ERM AND

M

ODIFICATION OF

IDM ...28

5.7 S

LOPE OF THE

I

^NERTIAL

S

^UBRANGE

...31

6. DISCUSSION AND CONCLUSIONS ...31

7. ACKNOWLEDGEMENTS...33

8. REFERENCES...34

1. Introduction

A common method to determine the turbulent flux of momentum over sea, is the use of the Inertial Dissipation Method. It however relies on several assumptions that have to be verified. One way to verify the method is to evaluate it over land, where theories of the layer of the atmosphere close to the surface have been more evaluated.

In the meteorological and climatological models, boundary conditions are needed.

These boundary values are found in the layer connected to the earth’s surface, the Atmospheric Boundary Layer (ABL), or even the lowest 10 % of the ABL, the Surface Layer. The ABL is characterised by its direct influence by the surface of the earth and can reach 1-3 km during daytime, depending on season and cloudiness. In the surface layer, turbulent fluxes vary only within 10 % with height, but are

practically assumed to be constant. Thus it is also called the Constant Flux Layer. The turbulent fluxes of momentum, heat and moisture provide the boundary conditions mentioned above.

Three methods are often used to determine the turbulent fluxes in the surface layer.

The most direct is the eddy-correlation method (ECM), using measurements from turbulence instruments. A major disadvantage is that ECM requires a fixed mounting, which makes this method less suitable over sea, if not having extremely accurate positioning systems, which can be used to remove the movements of the platform.

Another disadvantage is the sensitivity for flow distortion, where the instrument or tower disturbs the wind flow.

Another method is the so-called bulk method, where parameterisation of the flux is used together with finite differences in the respective variable. This method is in common use over water on ships or buoys. It is however somewhat uncertain, because it is based on differences between discrete heights and correct estimates of the wind speed and drag coefficient etc.

The third of these methods to determine the turbulent fluxes in the surface layer is the Inertial Dissipation Method (IDM). It is a very useful method and uses the

information from the dissipation rate of the turbulence, i.e. the speed at which the turbulent energy is broken down into heat motion. The major advantages with this method are the irrelevance of ship and buoy movements, because these are not included in the frequency range used by the method. There is also a very small influence of flow distortion in that frequency range. The classical form of this method is to neglect the transport terms, which are hard to measure, together referred as the imbalance term. There have, however, been attempts to parameterise this imbalance.

The IDM is, as the bulk method, in common use on ships and buoys. A disadvantage with the IDM is, as mentioned above, that it relies on many assumptions.

Several evaluations of the IDM have been made during the years, but only to the author’s knowledge over the sea. Large and Pond (1981) showed good agreements in the determination of the friction velocity between the ECM and IDM (within 20 %).

Dupuis et al. (1997) for example show the dependence of stability when they found

larger values of the friction velocity for larger instability using IDM. They also

parameterised the imbalance term. Sjöblom and Smedman (2003) show the influence

of swell, which could be coupled to the transport terms. During swell it was found

that IDM gives larger values than ECM. It could be due to that the theory has its shortcomings due to the influence of waves.

The purpose with this study is to evaluate the validity of the IDM, but over land. The focus will be on unstable stratification, common during daytime. While the surface layer theory is functioning quite well over land, the heterogeneity can instead become a problem.

Section 2 will describe the relevant theory and section 3 describes the site and the instrumentation. The data treatment and description is dealt with in section 4. The result is presented in section 5. In section 6 a discussion is held and there the conclusions are also drawn.

2. Theory

2.1 Surface-layer Theory and the Eddy-Correlation Method

In the surface layer, we assume constant turbulent fluxes of heat, moisture and momentum. Monin-Obukhov similarity theory describes the events in this layer with the following formulae. First we introduce the division in a mean part and a

perturbation or turbulent part of a variable:

x X

X = + ′ . (1)

The friction velocity is an important scale of velocity, which is part of most of the relations in the surface-layer theory.

[ ^{u w v w} ] ^{u w}

u

_∗

= = ( − ′ ′ )

²

+ ( − ′ ′ )

^{2 14}

≈ − ′ ′ ρ

τ _{, (2)}

where τ is the surface stress, u′ and w′ is the turbulent part of the wind component in the mean wind direction and in the vertical respectively. The w u ′ -term is also ′

referred to as the momentum flux. The w v ′ -term is usually much smaller than w ′ u ′ ′ and will be ignored during this study.

The way to determine u

_∗

using equation 2, is the basis of the Eddy-Correlation Method. This is the most direct and accurate method to determine u

_∗

. It requires that the instrument is fixed. This makes the method unsuitable for measurements over the sea, since moving platforms almost always are used. The movements of the platforms can be very evident, since ECM uses the whole frequency spectrum. ECM is therefore also very sensitive to flow distortion. In this study, ECM is used for reference values of u

_∗

.

There are similar scales for potential temperature θ ( T ) and specific moisture Q

*

( Q ), including

_*

u :

_∗

*

*

u

T = − w ′ θ ′ , (3)

*

*

u

q

Q = − w ′ ′ , (4)

where w ′ θ ′ and w ′ respectively is heat and humidity flux. q ′

An example of the use of u

_∗

, is the logarithmic law for the mean wind speed, U :

0

*

ln z

z k

U = u , (5)

with von Kárman’s constant k=0.40 suggested by Högström (1996), z is the height and z

₀

the roughness parameter. Equation 5 only applies in neutral conditions i.e. = 0

∂

∂ z

θ . To get the influence of stratification we introduce the Monin-Obukhov length:

w

v

gk T L u

θ ′

− ′

=

⁰

3

*

, (6)

where gravity g=9.82 m/s

²

(60° latitude) and T

0

is the mean absolute temperature in the layer of measurements. θ

v

= ^{( 1} + ^{0 . 61 Q )} θ is the virtual potential temperature, so

w ′ is the virtual heat flux. The expression z/L, or ζ, is called the stability θ

v

′ parameter. With ζ < 0 , we have unstable conditions, corresponding to < 0

∂

∂ z

θ _{. The} logarithmic wind law (equation 5) then becomes:

( ) ^ζ

φ φ

m

kz

m

u L z kz u z

U

_{* *}

=

 

 



= 

∂

∂ , (7)

where φ

m

is the dimensionless wind gradient expressed as a function of z/L. φ

m

has been determined during experiments. Högström (1996) suggests

ζ

φ

m

= ¹ + ^{5 . 3} , ζ > 0 ^{, (8)}

4

)

1

19 1

( −

⁻

= ζ

φ

m

, ζ < 0 ^{. (9)}

In the same manner for temperature and humidity we have:

( ) ^ζ

θ φ kz

h

T z

=

*

∂

∂ , (10)

( ) ^ζ

φ

w

kz Q z

Q

_*

∂ =

∂ , (11)

with the dimensionless gradients of temperature and humidity:

ζ φ

φ

h

=

w

= ¹ + ^{8 . 0} , ζ > 0 ^{, (12)}

( ^{1 11 . 6} )

¹²

95 .

0 −

⁻

=

= φ ζ

φ

h w

, ζ < 0 ^{, (13)}

as suggested by Högström (1996).

From this we can see that u

_∗

is an important parameter, not only used in the

determination of the surface stress τ and the wind profile, but also in the normalised profiles of temperature and humidity. Beside this parameter, the fluxes of heat and humidity are of course important as well. The next section shows how these three turbulent fluxes can be parameterised if there is no access to operative eddy- correlation measurements, as often is the case with measurements over sea.

2.2 The Bulk Aerodynamic Method

A way to determine the turbulent fluxes is to use the bulk method.

The drag coefficient, C

D,

relates wind speed to the momentum flux:

2 2

*

z z

D

U

w u U

C u  = − ′ ′



 



=  . (14)

The Stanton number, C

H,

is the heat transfer coefficient:

(

^{S z}

)

z

H

U T

C w

θ θ

−

′

= ′ . (15)

The Dalton number, C

E,

is the moisture transfer coefficient:

(

^{S z}

)

z

E

U Q Q

q C w

−

′

= ′ . (16)

S denotes surface values and z a standard reference height of 10 m. This height is also used for the turbulent fluxes, i.e. w u ′ , ′ w ′ θ ′ and w ′ . q ′

The values of C

D

, C

H

and C

E

have been measured during projects both over sea, and at land sites. The coefficients for the neutral case (C

DN

etc.) have been determined and are then used in formulae where the stability is taken into account. One example of a sea project is described in Large & Pond (1981, 1982), who for the drag coefficient found C

DN

= 1 . 2 ⋅ 10

⁻³

, for 4 ≤ U < 11 m/s and ^C

DN

⁼ ( ^{0 . 49 + 0 . 65 U} ) ^{⋅ 10}

⁻³

, for

25

11 ≤ U ≤ m/s.

Over land, Mahrt et al. (2001) determined the drag coefficient for different surfaces and stabilities. For grassland, they found a near neutral value of C

D

to be around

10

3

4 ⋅

⁻

for winds between 5 and 13 m/s.

In this study, the values of C

H

and C

E

was set to 2 ⋅ 10

⁻³

(Stull, 1988). The reason why C

D

from Mahrt is not used, is that C

D

is more sensitive to the roughness of the surface and plays a central role in the IDM.

The bulk method is often used over sea to give boundary conditions for the weather and climate models. It does, however, have its disadvantages, since a bulk formulation never gives more than an approximation.

2.3 The Turbulent Kinetic Energy Budget

The Turbulent Kinetic Energy (TKE) budget describes how turbulence is created or destroyed.

ρ ε

θ −

∂

′

∂ ′

∂ −

∂ ′

′ − + ′

∂

′ ∂

− ′

∂

′ ∂

− ′

=

∇

∂ +

∂

z w p z

e w w

T g z w V z v w U u e t U e

v

1

0

, (17)

(A) (B) (C) (D) (E) (F) (G) (H)

where ^{e = 0 . 5} ( ^{u ′}

²

^{+ v ′}

²

^{+ w ′}

²

) , i.e. the TKE, t is the time, ρ the air density, p´ the pressure fluctuation and ε the dissipation.

Term (A) is the local storage and (B) the advection. Concerning the local storage, it is often non-zero, but this term is most often neglected anyway. According to Stull (1988), we can neglect the advection term for land studies, at least for larger homogenous surfaces.

Terms (C) and (D) are the mechanical production terms. Within the surface layer, term (D) is negligible, since the lateral wind component is very small on average. The mechanical production is always a source of energy.

Term (E) is the buoyancy production, which is a source of energy for unstable conditions (z/L < 0) and a sink for stable (z/L > 0).

Term (F) is the turbulent transport and (G) the pressure transport. Those two are together sometimes referred to as the imbalance term. For the whole ABL, (F) is zero, but locally it could be either negative or positive (Stull, 1988). In other words, the turbulent transport neither creates nor destroys turbulent energy, but rather redistribute it within the ABL. The pressure transport is very difficult to measure, since the instrument itself affects the measurements and the fluctuations are very small compared to the total pressure. In this study, the transport terms will at first be neglected, but will later implicitly be a part of the so called effective Kolmogorov’s constant (see the next Section).

The dissipation (H) is the rate that the turbulence is broken down to heat motion.

Assuming stationary and homogeneous condition, the TKE budget can be expressed as:

ρ ε

θ −

∂

′

∂ ′

∂ −

∂ ′

′ − + ′

∂

′ ∂

− ′

= z

w p z

e w w

T g z w U

u

_v

1

0

0

. (18)

(C) (E) (F) (G) (H) Normalising with kz u

_*³

we get:

1 0

3

* 3

* 0 3

*

*

=

 −



 





∂

′

∂ ′

∂ +

∂ ′

′ − + ′

∂

∂

u kz z

w p z

e w u kz T

u w zgk z U u

kz

_v

ε

ρ

θ . (19)

(c) (e) (f & g) (h)

Using equations 2, 6 and 7, (19) can be expressed as φ-functions of ζ:

( ) ( )

3

u

*

kz

imb

m

ζ ζ φ ζ φ ε

φ − − =

_ε

= . (20)

(c) (e) (f & g) (h)

Here φ

^imb

is the normalised imbalance term. This imbalance term has been parameterised in studies both over sea (e.g. Dupuis et al., 1997), and over land.

Högström (1990) gives a suggestion of parameterisation for land conditions:

( ζ ) ζ

φ

_imb

= − 0 . 24 1 − 19

^{− 41}

− , ζ ≤ 0 ^{. (21)}

2.4 The Inertial Dissipation Method

The inertial dissipation method uses the dissipation to calculate u . Using

_*

Kolmogorov’s similarity and Taylor’s hypothesis, ε can be expressed as:

( )

rel u

U n n

nS π

ε α

²

²

3

 ⋅

 

=  , (22)

with frequency n, spectral density in the mean wind direction S

u

, Kolmogorov’s constant α and the relative wind speed U

^rel

, i.e. the flow past the instrument.

Equation 22 is valid in the inertial subrange, where no production or destruction of energy is active. Only the so-called energy cascade, where larger scale whirls become smaller and smaller until they pass to heat motion is active. The advantage with the IDM is, as mentioned in the Introduction, that it can be used on moving platforms at sea. In this study U

rel

is identical to the actual wind speed.

After insertion of (22) into (20) and re-arranging we get:

( )

^{12 13}

*

2 



 





−

⋅ −

 ⋅

 

= 

imb m

u

kz

U n n

u nS

φ ζ φ π

α ^{. (23)}

In this equation, u

_*

is present also on the right hand side, through the stability parameter ζ (see equation 6). There are a couple of methods to deal with this problem. One is to use an iterative process, where neutral stability is assumed as a start value. Or, since there are some suggestions of the parameterisation of φ

_ε

^{, one} can let the local minimum of φ

_ε

, be the start value (Dupuis et al., 1997). Dupuis et al.

use two different algorithms. Either to determine the stability with aid from the dissipation of virtual temperature energy in each step, or to use a bulk

parameterisation of the stability parameter.

In this study, the start value of u

_*

is determined with the aid of the drag coefficient C

D

. This avoids an iterative process. Since C

D

is not known for the measurement site, a parameterisation from wind speed and bulk heat flux is performed. See section 5.1.

Kolmogorov’s constant α is experimentally determined to 0.52 (Högström, 1996).

Practically, the “effective” constant α

eff

, where the imbalance term is implicitly included is often used. This effective constant seems to vary from about 0.50 up to above 0.60 (e.g. Högström, 1990). With α

^eff

, equation 23 becomes:

( )

^{12 13}

*

2 



 





⋅ −

 ⋅

 



 



= 

ζ φ π

α

eff m

u

kz

U n n

u nS . (24)

2.4.1 The Inertial Subrange

Kolmogorov predicted a –2/3 dependence of the normalised longitudinal wind

spectrum ( ln( nS

_u

) = f (ln n ) ) in the inertial subrange. However, this is not the only

criterion for an inertial subrange to exist. Local isotropy and homogeneity in space is

also needed (e.g. Henjes, 1998). To see if isotropy exists, one can for instance

examine if the ratio of spectral density in the vertical and the longitudinal direction

(S

w

(n)/S

u

(n)) is close 4/3 (e.g. Henjes, 1999).

3. Site and Measurements

Forest Arable fields Grassland

River Fyrisån N

Figure 1. 5 x 5 km map with Uppsala to the south and the measurement site Marsta in the centre of the figure. (From Halldin et al., 1999).

3.1 The site Marsta

The site where all measurements were made is called Marsta, which is situated about 8 km north from the centre of Uppsala and located in a flat agricultural region (59°

56’ N, 17° 35’ E)(figure 1). The fetch is limited by small forests more than 500 m to the north and south-west. In the other directions the fetch is 1-4 km. The measurement site is placed on a grass field with the dimensions about 100 x 150 m (figure 2). It is divided into four subsites (Lawn, Grassland, Roof and Arable-field), of which the Grassland subsite is used in this study. Both the turbulence and profile measurements are taken from a 30 m tower, triangular in cross-section with sides of 0.28 m and with 1 m long booms. The radiation measurements are taken from a ramp next to the tower. The data was collected from June to August in 1994 during one of the summer campaigns of the NOPEX project (Northern Hemisphere Climate Process Land Surface Experiment)(e.g. Halldin et al., 1998, Halldin et al., 1999).

3.2 Measurements

The turbulence data (wind and temperature) are taken from a Solent Ultrasonic Anemometer 1012R2 (Gill Instruments, Lymington, UK), sampled with 21 Hz and placed at the height of 10 m. It is mounted on a boom so that the instrument is directed towards 245°, which implies a disturbed sector of approximately 40°-80°.

The profile instruments sample with 1 Hz and are mounted on booms at the heights of

0.84, 1.95, 4.78, 10.1, 17.2 and 29.0 m. The wind speed and wind direction are

measured with a combined three-cup anemometer/wind vane (Bergström and Lundin,

1993) developed by MIUU, Uppsala. For the temperature, a radiation-shielded and

ventilated Pt500 thermocouple is used (Högström and Smedman, 1989). Humidity is

calculated using the psychrometric method, also here with shaded and ventilated Pt500 sensors. The ventilation speed for all these arrangements is 5-6 m/s. At the height of 0.84 m, an absolute dry temperature is recorded, used as a reference accurate temperature, since the other temperature measurements (both dry and wet) are based on the differences to this reference height. This method is described in Smedman- Högström and Högström (1973). The profile booms have the orientation to 305°, so the disturbed sector will be about 110°-150°.

Also a ground-facing Eppley PIR pyrgeometer (Newport, USA) mounted on the ramp next to the tower, measuring long-wave radiation from the ground, was used and its height above the ground was 1.2 m.

100 Snow depth Precipitation gauge Groundwater Soil profile Radiometry Tower

N

0 50

InSitu SMHI

Hellman 30m tower

Old tower

Flux station

Lawn subsite

Arable-field subsite Grassland subsite

Roof subsite

Figure 2. The site of Marsta (scale in meters). The instruments are placed on the Grassland subsite within the circle. (After Halldin et al., 1999).

4. Data

The data used are from periods in June, July and August 1994. The profile

measurements of temperature and wind were operating the entire period, while the moisture and turbulence measurements were down for some periods (respectively ca 30% and 40% of the whole period).

Measurements from the mean wind directions of 320°-80° and 110°-150°, was

removed due to distortion from the tower and the small forest to the north. Since this

was a study for unstable cases only, positive vertical potential temperature gradients

were removed. Also the upward momentum fluxes that arose at a few occasions were

removed. Mean wind speeds below 2 m/s at 10 m were rejected due to statistical uncertainties.

The heat flux measurements made by the sonic anemometer nearly coincide with the virtual heat flux.

The profile and turbulence data were averaged in 10-min sections and averaged to 50- min runs, to be correlated to the turbulence spectra.

The turbulence wind data were Fourier transformed and averaged into log-bins.

To find an inertial subrange, the condition 1.1 < S

_w

(n)/S

_u

(n) < 1.4 for n > 0 . 15 Hz was used. The values from the highest frequency were ignored to avoid instrument errors and spectra were rejected if there was less than two valid bins that satisfied these conditions. The n and nS

u

(n) for all valid bins in each spectrum were then used in the IDM.

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

10⁻³ 10⁻² 10⁻¹ 10⁰

n (Hz)

n S(n)

U−spectrum

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

10⁻³ 10⁻² 10⁻¹ 10⁰

n (Hz)

n S(n)

w−spectrum

Figure 3. Normalised spectra in a) mean wind direction and b) in the vertical. The line in the upper right corner represents the ideal slope in the inertial subrange.

The amount of data used in the parameterisation of C

D

was 1047 10 min values, using

u

*

from the turbulence measurements and U from profile measurements.

The final data amount of u

_*

used in the comparison with ECM was strongly reduced, containing only 88 data points, due to the need of simultaneous measurements.

Figure 4a displays the hourly distribution of wind speed and figure 4b the stability, expressed as z/L, for this data amount. Wind speeds below 7 m/s are well represented, and so also the case with the near neutral conditions ( z L ≈ 0 ). The highest wind speed was 7.8 m/s, while the most unstable case was z L = - 0.84 .

2 3 4 5 6 7 8

0 10 20 30

U (m/s)

number of hours

Distribution of wind speed and stability

−0.90 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 10

20 30 40

z/L

number of hours

a)

b)

Figure 4. Distribution of a) the wind speed and b) the stability expressed as z/L.

5. Result

5.1 The Drag Coefficient

The drag coefficient, C

D

, is needed, because in the inertial dissipation method, u in ζ

_*

on the right hand side (equation 24) is not known and must be calculated from C

D

. Since C

D

was not known for the site, a parameterisation was made.

C

D

was calculated using equation 14, with u from the turbulence measurements and

_*

U from the profile measurements at 10.1 m. In figure 5, the calculated C

D

is presented as a function of wind speed. The data was averaged in bins of wind speed for each 0.25 m/s and for five different stabilities z/L (see legend and figure text).

2 4 6 8 10 12 14

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

CD(z/L,U) June−Aug

U (m/s)

Cd

0.0−0.2 0.2−0.4 0.4−0.6 0.6−0.8 0.8−

Figure 5. Measured C

D

as a function of wind speed and stability. The symbols in the legend show –z/L.

The lines are values for grass suggested by Mahrt et al. (2001), with solid line for near neutral (-0.05

< z/L < 0.05), dashed-dotted line for weakly unstable (-0.5 < z/L < -0.05) and dashed for very unstable stratification (z/L < -0.5). The solid line at C

DN

= 0.0012, represents an established neutral sea value for growing sea (Large and Pond, 1981).

A comparison with results from Mahrt et al. (2001) was made and there is a similarity to their values for a grass field. However, Mahrt et al. (2001) give higher values for the more unstable cases, while it here shows the opposite. C

D

is a function of wind speed and ζ. To avoid possible spurious correlation and because ζ in itself is a function of C

D

through u , a parameterisation with the difference in potential

_*

temperature, ∆ , at 10 and 4.78 m will instead be made. The reason why the result of θ Mahrt et al. (2001) is not used here, is that the roughness of the field is not known for the measurements and that z/L is not known when the IDM is used.

Figure 6, with the stability parameter, z/L, plotted as a function of ∆ , shows the θ

uncertainty of the approach with temperature differences. The scatter is very large, but

there is some correlation (-0.13) between z/L and ∆ . The stability parameter, z/L θ

(equation 6), is not only a function of the heat flux and ∆ through equation (15), but θ

also of u

_*

. Later, it will be shown how this parameterisation affects the bulk formulation of z/L.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1

∆θ (K)

z/L

June − August

Figure 6. The dependence of the stability parameter z/L on the potential temperature difference

∆θ between the heights of 10 and 4.78 m. The line shows a linear least-square fit.

In figure 7, the same wind bins as in figure 5 are shown, but the “stratification”, expressed as ∆ , is divided into four classes. A best fit of the second degree is made θ for each of the four ∆θ:s for wind speeds below 7 m/s. For wind speeds above 6 m/s, a linear regression is performed instead. The error bars enclose 50% of the data points.

0 2 4 6 8 10 12

0 0.005 0.01 0.015 0.02 0.025

CD(∆θ,U) June−Aug

U (m/s)

C D

0−0.1 K 0.1−0.2 K 0.2−0.3 K 0.3−0.4 K Sea value (0.0012) 0−0.1 K

0.1−0.2 K 0.2−0.3 K 0.3−0.4 K U>6 m/s

Figure 7. C

D

as a function of wind speed and different ∆θ. Best fits of the second degree for wind speed

below 7 m/s are made for each bin. A linear regression for all stabilities at wind speeds higher than 6

m/s is also made. Within the error bars, 50% of the data points are found.

The fitted curves in figure 7 are somewhat smoothed in figure 8. The wind speed 6 m/s is chosen as a separation point, and above that wind speed, the curves collapse.

Further, for wind speeds above 9 m/s a constant value is used. The numerical parameterisation is shown in Table 1.

2 4 6 8 10 12

0 0.005 0.01 0.015 0.02 0.025

CD(∆θ,U) June−Aug

U (m/s)

C D

0−0.1 K 0.1−0.2 K 0.2−0.3 K 0.3−0.4 K 0−0.1 K 0.1−0.2 K 0.2−0.3 K 0.3−0.4 K U>6 m/s U>9 m/s

Figure 8. Calculated C

D

is plotted as in figure 7. The lines are the parameterised C

D

, with the largest difference giving the highest values of C

D

. The legend shows the temperature difference bins and the parameterised lines for the different wind speed ranges.

Table 1. The parameterised values of C

D

as a function of wind speed and ∆ θ ^. Wind speed Temperature

difference

1000·C

D

2<U<6 m/s ∆ <0.1 K θ 0.02U

²

-0.9U+9.7 0.1< ∆ <0.2 θ 0.18U

²

-2.8U+15.3 0.2< ∆ <0.3 θ 0.40U

²

-5.4U+23.0 0.3< ∆ <0.4 θ 0.65U

²

-8.4U+32.0

6<U<9 m/s -0.67U+9.0

U>9 m/s 3

5.2 The Roughness Parameter

The roughness parameter, z

0,

was not known during the project, but an attempt is made here to get an idea if the roughness was changing with time. Since the

measurement site is an agricultural area, one could assume a change in roughness due to the vegetation growth. The roughness parameter was calculated using the near neutral (-0.05<z/L<0.05) calculated drag coefficient C

D

(equation 14) and the relation:

CDN

e

k

z

0

= 10

⁻

. (25)

2 4 6 8 10 12 14 10⁻⁴

10⁻³ 10⁻² 10⁻¹ 10⁰

z0 (m)

80°−110°

150°−180°

180°−220°

220°−270°

270°−320°

3 4 5 6 7 8 9 10 11 12 13

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

z0 (m)

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

10⁻² 10⁻¹ 10⁰

z0 (m)

2 3 4 5 6 7 8 9

10⁻³ 10⁻² 10⁻¹ 10⁰

z 0 (m)

a) June − August

b) June

c) July

d) August

U (m/s)

Figure 9. The roughness parameter, z

0

, plotted against wind speed for a) all three months, b) June, c) July and d) August. The lines show the average logarithmic dependence on wind speed. The legend in a) is valid for all figures. The solid line: 80°-110°. Dashed, dashed – dotted, and dotted lines respectively 150°-180°, 180°-220° and 220°-270°. (Observe that some of the wind directions are less represented and therefore the corresponding average line is omitted.)

The result is shown in figure 9, where z

0

is plotted against wind speed and with

respect to different wind directions. This is done for all three summer months; a)

shows the summer months together, b) June, c) July and d) August. All wind

directions are not represented for each month, which makes it hard to draw direct conclusions. The lines represent the average logarithmic dependence on wind speed for the different directions. Some of the wind directions are not averaged because of too few data. However, there is a tendency that the roughness is higher in July and August, than in June, with an average of 2 cm in June and 6-8 cm in July and August.

Some difference is also seen in the different wind directions, suggesting some

heterogeneity. This gives an example of the uncertainty of the parameterisation of C

D

, not taking wind direction and changing surface conditions into account. This will be discussed more later.

5.3 The Heat Flux

The virtual heat flux used for z/L in IDM (equation 24) was calculated using bulk formula:

(

^{S z}

) ^{0 . 61}

^{0 E}

⁽

^{S z}

⁾

H

v

C U T T C U Q Q

w ′ θ ′ = − θ + − ^{, (26)}

where C

H

and C

E

both were set to 2 ⋅ 10

⁻³

(Stull, 1988). S denotes the surface values and z the reference height of 10 m. In this study, two different surface temperatures were evaluated. Figure 10 shows these bulk values of the heat flux plotted against the eddy-correlation values. One evaluation used the temperature at the height of 0.84 m ( •). The result was, however, discouraging, since the bulk formula of heat flux with surface temperature measured at 0.84 m gave about a magnitude less values than the eddy-correlation determined heat flux.

The second evaluation is instead based on the long-wave radiation measurements.

Assuming the ground to be a grey body (in the effective frequency range) with emissivity ε

lo

= 0 . 95 and using Stefan-Boltzmann’s law of blackbody radiation we get the surface temperature T

s-lo

:

15 . 273

4

−

−

= ε

lo

σ

lo s

T R , (27)

where temperature is in °C, R the outgoing long wave radiation (W/m

²

) and Stefan- Boltzmann’s constant σ = 5.6708·10

^-8

W/m

²

K

⁴

. The result is shown in figure 10 ( ∗) proving this method to be better than using temperature measurements at 0.84 m, although the scatter is large. Figure 10 shows an overestimation for low fluxes and an underestimation for higher fluxes. For statistics see table 2, where the ratio of the bulk and the eddy-correlation determined heat flux is presented. The average of 1.21, suggests that the emissivity was slightly too small. Further, the large scatter, with relative standard deviation of about 36 %, shows the shortcoming of the bulk method and probably also the uncertainty of a radiation approach.

The difference in humidity between the surface and 0.84 m did not seem to influence

the calculations significantly. (Not shown here.)

0 0.05 0.1 0.15 0.2 0.25

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

w´θ´_ECM (Km/s) w´θ´bulk (Km/s)

Heat flux

Ts (radiation) Ts (0.84 m)

Figure 10. Virtual heat flux calculated using bulk formula, plotted against eddy-correlation virtual heat flux. Dots: assuming surface temperature equal to the temperature at 0.84 m. Stars: using Stefan- Boltzmann’s grey-body radiation law.

5.4 The Stability Parameter

The bulk calculation of the stability parameter, ζ, used in the IDM (equation 24) is defined as:

0 2 3

3

U T

C w k g z

D

vbulk

ζ = − ^{⋅ ⋅ ⋅ ′} θ ^{′ ,} (28)

with z = 10 m, g = 9.82 m/s

²

and k = 0.4.

Table 2. Some statistics of the heat flux and the stability parameter.

Variable Average Standard

deviation Correlation coefficient

 

 





′

′

′

′

vECM vbulk

w w

θ

θ ^{1.21 0.44 0.70}

 

 





ECM bulk

ζ

ζ _-

parameterised C

D

1.10 0.66 0.62

 

 





ECM bulk

ζ

ζ _-

measured C

D

1.21 0.44 0.80

Figure 11a displays ζ from equation 28 plotted against the eddy-correlation value of ζ ( ζ

_ECM

, equation 6). Figure 11b shows ζ

_bulk

ζ

_ECM

plotted against the ζ

_ECM

^.

ζ

bulk

calculated from the parameterised C

^D

( ∗) gives too low values for higher instability, 2 ζ < − 0 . ^{, while –} ζ

bulk

is overestimated, up to 4 times the “correct” value, for more neutral conditions. ζ

_bulk

calculated from the measured C

^D

( •)(equation 14) gives ζ

_bulk

-values closer to the 1:1 line. We also see that ζ

_bulk

never becomes lower than − 0 . 45 . In table 2, the statistics of ζ

_bulk

ζ

_ECM

is shown. The measured C

D

gives the same average and standard deviation of ζ

_bulk

ζ

_ECM

as the heat flux, since the heat flux is the only variable that gives scatter. The scatter is even larger with the

parameterised C

D

, with 60 % relative standard deviation. This shows the importance of a good parameterisation of C

D

.

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1 0

z/L (ECM)

z/L (bulk)

C_D = (u_*/U)² C_D = parameterised a)

−0.90 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.5

1 1.5 2 2.5 3 3.5 4

z/L (ECM)

[z/L(bulk)]/[z/L(ECM)]

b)

Figure 11. (a) Bulk calculated stability parameter, z/L, plotted against the “real” z/L determined with

the eddy-correlation method. Note that one data point (measured C

D

) lies outside the graph at [-0.8, -

1.7]. (b) The ratio of bulk calculated and eddy-correlation determined stability parameter, z/L, plotted

against the “real” z/L determined with the eddy-correlation method. The legend in figure a) is valid for

both figures.

5.5 The Friction Velocity

Using IDM, the friction velocity, u

_*

, was calculated without the imbalance term, both without consideration of stability and with stability.

For the method without stability consideration, also called the neutral method, the following formula for u

_*

was used:

( )

^{12 13}

u IDMne

*

n 2

u nS  

  ⋅

 





 



= 

U n kz

eff

π

α ^{, (29)}

and for u

_*

with the method considering the stability:

( ) ( )

3 2 1

1 u IDMst

*

2 n

u nS 



 





−

⋅

⋅

 





 



= 

ζ φ

π

α

eff

U

_m

n

kz . (30)

In order to account for a possible imbalance between local production and local dissipation, two different values of Kolmogorov’s constant were used, α

eff

= 0 . 52 and 55 α

eff

= 0 . ^{. Also} α

eff

= ^{0 . 60} was tried for one month, but gave too much discrepancy. The reason why different α

eff

was tested, was to get better agreement to the eddy-correlation determined u

_*

.

The result is shown in figure 12, where u determined with IDM, both with the

_*

neutral assumption and stability consideration versions, are plotted against u

_*

using ECM (equation 2). In figure 12a, α

eff

= 0 . 52 is used and in figure 12b α

eff

= 0 . 55 ^. The figures show that using the higher α

eff

, the average gets below the 1:1-line. Also, there is not so much difference between the neutral assumption ( ∗) and the stability consideration (o). This is due to that φ

^m

decrease for more unstable conditions, whereas - ζ increase. Figure 13 displays the sum φ

^m

- ζ, as a function of ζ. When

= 1

− ζ

φ

m

, the neutral assumption method and the stability consideration method are similar. With φ

m

− ζ > 1 , i.e. for ζ < − 0 . 4 , the neutral assumption gives higher values.

Table 3 shows some statistics of the ratio of u determined with IDM and with ECM,

_*

respectively. In the following, the numbers after u

_*IDMne

and u

_*IDMst

denotes

52 .

= 0

α

eff

^and α

eff

= ^{0 . 55} . The eddy-correlation determined u

_*

is called u

_*ECM

. The average of u

_*IDMne

/ u

_*ECM

is lower than u

_*IDMst

/ u

_*ECM

, because φ

^m

- ζ was almost always less than 1. The relative standard deviations were the same for all approaches, 10 %, and so also the correlation coefficient (0.95).

In the following evaluations, α

^eff

is set to 0.52, because u

_*IDM52

/ u

_*ECM

is closest to 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0

0.1 0.2 0.3 0.4 0.5

June−Aug (α_eff = 0.52, T

s−lo)

u*ECM (m/s) u*IDM (m/s)

neutral stability a)

0 0.1 0.2 0.3 0.4 0.5 0.6

0 0.1 0.2 0.3 0.4 0.5

June−Aug (α_eff = 0.55, T

s−lo)

u*ECM (m/s) u *IDM (m/s)

neutral stability b)

Figure 12. A comparison between u

_*

determined by IDM and ECM. In a) α

eff

= 0.52, in b) α

eff

= 0.55.

The legends show the neutral assumption and the stability consideration methods.

In figure 14a-c u

_*IDMst52

u

_*ECM

is plotted against wind speed, stability and heat flux respectively. Circles are bin averages, with standard deviations.

The data do not show any wind speed dependence (figure 14a) but the scatter is larger

for low wind speeds. The data were averaged into bins for 1 m/s.

−1 −0.8 −0.6 −0.4 −0.2 0 0.8

0.9 1 1.1 1.2 1.3 1.4 1.5

φ m−ζ

ζ

Figure 13. The stability dependent terms, φ

m

- ζ, in equation 24, plotted against stability ζ.

Table 3. Some statistics of the ratio of the IDM- and ECM-determined * u .

u * Average Standard

deviation

Correlation coefficient

IDMst52/ECM 1.01 0.10 0.95

IDMne52/ECM 0.97 0.10 0.95

IDMst55/ECM 0.98 0.10 0.95

IDMne55/ECM 0.94 0.10 0.95

The ratio has a much larger dependence on the stability (14b), with higher values for the more unstable cases and a little bit less than 1.0 for cases near neutral. This is probably coupled to the imbalance term. In the next section, we will therefore instead use an effective Kolmogorov’s constant varying with stability. Observe that z/L here is ECM-determined. In figure 14b, the data was averaged in bins for each 0.1 above

3 .

− 0

= L

z , and for every 0.2 for more unstable cases.

In figure 14c, we see no significant dependence on the heat flux, but again there is more scatter for low fluxes. Here, the bins consists of data for each 0.02 Km/s for

08 .

< 0

′

′

_v

w θ Km/s, otherwise 0.04 Km/s.

Similar graphs for u

_*IDMne52

were also analysed, but the result was very similar to

52

*IDMst

u , so this is not shown here.

Concerning the possibility of instationarity, the change of 10-min mean wind

direction during a 50-min period was always below 50° in the data. There was not

more scatter for the higher values, suggesting that the data could be treated as

stationary, at least regarding the wind direction.

2 3 4 5 6 7 8 0.8

1 1.2 1.4 1.6

June − Aug (α_eff = 0.52, T

s−lo)

u *IDMst/u *ECM

U (m/s)

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.8

1 1.2 1.4 1.6

u *IDMst/u *ECM

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0.8 1 1.2 1.4 1.6

u *IDMst/u *ECM

w´θ´

v (Km/s) a)

b)

c)

z/L

Figure 14. The ratio of * u with stability considered IDM and ECM as a function of: a) wind speed, b) stability (ECM) and c) heat flux. Circles mark the bin averages and error bars the standard deviation.

5.6 The Imbalance Term and Modification of IDM

In this section, the different terms in the normalised TKE budget

( ) ( )

3

u

*

kz

imb m

φ ε ζ φ ζ ζ

φ − − =

_ε

= (20)

are examined. There is not much to say about φ

^m

and ζ, but so much more about the dissipation and imbalance terms.

Assuming homogeneity and stationary conditions, as in equation 20, we can assume that the “correct” u

_*

will satisfy this equation:

imb m

ECM

u kz

φ ζ

φ ε

−

= −

3

*

. (31)

For the u

_*IDMst

, without the imbalance term, the corresponding equation becomes:

ζ φ

ε

= −

m IDM

u

_*³

kz . (32)

Thus, we can from these two equations solve for φ

^imb

:

( ) [ ( ) ] 



 



 −

−

=

ECM IDM m

imb

u

u

*

1

*

ζ ζ φ ζ

φ ^{. (33)}

The normalised dissipation, φ

ε

, can then be determined, using equation 20.

In figure 15, the different terms in the TKE budget are plotted against z/L(ECM). The terms are plotted so that there is a balance in the budget. The stability z/L and the dimensionless wind gradient φ

^m

(equation 8) are of course neatly plotted, since they are numerically exact determined. The scatter is of course large for the imbalance and dissipation terms. The normalised dissipation is always a sink, but tend to have higher absolute values when moving towards more unstable conditions. The imbalance term is close to and even below zero for near neutral conditions and tends to higher values with increasing instability.

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

−4

−3

−2

−1 0 1 2 3

Normalised TKE budget

ζ

φ_m

−ζ−φ_imb

−φ_ε

Figure 15. The terms in the normalised TKE budget, plotted as functions of stability z/L (ECM).

Positive values represent gain in energy. The line represents Högström’s equation 34.

This figure can be compared to figure 8 in Högström (1990), where several results

from different projects were compared. The normalised dissipation in this study seem

to show the same tendency as Högström’s results, but the scatter is too large to say if

it confirms Högström’s equation:

( )

[ ^{ζ ζ} ]

φ

_ε

= 1 . 24 1 − 19

^{− 41}

− , ζ ≤ 0 ^{. (34)}

The discrepancy is presumably due to the many parameterisations used in the IDM.

The imbalance term has of course the same scatter as the normalised dissipation, as φ

^m

and ζ have no scatter.

Using equations 23 and 24 and solving for α

^eff

leads to the determination of a varying effective Kolmogorov’s constant:

ζ α φ

φ ζ α φ

3 2

 

 





−

−

= −

m imb m

eff

, (35)

where α = 0.52. This equation is plotted in figure 16 against z/L(ECM). A second order best-fit curve is also shown, giving:

496 . 0 051 . 0 357 .

0

²

− +

= ζ ζ

α

eff

. (36)

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.3

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

ζ (ECM) α eff

Varying α_eff

Figure 16. The effective Kolmogorov’s constant, α

eff

, plotted as a function of stability ζ (ECM). The curve represents the second-order best fit.

Using this varying α

^eff

in the IDM would remove the dependence of stability if a

correct stability parameter were used. But since there are uncertainties in the

determination of the stability, which is bulk parameterised, in the IDM, the scatter

was still the same. The relative standard deviation is about 10 %.

5.7 Slope of the Inertial Subrange

The slope of the inertial subrange varied between –0.55 and –0.85. In figure 17,

52

u

*IDM

/ u

_*ECM

with varying α

^eff

(equation 36) is plotted as a function of the slope of the inertial subrange. The scatter is here somewhat higher for the slopes different from

3

− 2 .

−0.9 −0.85 −0.8 −0.75 −0.7 −0.65 −0.6 −0.55 −0.5 0.8

0.9 1 1.1 1.2 1.3 1.4 1.5

Slope u *IDM/u *ECM

Importance of the Slope of the Inertial Subrange

Figure 17. Ratio of the IDMst52 and ECM for u

_*

with varying α

eff

plotted against the inertial subrange slope.

6. Discussion and Conclusions

The topic was to evaluate the Inertial Dissipation Method (IDM) over land for unstable conditions. The site where the measurements were taken is situated in a flat agricultural area, with its advantages and disadvantages. The open area gives quite homogeneous conditions, whereas the growth of the vegetation changes the roughness of the surface during the measurements.

The measurements were done with both profile and turbulence instruments. The turbulence measurements were principally used for producing wind spectra and calculating the dissipation, but also as reference measurements of the heat flux and friction velocity. Over sea one can not calculate the heat flux and friction velocity with the eddy-correlation method (ECM), if not having accurate positioning systems, because of the movements of the ships or buoys. Therefore, profile measurements are often used there. To imitate these conditions, bulk formulations and

parameterisations, using the profile measurements, are used here.

The IDM was used to calculate the friction velocity u

_*

and the result was compared to

the ECM-calculated u

_*

. Since u

_*

is on both sides of the equation determining u

_*

(equation 6, 9 and 24) and to avoid iteration, an estimation of u

_*

was calculated using the drag coefficient C

D

. C

D

was not known for the site, so therefore it was

parameterised with wind speed and temperature difference, due to that the stability parameter, ζ, was not known at this stage. This led to some uncertainties, which for example gave overestimated values of the parameterised C

D

, compared to the measured C

D

for unstable stratification ( ζ < − 0 . 2 ^).

Over sea, several projects have tried to determine parameterisations of C

D

(e.g. Smith, 1988, Large and Pond, 1981). C

D

is usually determined as a function of stability, wind speed and sometimes of wave age c

_p

u

_*

, where c

p

is the speed of the waves at the peak of wave spectrum. Over land, C

D

is a function of wind speed and stability, but also of the roughness, which changes for different vegetation and also in some cases with vegetation growth and wind direction. Probably it is not easier to determine C

D

over sea, but more studies have been made over sea, so there are more parameterisations available.

The heat flux (included in the stability parameter ζ) was calculated with bulk-

parameterisations, using long-wave outgoing radiation measurements to determine the surface temperature. The scatter was large (relative standard deviation 36%), which might be an indication that a bulk approach never fully can express the true fluxes.

The heat flux is underestimated for high heat fluxes, while it is overestimated for small fluxes.

Since both the bulk-calculated u

_*

(using parameterised C

D

) and the bulk heat flux were included in the calculation of the stability parameter ζ, there was even more scatter (relative standard deviation 60%), comparing ζ

_bulk

^and ζ

ECM

. For near neutral cases the ζ

_bulk

almost always gave too large values compared to ζ

_ECM

. For more unstable stratifications, approximately ζ < − 0 . 2 ^{, the} ζ

bulk

instead gave too neutral stratification, compared to the eddy-correlation calculations.

Over sea, Large and Pond (1981) got typical deviations of 0.05, comparing ζ determined with the bulk method with the eddy-correlation method. This is indeed smaller than the results here. Sjöblom and Smedman (2003), however, got similar results as this study, with larger deviations (under-estimations) for more unstable conditions and a standard deviation of 0.16 ( ^{- 1} < ζ < 0.5 ). Still, their result contains less scatter, which could be due to the uncertainty of the parameterised C

D

in this study.

For an inertial subrange with isotropy to exist in this study, several criteria had to be fulfilled, i) frequency above 0.15 Hz; ii) the ratio of vertical and longitudinal spectral density ^{1 . 1} < S

_w

( ) ( ) n S

_u

n < ^{1 . 4} . Further, the slope of the inertial subrange in the spectra that fulfilled these criteria was not exactly -2/3 in all spectra, but varied between − 0 . 55 and − 0 . 85 . The scatter of u

_*IDM

u

_*ECM

was larger for the larger slopes (see figure 17), which was expected. A variation within 10 % does not seem to affect the result appreciably.

The scatter of u

_*IDM

u

_*ECM

(standard deviation about 10 %) was the same, when

assuming neutral conditions and when the stability was considered. The difference

between those stability consideration methods was small, below 5% for ζ > − 0 . 6 ^.