The Influence of Waves on the Heat Exchange over Sea

Examensarbete vid Institutionen för geovetenskaper ISSN 1650-6553 Nr 20

The Influence of Waves on the Heat Exchange over Sea

Erik Sahlée

The Influence of Waves on the Heat Exchange over Sea

Abstract

The main focus of this study is the influence of waves on the heat transfer over sea. In particular, the bulk transfer coefficient CH (the Stanton number), has been investigated for possible wave influence. Measurements from the site Östergarnsholm in the Baltic Sea have been used. The site has a large sector with undisturbed over water fetch.

Data during the period 1995-1999 have been used.

It is shown that CH behaves differently as it approach z/L=0 from the unstable side depending on the wave state. During growing sea, CH makes a rapid drop as it passes over neutrality, strikingly different from swell conditions where CH makes a much

’smoother’ transition. This difference is also shown to exist for the kinematic heat flux. Based on the definition of CH, it is suggested that one of the reasons of CH’s different behaviour for different stratification and wave state, is ought to be sought in the kinematic heat flux itself.

A comparison of the w,θ cospectra during growing sea and swell conditions, showed differences. For growing sea, the larger size eddies dominates the heat flux during unstable conditions. There is no significant difference in peak frequency for different grade of instability. The swell cases showed a more inconsistent behaviour as it approached neutrality, with the peak frequency shifting for different stability ranges.

The correlation coefficient between u, the longitudinal wind component, and w, the vertical wind component, Ru,w is also investigated in this study. It is shown that Ru,w is exposed to some wave influence. A comparison of Ru,w as a function of wave age, for neutral and non-neutral stratification is made. For swell cases and non-neutral

stratification Ru,w makes a rapid drop and assumes values close to zero. This is not seen for the neutral cases although there is a slight decrease. It is concluded that a certain amount of positive heat flux and inactive turbulence is needed to see this drop in the correlation coefficient.

Sammanfattning av ”Våginflytandet på värmeutbytet över hav”

Syftet med studien är att undersöka om det existerar ett våginflytande för

värmeutbytet över hav och speciellt eventuellt våginflytande på utbyteskoefficienten CH (Stantons tal). Mätdata från Östergarnsholm utanför Gotland har använts. Denna mätstation har en stor sektor i vilken vindens anloppssträcka ostört är påverkad av hav. Data från perioden 1995–1999 har använts.

Stantons tal CH beter sig annorlunda vid övergången från instabil till stabil skiktning beroende på havsytans tillstånd. Vid uppbyggande sjö gör CH ett ’hopp’ då det passerar neutral skiktning. För dyning finns inte detta hopp utan övergången är mycket mjukare. Denna skillnad observeras också hos det turbulenta värmeflödet.

Baserat på definitionen av CH föreslås det att dess olika beteende för olika skiktning och vågtillstånd finns att söka i beteendet hos det turbulenta värmeflödet.

En jämförelse av cospektrat för vertikal vind, w, och potentiell temperatur, θ, visar att där finns olikheter mellan uppbyggande sjö och dyning. Under instabila förhållanden och uppbyggande sjö domineras värmeflödet av storskaliga virvlar. Det existerar ingen signifikant skillnad i maximal värdets frekvens för olika grad av instabilitet.

Dyningsfallen visar ett mer varierat beteende med en maximalvärdes frekvens som skiftar för olika stabilitetsområden.

I studien undersöks också korrelationskoefficienten mellan longitudinal vind u, och vertikal vind w, Ru,w. Det visas att Ru,w är utsatt för ett visst våginflytande. Ru,w som en funktion av vågålder jämförs för neutral och icke-neutral skiktning. För dyning och icke-neutral skiktning så faller Ru,w snabbt till små värden nära noll. Detta resultat skiljer sig för neutral skiktning där Ru,w bara gör en svag minskning. Slutsatsen är att det krävs en viss mängd positivt värmeflöde och inaktiv turbulens för att se det kraftiga avtagandet hos korrelationskoefficienten.

1 INTRODUCTION...7

2 THEORY ...7

2.1 SWELL...7

2.2 CORRELATION COEFFICIENTS...8

2.3 THE STANTON NUMBER,CH ...9

2.4 CHARACTERISING THE SEA STATE...11

2.4.1 Phase speed ...12

2.5 WAVE INFLUENCE ON Z/L ...13

3 SITE AND MEASUREMENTS...13

3.1 T^{HE SITE} ÖSTERGARNSHOLM...13

3.2 INSTRUMENTATION...14

3.2.1 Correction of the heat flux...15

3.3 DATA CRITERIA...16

4 RESULTS ...17

4.1 THE VARIATION OF THE CORRELATION COEFFICIENTS,RU,W AND RW,θ, WITH STABILITY AND WAVE AGE...17

4.2 THE WAVE STATE INFLUENCE ON CH...20

4.3 THE STABILITY INFLUENCE ON CH...20

4.4 THE STABILITY INFLUENCE ON THE HEAT FLUX...22

4.5 SPECTRAL CHARACTERISTICS...24

4.5.1 Stability influence on the w,θ cospectra ...24

4.5.2 Wave influence on the w,θ cospectra...25

4.6 REASONS OF SCATTER; ADVECTIVE EFFECTS...26

5 SUMMARY AND CONCLUSIONS ...29

5.1 THE CORRELATION COEFFICIENTS...29

5.2 STABILITY AND WAVE INFLUENCE ON CH...29

6 ACKNOWLEDGEMENTS...30

7 REFERENCES...30

1 Introduction

The interaction between the sea and the atmosphere is less well studied and

understood than the land-atmosphere interaction. Since oceans cover the majority of the earth’s surface, the importance of understanding the air-sea interaction couldn’t be enough emphasised.

Models used to predict the future climate are dependent on understanding the air-sea interaction. The better this interaction is described the better the predictions for our climate will be. The difference between the air-sea and air-land interaction appears in lowest level of the atmosphere, the boundary layer.

The main difference between the marine boundary layer (MBL) and the boundary layer over land is that in the MBL the surface is constantly moving as a response to the meteorological forcing. As the waves evolve, the roughness changes. Since the meteorological forcing (the fluxes of momentum, heat and moisture) are dependent on the roughness, the flow is coupled to the evolution of the surface itself.

The air-sea fluxes of momentum, heat and mass in the MBL are important parameters for atmospheric, oceanic and wave models. Direct measurements of these fluxes in the marine environment are difficult to make and therefor they are sparse. Instead, the fluxes are estimated from more easily measured mean variables together with bulk transfer coefficients. The bulk transfer coefficient used to parameterise the sensible heat flux is called the Stanton number, or CH.

Earlier studies of CH conclude that the Stanton number has more or less a fixed value although with large scatter (Large and Pond, 1982; DeCosmo et al., 1996; Oost et al., 1999) . The fixed value for CH is different for stable and unstable stratification. A previously study has been made for CH at the Östergarnsholm site (Rutgersson et al., 2000) although the wave-state point of view was not applied. The current paper will however focus on the connection between wave age and the Stanton number.

Existence empirical relationships for the turbulent fluxes follows from Monin- Obukhov’s similarity theory, see Section 2.

2 Theory 2.1 Swell

Waves travelling faster than the wind (swell) are a common feature over the ocean.

Air-sea interaction during these circumstances is much less well studied than

situations where the wind speed is much higher than the speed of the waves. If swell has any influence on the transfer of momentum and heat, it will have a large impact on the prediction of the future climate.

A commonly used definition of swell is when the wave age is larger than 1.2. Wave age, in this case, is defined as the speed of the dominant waves (the speed of the waves at the peak of the wave spectrum, c0) divided by the wind speed at 10m, U10:

10 0

U

c (2.1)

Another way of describing wave age, the grade of development of the wave field, is:

* 0

u

c (2.2)

where u* is the friction velocity u_* = =((u′w′)² +(v′w′)²)^1/4 ρ

τ , u ′′w and v′w′ are

the kinematic fluxes of momentum, τ is shearing stress and ρ is the density of air.

Swell is then defined as 30

* 0 >

u

c .

2.2 Correlation coefficients

Definition of a correlation coefficient between two variables, for instance, the longitudinal wind component, u, and the vertical wind component, w, is,

w u w u

w R u

σ σ

′

= ′

, (2.3)

where u′w′ is the momentum flux, and σu and σw are the standard deviation of the longitudinal and vertical wind component respectively.

According to Schwartz’s inequality uw ≤σ_uσ_w Ru,w is a number between –1 and 1, and it states how well the two variables are correlated (Arya, 1988). A value of 1 or -1, means that the variables are totally correlated, or counter-correlated (vary together or oppositely) and a value of 0 means that there are no correlation at all.

In an earlier study (Smedman et al., 1999) of a single case, the correlation coefficient was found to drop to very low value during swell conditions. It was suggested that this was due to wave induced pressure transport of inactive turbulence from the top of the boundary layer to the lower levels. In this paper, Ru,w has been investigated with a larger data set, from 1995-1999, and it is compared to the correlation coefficient between vertical wind and temperature Rw,θ The correlation coefficients are also studied during neutral conditions, see Section 4.

2.3 The Stanton number, CH

The sensible heat flux, H, can be written as:

θ ρ ′ ′

= c w

H _p (2.4)

where ρ is the density of air, cp is the air specific heat for constant pressure, w′and θ′ are deviations of the mean values for vertical wind and potential temperature and

θ′

w′ are the kinematic heat flux. w′θ′ is then simplified by a parameterisation with a bulk formula:

) )(

( ₀ θ θ₀

θ′= − −

′ C U U

w _H (2.5)

CH is the heat bulk transfer coefficient (the Stanton number), U and U0 are the mean wind speeds at a height of z (order of 10m) and at the sea surface respectively and θ and θ0 are the potential temperatures at a height of z and at the sea surface

respectively.

This parameterisation ignores any fluctuations in the density of the air but these fluctuations make a very small contribution to the total heat flux. However, by just considering the turbulent transport in the parameterisation will provide uncertainty in the flux estimates (Geernaert, 1999).

As mentioned in the Introduction, CH is derived with the Monin-Obukhov's similarity theory, which predicts that four fundamental parameters are sufficient to describe the relation between the mean gradients and the turbulent fluxes in the surface layer. The four parameters are: 1) friction velocity u_* =((u′w′)² +(v′w′)²)^1/4 2) the heat flux

θ′

w′ , 3) height above the surface z and 4) the buoyancy parameter T0

g , where g is the

acceleration of gravity and T0 is the mean absolute temperature (Geernaert, 1999).

With the aid of these, the scaling parameters for temperature and humidity can be written as:

Temperature scale

*

* u

T −w′θ′

= (2.6)

Humidity scale

*

* u

q q w′ ′

= (2.7)

The Monin-Obukhovs length is defined as:

w v

gk T L u

θ′

′

= − ^{*3 0} (2.8)

Where w′θ_v′ is the turbulent flux of virtual potential temperature and k is the von Karmáns constant equal to 0.40 (Högström, 1996).

The physical interpretation of L is that it’s proportional to the height at which the buoyant production of turbulence starts to dominate over the mechanical production of turbulence (Stull, 1988). During neutral conditions, w′θ_v′ =0, making L equal

, and making the ratio z/L equal 0. z/L is a commonly used stability parameter measuring the relative importance of buoyant and mechanical production of turbulence. For heights close to the surface (

∞

±

L

z<< ) the wind shear effects (mechanical production) dominates over the buoyancy effects and the other way around for z>> L (Arya, 1988). Unstable conditions, w′θ_v′ >0, corresponds to negative values of z/L and stable conditions w′θ_v′ <0 corresponds to positive values of z/L.

The non-dimensional profile functions, are, according to the similarity theory, universal functions of z/L. For temperature and momentum, the profile functions are written as:

θ φ θ

′

∂ ′

= ∂ w kzu L z

h(z/ ) * (2.9)

*

) /

( u

kz z L u

m z ∂

=∂

φ (2.10)

Vertical integration of eq. 2.9 and eq. 2.10 gives the surface layer profiles for wind speed and temperature:

) ) / (ln( ₀

*

0 z z m

k U u

U − = −ψ (2.11)

) ) / (ln( ₀

*

0 z z T h

k u

wθ ψ

θ

θ ′ ′ −

=

− (2.12)

where z0 and z0T are the roughness lengths for momentum and heat respectively. That is, the height at which the wind and the temperature approach their surface values. ψm

and ψh are the integral of the non-dimensional profile functions,

(2.13)

( )

L

z 1 ( / /

)

( ^/

0 − ⋅ =

=

∫

ζ ψ

They are also known as the stability functions since they give a correction for non- neutral stratification. For neutral conditions the stability functions are equal to zero (integration from 0 to 0).

With substitution of eq. 2.11 and 2.12 into eq. 2.5 CH can be expressed as:



 





 −



 





= −

−

−

′

= ′

h H m

H z z

k z

z k U

U C w

ψ ψ

θ θ θ

) / ln(

) / ln(

) )(

( _{0 0 0 0} (2.14)

This derivation requires that conditions for the Monin-Obukhov's similarity theory in the surface layer are satisfied (height constant fluxes, stationary- and horizontal homogeneous- flow) Deviations from these conditions lead to erroneous values of CH.

2.4 Characterising the sea state

The same approach as made by Guo-Larsén et al., 2002, to characterise sea state with one dimensional wave spectra, is adopted in this study. A division of the wave

spectrum into two parts is made:

(2.15)

∫

= ¹

0

1 ( )

n

dn n S E

(2.16)

∫

=

1

)

2 (

n

dn n S E

ϕ π cos 2 ₁₀

1 U

n = g (2.17)

where, n is the frequency, S(n) is the one-dimensional wave spectrum and ϕ is the angel between the wind at 10 m and the waves at the peak frequency. The frequency n1, is the frequency at which the wind component in the wave direction and the phase speed c is equal. Lower frequencies than n1 correspond to the long-wave part of the spectrum, E1 and higher frequencies correspond to the short wave part E2.

Figure 1(a) (from Guo-Larsén et al., 2002) shows wind direction (horizontal line) compared to wave direction at the different frequencies, 1(b); the logarithm of n⁴S(n), 1(c); An example of a wave spectrum S(n) and 1(d) shows U10 (horizontal line) compared to Uc=U10cos φ (dashed line), and c (solid line). Thus, E1 is the area left of the solid line in Figure 1(c) and E2 is the area to right.

When swell is present it will appear in the low-frequent part E1 of the spectrum. It was concluded in Guo-Larsén et al., 2002, that this method, dividing the one-

dimensional wave spectrum in two parts , one ”… ’wind-speed-independent part’ E1

and a ’wind sea part’, E2, is likely to give a reasonably correct result”.

0 0.1 0.2 0.3 0.4 0.5 0.6 0

50 100 150 200 250 300 350

n (Hz)

a

0 0.1 0.2 0.3 0.4 0.5 0.6

0 0.05 0.1 0.15 0.2

n (Hz) S(n) (m2 Hz−1 )

c

0 0.1 0.2 0.3 0.4 0.5 0.6

−25

−20

−15

−10

−5

n (Hz) ln( n4 S(n))

b

0 0.1 0.2 0.3 0.4 0.5 0.6

0 5 10 15

n (Hz) U c,C,U (ms−1 )

d

mean wind direction wave direction ( ^o )

longer waves shorter waves

E1 E

2

C

Uc

U

Figure 1. Example of wave-spectrum analysis (from Guo-Larsén et al., 2002) a) wind direction compared to wave direction b) the logarithm of n⁴S(n) c) example of a wave spectrum d) U10

(horizontal line) compared to Uc (dashed line) and the wave speed c (solid line)

2.4.1 Phase speed

The phase speed of the dominant waves, cp, are calculated according to deep water dispersion relation:

p

p n

c g π

= 2 (2.18)

(Arya, 1988), where np is the peak frequency of the wave spectrum. Although no major shallow water effects are present, as discussed in the Introduction, a slight correction for waves travelling faster than 6,5 ms^-1 is done. The correction is given by the following equation (Guo-Larsén, 2002, pers. comm) :

(2.19) 98487

. 0 3858 . 1 037433 .

0 ² + −

−

= _{p p}

pc c c

c

where cpc is the wave phase velocity derived with consideration of water depth, sea- bottom topography and wind direction.

2.5 Wave influence on z/L

In the marine surface layer, the total stress τ can be written as a sum of three parts:

visc wave

turb τ τ

τ

τ = + + (2.20)

where τ_turb is the turbulent stress, τ_wave is the wave induced stress and τ_visc the viscous (molecular) stress. The viscous part is important only in the lowest

millimetres. The height of the wave influence is currently under debate. For growing sea, the wave influence is believed to be in the order of 0.1m. Although, Smedman et al., 1999, found that during swell, the wave influence extend far up in the boundary layer.

The wave induced stress is negative during swell, thus reducing the total stress. This reduction results in very low values of τ, close to zero or even negative.

Remembering the definition of u*:

ρ

= τ

u* , and z/L:

0 3

*T u

w zgk L

z − ′θ_v′

= we see that if τ is reduced then u* is also reduced. For marine conditions the heat flux is usually small, making u* the most important parameter in determine z/L. Thus, a small friction velocity leads to large values of z/L (Sjöblom and Smedman, 2002). To conclude: During swell, the total stress is reduced, resulting in large values of the stability parameter z/L. Thus, using the ψ–functions to reduce CH to its neutral value, as done by many studies, doesn’t have any physical meaning.

3 Site and Measurements 3.1 The site Östergarnsholm

The main measuring site, Östergarnsholm, is a small, flat island situated 4 km east of Gotland (Figure 2). The tower is situated at the south end of the island and with its base just 1m above the mean sea level. The actual sea level isn’t constant, it’s changing with changing wind conditions, and it’s derived with the aid of measurements of the sea levels in Visby harbour.

Tower

Wave rider Gotland buoy

Östergarnsholm Tower

0 1 2 3

km

5 10°

55°

60°

20°

5 6

6 6

23

15

30 10

Baltic Sea

Figure 2. Map of the measuring site with surroundings

In Smedman et al., 1999 it was concluded that there isn’t any serious influence of limited water depth. It was found that, for the 10m level, 90% of the fluxes originated from 250m beyond the shoreline and that during light-wind conditions, the phase speed of the dominating waves cp varied between 92-99% of the deep-water value.

Although no significant effect of shallow water is assumed to be present, a slight correction for waves travelling faster than 6,5 ms^-1 is made (Guo-Larsén, 2002, pers.

comm.) as mentioned Section 2.

Wave data is received from a wave rider buoy, run and owned by the Finnish Institute for Marine Research, moored 4km ESE of Östergarnsholm at 36m depth. This means that the buoy is representing the wave conditions for wind coming from the south sector (80-220°). Also, in this direction, there is undisturbed water fetch for at least 150km, only data from this sector is used. Fetch can be described as the surface area which influence the measured fluxes.

3.2 Instrumentation

The tower is equipped with slow response sensors for profile measurements of temperature, wind speed and wind direction. These sensors are placed at 7, 11.5, 14, 20 and 29m above the tower base. Humidity is measured at 7m above the tower base.

Turbulent fluctuations were recorded with SOLENT 101R2 (Gill Instrument,

Lymington, UK) sonic anemometers placed at 9, 16.5 and 25m above the tower base.

The sonic gives the three wind components (u, v and w) and the virtual temperature.

Turbulence data from the lowest level is only used in this study.

Profile data is recorded at 1 Hz and turbulence data at 20 Hz and are averaged over 60 minutes periods. Wave data is recorded once an hour.

3.2.1 Correction of the heat flux

The sonic anemometer is influenced by the density changes of the air caused by the water vapour and thus gives the virtual sensible heat flux. In marine conditions, with small heat fluxes and a high humidity level, the difference between virtual and sensible heat flux can be quite large. Since the calculation of CH requires the sensible heat flux a correction for humidity is needed. Lumley and Panofsky, 1964, give the following relation as a good approximation:

β θ θ

07 . 1+0

′

= ′

′

′ w ^v

w (3.1)

where β is the Bowen ratio, the ratio of sensible heat flux and latent heat flux = λ E

H , λ is the latent heat of vaporisation. With the assumption that the bulk transfer

coefficients for heat and humidity, CH and CE, are equal, the Bowen ratio can be calculated from the temperature and humidity gradients:

z qz c_p

∂

∂∂

∂

= λ

θ

β (3.2)

q is the mixing ration, (kg water)/ (kg air) and is written:

s s H

e p

e q r

) 1

( ε

ε

−

= − (3.3)

where rH is the relative humidity, p is air pressure, ε is the ratio of the specific gas constant for dry air, Rd, and the specific gas constant for water vapour, Rv,

622 .

≈0

v d

R

R , and es is the saturated water vapour pressure which can be calculated

according to Magnus’ exact formula:

T T

e_s 6808,475 5,088336ln

281723 ,

55

exp( − −

= (3.4)

A second correction for heat flux is also needed. This is due to crosswind

contamination. The crosswind (the perpendicular wind component) creates a flow distortion causing signal deflection of the sound path in the instrument. Kaimal and Gaynor’s correction (Kaimal and Gaynor, 1991), is used in this study:

403 2U uw w

w corrected uncorrected ^z

′ + ′

′

= ′

′

′θ θ (3.5)

3.3 Data Criteria

In this paper data from 1995-1999 has been used. The data used for the analysis of CH

has been selected according to the following criteria:

1. Wind from the south-sector (80-220°) 2. Wind speed at 10m, U10 > 2 ms^-1

3. φ, angel between the wave direction and wind direction at 10m < 40°

4. Wave spectra with a single peak

5. θ₁₀ −θ_s (the temperature difference between the sea surface and the air at 10m)>1.5K

6. w′θ′>0.01 ms^-1K

7. Complete meteorological and wave data

Wind from the south sector is needed to get undisturbed sea-fetch as discussed above.

The second criterion is due to ’low wind effects’. With low wind speeds, the wind direction is hard to determine, it can change between two extreme values during a short period of time. The third criterion is based on the fact that it is mainly the wind component in the wave direction that is related to the exchange with the waves.

Equation 3.6 is therefor a better definition of wave-age than equation 2.1 (Rutgersson, et al., 2001):

φ

10cos

0

U

c (3.6)

This definition will be used in combination with another definition as discussed in Section 2. Wave spectra are also discussed in Section 2. Criteria number 5 and 6, are needed to get meaningful calculations of CH (see Section 2, at neutral stability both the heat flux and the temperature difference approaches zero).

Another data set is used to analyse Ru,w and the heat flux during neutral conditions.

The same limits as above are applied to this set but criteria number 5 and 6 are replaced with a criteria for near neutrality based on the magnitude of the heat flux:

K ms w 0.01 ¹ 002

.

0 < ′ ′< ⁻

− θ .

Normally, near neutrality is defined in terms of z/L (see eq. 2.8 for a definition of L).

However, the wave state has a large influence on z/L during swell conditions

(discussed in Section 2). Defining neutrality based on the heat-flux has a flaw. Cases with w′θ′ >0.01ms⁻¹K could be neutral in terms of z/L if the heat flux is combined high wind speeds.

4 Results

4.1 The variation of the correlation coefficients, Ru,w and Rw,θ , with stability and wave age

The variation of Ru,w with stability and wave age can be seen in Figures 3a and 3b respectively. The variation of Rw,θ with stability can be seen in Figure 4. Different colours and symbols represent different values of E1/E2. E1/E2<0.2 (blue squares) represent ‘growing sea’, pure wind sea conditions. E1/E2 > 4.0 (magenta diamonds) represent swell conditions. Values in between growing sea and swell, represent different states of ‘mixed sea’.

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2

z/L R u,w

E₁/E₂ <0.2 0.2 <= E₁/E₂ <1.0 1.0 <= E₁/E₂ <2.0 2.0 <= E₁/E₂ <4.0 E₁/E₂ > 4.0

Figure 3a. The correlation coefficient Ru,w as a function of z/L

0.5 1 1.5 2 2.5 3 -0.5

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2

R_{u´w ´}=-0.32806

R_{u´w ´}=-0.13583

c_p /(u₁₀*cos(φ) ) R u´w´

Figure 3b. The correlation coefficient Ru,w as a function of wave-age. Bars represents ±1 standard deviation. Stars represents mean values of each wave-state group.

-1.5 -1 -0.5 0 0.5

-0.4 -0.2 0 0.2 0.4 0.6 0.8

z / L R w´θ´

Figure 4. The correlation coefficient of temperature and vertical wind, Rw,θ as a function of z/L

When plotting Rw,θ and Ru,w with respect to stability, the correlation coefficients almost have the same value close to neutrality. As the atmosphere becomes more unstable the coefficients diverge. Rw,θ increase to values around 0.6, Ru,w decrease to values around zero. When plotting Ru,w as a function of wave age, we note that the decrease with increasing instability (as shown in Figure 3a) also seem to be coupled

with a decrease with increasing wave age. This coupling is probably due to wave influence on z/L as discussed in Section 2. In Figure 3b a rapid drop for the swell cases is seen.

Ru,w is also plotted as a function of wave-age for the neutral cases, in Figure 5. This Figure is divided into two sub graphs; one representing slightly stable cases, Figure 5a and the other representing slightly unstable cases, Figure 5b. There is a slight decrease of the coefficient with increasing wave-age although not as rapid for the non-neutral cases. The decrease is slightly more obvious for the unstable cases than for the stable.

The decrease in Ru,w , and the divergence of Ru,w and Rw,θ with increasing instability has been noted before by Haugen et al., 1971. It was concluded that this was an expected result as the atmosphere approaches free convection. The result is easily understood, on a hot day when the surface is heated, warmer than average air rise (θ′>0, w′>0) and cooler than average air sink (θ′<0, w′<0), thus the product is on the average positive and indicating that the variables are varying together (Stull, 1988).

The decrease of Ru,w with increasing wave-age has also been noted before by Smedman et al., 1999 as discussed in the Introduction.

The behaviour of Ru,w during neutral conditions has not been studied before. This is further discussed in Section 5.

0.5 1 1.5 2 2.5 3 3.5

-0.4 -0.2 0 0.2

R_{u´w ´}=-0.33452

R_{u´w ´}=-0.30084 c_p /(u₁₀*cos(φ) ) R u´w´

0.5 1 1.5 2 2.5 3 3.5

-0.4 -0.2 0 0.2

R_{u´w ´}=-0.35452 R_{u´w ´}=-0.30336

c_p /(u₁₀*cos(φ) ) R u´w´

E₁/E₂ <0.2 0.2 <= E₁/E₂ <1.0 1.0 <= E₁/E₂ <2.0 2.0 <= E₁/E₂ <4.0 E₁/E₂ > 4.0 a

b

Figure 5 The correlation coefficient, Ru,w as a function of (wave age), neutral cases. Different symbols and colours represent different values of E1/E2 (see figure legend) a. slightly stable stratification b.

slightly unstable stratification

4.2 The wave state influence on CH

Figure 6 shows CH as a function of wave age. There is a slight increase in CH with increasing wave age, although there is large scatter. This result is somewhat

contradicting an earlier study, Mahrt et al., 1998, who found a slight decrease of CH

with increasing wave age, further discussed in Section 5.

0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5

3x 10^-3

C_H=1.0639*10^-3

C_H=1.2399*10^-3

c_p /(u₁₀*cos(φ^{) )}

CH

Figure 6. The Stanton number, CH, as a function of wave age. Different symbols and colours represent different values of E1/E2 (see legend in Figure 5)

4.3 The stability influence on CH

The stability dependence of CH is shown in Figures 7a and 7b. Figure 7a shows the stability dependence for the growing sea cases (E1/E2<0.2) and for the cases where 0.2<E1/E2<1.0. Figure 7b shows the rest of the data set, including the swell cases.

There is a difference in the behaviour of the curves between the growing sea cases and the swell cases. Figure 8 stresses this difference by just showing these two cases and just mean values. For growing sea, CH falls rapidly as it passes over z/L=0, for swell, the transition from unstable to stable stratification is much ‘smoother’. Also, there seems to be a slight enhancement of CH for the growing sea cases just before the rapid drop on the unstable side.

-0.50 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

1 1.5

2x 10^-3

z/L C H

E₁/E₂ <0.2 0.2 <= E₁/E₂ <1.0

-1.20 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

0.5 1 1.5 2 2.5

3x 10^-3

z/L C H

1.0 <= E₁/E₂ <2.0 2.0 <= E₁/E₂ <4.0 E₁/E₂ > 4.0 a

b

Figure 7. CH as a function of stability (z/L). The figure is divided into two plots depending on the value of E1/E2 a. 0.2<E1/E2<1.0 b.E1/E2 >= 1.0

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8x 10^-3

z/L

C H

E₁/E₂ <0.2 E₁/E₂ > 4.0

Figure 8. CH as a function of z/L. Blue line represent mean values of CH for growing sea cases, magenta line represent mean values of CH for swell cases. Note the more continuos transition over

Earlier studies have noted the difference in CH for unstable and stable stratification (Large and Pond, 1982, Oost et al., 1999). Their results are very similar to the behaviour of the growing sea cases in this study, with a rapid decrease of the transfer coefficient as it passes neutrality. The more continuos decrease in CH for swell cases haven’t been noticed before.

4.4 The stability influence on the heat flux

Figure 9, shows the turbulent heat flux as a function of z/L. The data set used is the same as the one used to calculate CH (Section 3.3). In this case the heat flux seems to approach neutrality along different lines depending on the wave state. Keep in mind that the near neutral cases have been excluded.

When plotting just the averages of the heat flux for growing sea and swell, in Figure 10a, the result emerged is strikingly similar to Figure 8. The heat flux seems to approach neutrality along lines, as described above, where the tilt increases as the wave age decrease. Similar to CH’s dependence on stability, it is the growing sea cases who has the most rapid decrease approaching z/L=0. Another similarity is the slight increase of the turbulent heat flux before the rapid drop, this is also seen for the growing sea cases in Figure 8.

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.06

-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

z / L w´θ´

Figure 9. Turbulent heat flux, w′θ′, as a function of z/L. Remember that data is selected according to

01 ,

>0

′

′θ

w .

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -0.02

-0.01 0 0.01 0.02 0.03 0.04 0.05

z/L w´θ´

E₁/E₂ <0.2 E₁/E₂ > 4.0

Figure 10a. The turbulent heat flux,w′θ′, as a function of z/L, mean values. Blue line represent growing sea cases, magenta line represent swell cases.

For the neutral data set, fig 10b, the plot of the mean value of the heat flux as a function of stability, is similar to 10a, with a clear stratification of the different wave ages, growing sea making the most rapid transition over neutral stratification.

-0.1 -0.05 0 0.05 0.1 0.15

-2 -1 0 1 2 3 4 5 6x 10^-3

z/L w´θ´

E₁/E₂ <0.2 0.2 <= E₁/E₂ <1.0 1.0 <= E₁/E₂ <2.0 2.0 <= E₁/E₂ <4.0 E₁/E₂ > 4.0

Figure 10b. Heat flux as a function z/L, neutral cases, mean values. Different colours represent different values of E1/E2.

This result states that there is more efficient turbulent transport of sensible heat (larger exchange coefficient) for increasing z/L and that the increase is larger for growing sea cases than for swell cases.

4.5 Spectral characteristics

A cospectral graph for w and θ shows how much each frequency contributes to the total heat flux. The area confined in the Figure sums up to the total heat flux during the corresponding time period, which in this analyse is one hour. The relation used to describe wavelength is:

f

= v

λ (5.1)

where λ is wavelength, v is velocity and f is frequency. If the velocity is constant for each frequency, a higher frequency corresponds to smaller wavelengths, or eddies.

Seeing the difference in the heat flux behaviour for different wave states, it could be worth studying the cospectra for w and θ. This could determine the dominating frequencies for different wave states and different stabilities. The spectral analysis could also give clues to why CH behaves differently for swell and growing sea as it approach neutrality from the unstable side.

4.5.1 Stability influence on the w,θ cospectra

Figure 11a and 11b, show cospectra of w,θ normalised with the heat flux itself, plotted versus frequency, n. The normalisation is done to be able to study the relative contribution to the heat flux by each frequency. Figure 11a shows the cospectra for unstable cases (66 runs) and Figure 11b, shows the cospectra for stable cases (26 runs). For the stable case, there’s only one peak at n≈0.25 Hz, whereas for the

unstable case there’s a saddle shape structure with a well defined peak at, n≈0.015 Hz and a second, less well defined peak at n≈0.1 Hz

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹

0 0.05 0.1 0.15 0.2

n (Hz) Co wθ / w´θ´

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹

-0.1 0 0.1 0.2 0.3

n (Hz) Co wθ / w´θ´

a

b

Figure 11. Cospectra of w,θ normalised with the heat flux itself. The figure is divided into two sub plots depending on the stratification a. unstable stratification b. stable stratification

4.5.2 Wave influence on the w,θ cospectra

Figure 12 is divided into two sub plots; Figure12a shows the cospectrum, normalised with the heat flux, plotted versus frequency n for cases with the wave parameter E1/E2

<1.0. The different lines represent different stability ranges, see figure legend. In Figure 12b is the same analysis is made for wave parameters with E1/E2>=2.0, although the stability ranges are somewhat different from 12a (see figure legend).

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹

-0.4 -0.2 0 0.2 0.4

n (Hz) Co wθ / w´θ´

-0.3<z/L<-0.2 -0.2<=z/L<-0.15 -0.15<=z/L<0 z/L >0

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹

-0.1 0 0.1 0.2 0.3

n (Hz) Co wθ / w´θ´

-1.1<=z/L<-0.55 -0.55<=z/L<-0.35 -0.35<=z/L<0 z/L >0 a

b

Figure 12. Stability dependence of the w,θ cospectra. The different colours represent different stability ranges. The two plots show the w,θ cospectra for different ranges of E₁/E2. a. E1/E2 <1.0

b. E1/E2>=2.0

For the younger waves, Figure 12a, the curves for the unstable cases have similar behaviour with no obvious difference between the most unstable curve (blue, squared line) and the curve closest to neutrality (green, dotted line). The stable case,

represented by the magenta, triangle line, describes a completely different regime where the heat flux is dominated by higher frequencies (smaller scale eddies).

The older waves in Figure 12b, show a more inconsistent behaviour. For the unstable cases, the peak jumps between high and low frequencies, strikingly different from the younger waves. For the stable cases (the magenta triangle curve), the curve suggests that there still is one peak at low frequency, and one peak at a higher frequency, similar to the unstable cases, although the mean is calculated from only 4 data. This result is also quite different compared to the younger waves.

4.6 Reasons of scatter; advective effects

As mentioned in Section 2, the definition of CH requires that the conditions for the Monin-Obukhov’s similarity theory in the surface layer are satisfied. Reasons of scatter in CH could be due to deviations from these conditions. One particular case has been identified and studied.

The marked data points in Figure 13 (Fig. 13 is a close-up version of Fig. 8) are all from consecutive time periods. CH is constantly increasing during this time series and it culminates at a very large value in stable stratification. Then the stratification changes to unstable. The wave state changes as well, to swell values due to lower wind speed.

0.010 0.02 0.03 0.04 0.05 0.06 0.07

0.5 1

x 10^-3

z/L C H

-0.50 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2

1 2 3x 10^-3

z/L

C H

298

299

300 301

302 303

304

305

306

Figure 13. Close up version of Figure 8. CH as a function of z/L. Numbered data points marks consecutive measurements during the time period 991002-991003.

Figure 14, shows, in detail, some different parameters during the time period (991002 16:30 – 991003 15:30), a) shows the turbulent heat flux w′θ′ , b ) shows the wind speed at 10m c) the temperature difference between the sea surface and 10m d) shows CH e) shows the wind direction and f) shows the stability parameter z/L.

This figure illustrates three things:

1) During frontal movement over the measuring site, warm or cold advection, the Monin-Obukhovs conditions for the surface layer is not satisfied. The main violation is the assumption that the fluxes are height independent; during stationary temperature advection ( =0

∂

∂ t

θ ) we obtain from the thermodynamic energy equation:

z w U x

∂

′

∂ ′

−

∂ ≈

∂θ θ

(5.2) (Stull, 1988, Mahrt et al., 1998). This means that the vertical integration of the

nondimensional gradients φ, made in Section 4, to obtain the stability functions ψ is not valid, which in turn leads to large scatter of CH.

2) It illustrates why the magnitude of the temperature difference and the heat flux is applied as a criterion in the data selection. During times with very small

temperature differences, close to zero, combined with a very small turbulent heat flux, we note that CH takes on unreasonable values and sometimes even negative values.

3) Figure a) and c) illustrate that there is a ‘time-lag’ between the temperature difference and the turbulent heat flux. It takes a certain amount of hours for the heat flux to adjust to the changing temperatures. For instance: The temperature difference reaches values close to zero around 5 a.m. but this is not seen in the heat flux until 8 a.m. This lag also causes CH to assume meaningless values.

15:00-5 18:00 21:00 00:00 03:00 06:00 09:00 12:00 15:00 18:00 0

5

15:00 18:00 21:00 00:00 03:00 06:00 09:00 12:00 15:00 18:00

-0.1 0 0.1

15:00-5 18:00 21:00 00:00 03:00 06:00 09:00 12:00 15:00 18:00 0

5

15:000 18:00 21:00 00:00 03:00 06:00 09:00 12:00 15:00 18:00 5

10 15

15:00 18:00 21:00 00:00 03:00 06:00 09:00 12:00 15:00 18:00

150 200 250 300

15:00 18:00 21:00 00:00 03:00 06:00 09:00 12:00 15:00 18:00

-0.02 0 0.02

a) w´θ^´

b) U 10m

c) T₀-T₁₀

d) C_H

e) Wind direction

f) z/L

Figure 14. Different parameters during cold advection over the measuring site. 991002-991003.

5 Summary and conclusions 5.1 The correlation coefficients

The main result is the difference in behaviour of Ru,w during neutral and non-neutral stratification. When investigating Ru,w during neutral conditions we isolate it from any stability influence, just studying the wave influence. The result showed that there is a small wave influence on the correlation coefficient. To reach values of Ru,w close to zero, it requires a certain amount of positive heat flux in combination of transport of inactive turbulence as discussed by Smedman et al., 1999.

Another evidence of wave influence on Ru,w is that the correlation coefficient seems to get even lower values during unstable stratification over sea than over land. In this investigation Ru,w even assumed positive values. Compared to Haugen et al., 1971, this study had overall slightly lower values, which is taken as evidence for some sort of wave influence on the correlation coefficient Ru,w.

5.2 Stability and wave influence on CH

The difference of the wave-age dependence of CH compared to Mahrt et al., 1998, is hard to explain. Although, in that paper, wave-age is defined as cp/u*. This definition has also been applied to the data in this study (not shown) but the difference remains.

One possible explanation may be due to the fact that the measurements in Mahrt et al., 1998, were made on the west coast of Denmark in shallow water (<4m). This could lead to deviations of the wave phase speed from its deepwater value. Another plausible explanation is that Mahrt et al., 1998, has different stability conditions than observed in this study.

The lower value of CH during stable conditions compared to unstable conditions over sea (as observed in many studies before) is also observed over land (Johansson, 2002, pers. comm.) Although the exact behaviour of CH over land is hard to describe since there is difficulties in determining the correct surface temperatures .

The difference of CH during different stratification physically means that the transfer of sensible heat is less efficient during stable conditions. The comparison of the cospectrum representing the heat flux, shows that the high frequencies (small eddies), with a peak at 0.25 Hz, dominate the heat flux during stable stratification as opposed to unstable stratification where lower frequencies (larger eddies) dominate the heat flux with a peak at 0.015 Hz.

In the cospectrum representing unstable cases, there is some kind of ’saddle

structure’, with a second, smaller peak, at n=0.1 Hz. Thus, the turbulent transport of heat consists of both larger and smaller scale eddies. During the shift to stable stratification, the larger size eddies are abruptly suppressed. This result suggests that the smaller scale eddies are less efficient in transferring heat compared to the larger scale eddies.

The main result emerged from this study is however the difference of how CH, approach neutrality depending on the wave state. During growing sea, CH decreases

approaching z/L=0. The investigation of the heat flux showed similar behaviour suggesting that the answer lies in the heat flux itself. The w′θ′ cospectra divided into different stability ranges and wave states showed differences.

For growing sea, there is very little stability influence on the cospectra during

unstable stratification. The shift to stable stratification corresponds to a rapid jump to higher frequencies in the cospectra. This could be one of the reasons explaining the rapid decrease of CH during the same shift from unstable to stable stratification.

For swell conditions, there seem to be some kind of stability influence on the cospectra during unstable stratification. The peaks in the spectrum can be found at different frequencies depending on the stability range. During stable stratification it’s hard to make any conclusion since the data set only consist of 4 data with E1/E2>=2.

Different stability dependence during different wave states has not been observed before. One of the reasons could be due to the fact that most of the earlier studies’

data sets consist of growing sea data, or non-swell dominant cases. If no division is made into different wave-states when plotting CH as a function of z/L, the swell cases will just be seen as scatter, and the result will be a CH with two different mean values depending on the stratification, with large scatter.

The spectral analysis could be further expanded to get more reliable means. The conclusion made here is that there is a difference in how CH approach neutrality from the unstable side depending on the wave state and that this difference also can be seen in the cospectra for the turbulent heat flux. The reason of the difference is yet to be explained.

6 Acknowledgements

I would like to thank my supervisor Ann-Sofi Smedman and Ph.D. student Xiaoli Guo Larsén for the help and ideas provided to this study. I would also like to thank

Andreas Svensson for the support with MATLAB and Word, and Jonas Höglund for valuable comments. Thanks to Cecilia Johansson for letting me use her picture of Östergarnsholm with surroundings (Section 3). At last I would like to thank the staff and students at MIUU for the contribution of good advice, practical help and a great working climate.

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