Some aspects of the viscous sublayer

Meteorologische Zeitschrift, Vol. 11, No. 4, 267-272 (August 2002) © by Gebrtider Borntraeger 2002

Article

Some aspects of the viscous sublayer THOMAS FOKEN

Department of Micrometeorology, University of Bayreuth, Germany (Manuscript received June 19, 2001; in revised form January 18, 2002; accepted January 20, 2002)

Abstract After an historical introduction into the research activities of Prof. Hinzpeter and the author, because of their roots in the Meteorological Observatory Potsdam the investigation of the molecular temperature sublayer over the ocean was selected as one field of a common interest. The experimental investigations made in Kiel and Leipzig are presented. It is shown that the thickness of the molecular sublayer depends on the friction velocity and the conditions of the wavy surface. Based on this data, parametrizations of the viscous sublayer on the basis of these investigations and of hydrodynamical measurements are discussed. Possibilities to use this parametrizations in air-sea interaction models are considered.

Zusammenfassung Nach einer historischen Einleitung zu den wissenschaftlichen Wurzeln von Prof. Hinzpeter und dem Autor im Meteorologischen Observatorium Potsdam wird die Untersuchung der molekularen Grenzschicht iiber dem Ozean als ein gemeinsames wissenschaftliches Interessengebiet nliher betrachtet. Dabei werden die in Kiel und Leipzig durchgefiihrten Untersuchungsergebnisse dargestellt. Es wird gezeigt, dass die Dicke der molekularen Temperaturgrenzschicht von der Schubspannungsgeschwindigkeit und von der Entwicklung der Oberfllichenwellen abhlingt. Auf Grund dieser experimentellen Befunde werden Parametrisierungen fiir die viskose Unterschicht auf der Basis dieser Daten und hydrodynamischer Untersuchungen sowie Moglichkeiten zu deren Einbeziehung in Wechselwirkungsmodelle Mcer-Atmosphlire diskutiert.

1 Introduction The author was invited to present a paper at the Hinzpeter colloquium about the energetic of the atmospheric boundary layer. Because I met Prof. Hinzpeter first in 1990 it was impossible to comprehensively review a long time period of common research activities. Nevertheless, to find common scientific interests a short historical review is necessary, because there are three interesting topics of similar activities of Prof. Hinzpeter and myself which I reviewed in my presentation at Hamburg. First, we both worked in the Potsdam Meteorological Observatory (1893-2000, KORBER, 1993) from the end of the 1940s to 1958 and 1978-1994, respectively, and our scientific ideas were grounded in the scientific tradition of this observatory, partly in the works by Albrecht (FOKEN and SPANKUCH, 1996). Secondly, our interest was focussed on radiation processes, particularly that of Prof. Hinzpeter during the solar eclipse in 1954 (HINZPETER and wORNER, 1955) and by completing HOLPER's (1893-1944) manuscript to a monograph on atmospheric radiation (FOITZIK and HINZPETER, 1958). This influenced his activities to focus the research activities of the Potsdam Meteorological Observatory on radiation research after the unifica• Author's address: Thomas Foken, Universitiit Bayreuth, Abt. Mikrometeorologie, 95440 Bayreuth, e-mail: [email protected]

tion of Germany, mainly since 1994 (HEMPEL, 2002). My personal interest in radiation started very late, in the 1990s, closely connected with the problem of the energy balance closure at the surface (FOKEN, 1998; FOKEN and ONCLEY, 1995). This topic was reviewed during the colloquium, but because of several other papers (ONCLEY et al., 2000) I want to focus this paper on the third problem: In 1939 BRUCH (1940) investigated on the 'Sacrow' lake near Potsdam and later at the Baltic Sea near the island 'Greifswalder Oie' the temperature profile from 1.3 m above the water down to some centimetres below the surface. This experimental work was mainly influenced by Albrecht and supported by the Potsdam Meteorological Observatory. He found very large and partly linear gradients near the surface. This indicates a nearly molecular heat exchange near the surface with linear temperature gradients and offers the possibility to determine the sensible heat flux directly. This work inspired Hinzpeter in Kiel at the end of the 1960s to investigate the molecular temperature sublayer above the ocean (CLAUSS et al., 1970; HINZPETER and LOBEMEYER, 1969), which is comparable to the laminar sublayer of the wind field. On the other hand this work was also the basis of Prof. Hupfer's (Leipzig, later Berlin) investigations of meteorological microstructure in the near shore zone of the Baltic Sea (HUPPER, 1970) and he focussed my interest on this topic in my diploma work and dissertation (FOKEN, 1978a). In the follow0941-2948/02/0011-0267 $ 02. 70

DOI: 10.1127/0941-2948/200210011-0267

© Gebriider Borntraeger, Berlin, Stuttgart 2002

Th. Foken: Viscous sublayer aspects

268

Meteorol. Z., 11, 2002

Table 1: Successful measurements of the molecular sublayer (FOKEN, 1978a).

Year

Place

1967 1968 1972-74,1976 1975,76

Baltic Sea, Bay of Kiel Atlantic Baltic Sea, coastal region near Zingst Caspian Sea

Number of Measurements

Reference

14

HINZPETER and LOBEMEYER (1969) CLAUSS et al. (1970)

38 202 178

FOKEN {1978a) FOKEN (1978a), FOKEN

et al. (1978)

ing I want to review both investigations of the molecular 1 or 2 µm diameter. Fortunately, this self-made sensor temperature sublayer and discuss these investigations in was not broken after every measurement. Such a sensor the context of the present parametrizations of this ex- has a time constant of approx. 0.001 sand no radiation change process. error (FOKEN, 1979) and is able to measure the temperature within the molecular sublayer nearly unmodified if the vertical velocity of the waves is small in comparison 2 Experimental investigations of the to velocity of the dropsonde. The registration was made molecular temperature sublayer with a high speed UV-paper recorder (approx. 5 m s- 1). An example of such a registration is shown in Fig. 1. Because my investigations about the molecular temperThe measurements initiated by Hinzpeter were done ature sublayer started some years after Hinzpeter's in- in the Baltic Sea and the Atlantic Ocean while my own vestigations and all papers as well as the diploma work measurements were done in the coastal zone of the by Lobemeyer were known, there are no significant dif- Baltic Sea and mainly in the Caspian sea. There are ferences in the measuring procedure of both investiga- 432 successful measurements of both groups available tions. The main benefit of the works done in Leipzig (Tab. 1) were new types of micro-amplifiers with a higher staFrom hydrodynamic boundary layer investigations bility (FOKEN, 1975) and some new features to realize a (LANDAU and LIFSCHITZ, 1974) it follows for the higher number of registration. Instead of a lift (BRUCH, thickness of the molecular temperature sublayer (u* f:. 1940) the concept of a dropsonde was used with aver- 0) tical velocity of approx. 1 m s- 1 • At the lowest part of the dropsonde was the sensor, a fine Platinum wire of (2.1)

o'f

20 18

16

o'f

14

12

'E

.§.

is the dimensionless thickness of the molecuwhere lar temperature sublayer, U* is the friction velocity and v is the kinematic viscosity of air. The vertical temperature gradient of this layer is linear and is of the order of 102 to 103 K m- 1• Hinzpeter found in his first investiga= 7 (HINZPETER tions over the Baltic Sea a value of and LOBEMEYER, 1969). For my measurements in the coastal zone of the Baltic Sea a value of oj: = 7.7±4.6 was found (HUPPER et al., 1975). All available data according to Table 1 are presented as a function of the thickness of the molecular temperature sublayer on the friction velocity in Fig. 2. The scatter of these unselected data is remarkable. Therefore the data from experiments in the Caspian Sea were selected depending on a roughness Reynolds number with the standard deviation of the elongation of the waves Res = cr11 • u * /v (FOKEN et al., 1978). For roughness Reynolds numbers Res < 300( U* < 0.2 ... 0.3 m S-l) values of Of = 6.0 ± 3.4 were found. For larger Reynolds numbers a significant dependence on the position of the measurements on the waves was found with small values at the windward site and high values on the lee site and a mean value of oj: ~ 12. As a parametrization of the dimensionless thickness of the molecular temperature sublayer it follows that(~= 0: windward site;~= 1t: lee site):

10

N

8 6

4 2

0

K Figure 1: Temperature profile with molecular sublayer over the Caspian Sea on April 15, 1975, 5:40 p.m. (FOKEN et al., 1978).

Meteorol. Z, 11, 2002

Th. Foken: Viscous sublayer aspects

269

6 0

5

:t

0 0

4

:t

:t

0

:t

x 0

-

0

0

E

.§. 3

0

I()

0

...

• CLAUSS et al. (1970)

o EKAM-73

x

KASPEX ·75

+

KASPEX76

0

t.

:t

0 0

:t

~ ~

0

Jt:. 0

0

co 0

t. Zing st 1974

t.

+

2

• HINZP. & LOB. (1969)

0

+

•

A

:j: +

1

+

+Oo+A +

+

0~++ + +

+

0 0

*+

:t

+ +

• '

+

++

20

10

••

+

x

+

* :j:

• 40

30

U•[Cm/S)

Figure 2: Dependence of the thickness of the molecular temperature sublayer on the friction velocity with respect to all available data (FOKEN,

1978a).

:* ·[

2 +sin ( ~ -

~)]

or u+ ~ z+. Above the viscous sublayer is the turbulent flow with r+ ~ lnz+ or u+ ~ lnz+, the typical logarithThese parametrizations are very similar to those mical temperature or wind profile. The main problem is found for the laminar boundary layer in hydrodynamic a parametrization of the buffer layer. The similarity of investigations of o+ ~ 6. For a smooth surface the ratio temperature profiles of the viscous sublayer measured of the molecular temperature sublayer and the laminar over a natural surface and in laboratory experiments is boundary layer is dependent on the Prandtl number of shown in Fig. 3. air (VAN DRIEST, 1959) The parametrizations of the energy exchange between the atmosphere and the surface, with an explicit 0 ~ VPf = 0.85. (2.3) parametrization of the viscous sublayer base on the bulk approach with an integration of the profile coefficient r for all layers one as follows 3 Parametrization of the molecular H (3.1) = r(To -T(z)) temperature sublayer oj: = 7 .5

(2.2)

o

or

p·Cp

In the last paragraph only the molecular temperature sublayer was investigated. Normally in meteorology the part of the atmospheric boundary layer which is influenced by molecular exchange conditions is called the viscous sublayer. This includes the molecular or laminar sublayer [both are not identical because of Eqn. (2.3)] and the so-called buffer layer with molecular and turbulent exchange processes. For boundary layer flows in hydrodynamics these layers can be shown in a graph with a dimensionless height z+ = z · u * /v and a dimensionless temperature r+ = T /T* (T: temperature, T*: dynamical temperature scale) or velocity u+ = u/u*, u: wind velocity (LANDAU and LIFSCHITZ, 1974, CSANADY, 2001). In the molecular or laminar sublayer r+ ~ z+

r-

z

]-1

dz

- [ [ Kr+vTt +vr

(3.2)

where His the sensible heat flux, pis the air density, Cp is the specific heat for constant pressure, z is the height, Kr is the turbulent diffusion coefficient for heat, VTt is the molecular-turbulent exchange coefficient for heat in the buffer layer and vr is the molecular exchange coefficient for heat. First integrations were done with an unseparated viscous sublayer (MONTGOMERY, 1940; SVERDRUP, 1937). For the viscous sublayer an assumption of a larger value of ~ 27 .5 and for the turbulent layer the log-linear wind profile with the roughness

otr

Th. Foken: Viscous sublayer aspects

270 15

•

Foken (1978), sand

+ o

Zukauskas & Slanciauskas (1973), example 81

¢

ZUkauskas & Slanciauskas (1973), example 83

Foken (1978), water

.. . ,,.

- - - -T+• Pr*z+ - - T+•4,2*1nz++2,5

10

Meteorol. Z., 11, 2002

.....

--Reichard (1951)

I'--------------------------------------'' +

t-

........... ....... •••

5

-- 1

10

100

z+ Figure 3: Dimensionless temperature profile near the surface in the viscous sublayer by ex-perimental investigations over natural surfaces (FOKEN, 1978b), by hydrodynamical investigations (ZUKAUSKAS and SLANCrAUSKAS, 1973) and by parametrization (REICHARDT,

1951).

length zo were used. For smooth surfaces this value was used as an integration constants instead of the roughness length according to VON KARMAN (1934). An integral approach for the turbulent, buffer and molecular (laminar) layer was given by REICHARDT (1951) for the ratio of the exchange coefficient for momentum Km and the kinematic viscosity v:

ture difference was found for the dimensionless temperature difference within the buffer layer. dT+ ~ 4 (FoKEN, 1978a; FOKEN, 1984). This was the basis of a new parametrization of the profile coefficient lC· U*

r

= [K·Pr-

~]·Of +5 +In~

34 ( . )

The dimensionless thickness of the molecular temperature sublayer can be parametrized according to v zf Eqn. (2.2) depending on the roughness Reynolds numwhere K is the von-Kanmin-constant. This approach is ber and the wavy structure of the surface. This model on in good agreement with the experimental data given in the basis of experimental data shows results in a good Fig. 3 and is a useful tool for parametrization of the ex- agreement with the models given above and experimenchange processes between the atmosphere and the sur- tal data. face (KRAMM et al., 1996). Unfortunately these parametrisations of the molecuIn the 1960s and 1970s several papers with a separate lar temperature sublayer or their use in three-layer exintegration for all three layers were published (BJUT- change models for air-sea interaction modelling did not NER, 1974; KITAJGORODSKIJ and VOLKOV, 1965; really gain practical use and further investigations since MANGARELLA et al., 1972; 1973). These investigations the 1970s are unknown to the author. This concerns included new experimental data sets from hydrodynam- even simpler analytical expressions like REICHARDT's ical investigations, and parametrized the influence of the (1951) approach. One reason may be the completely wavy structure of the surface (FOKEN, 1978a). Never- different approach used in meteorological modelling theless, the exact parametrization of the buffer layer was (GEERNAERT, 1999): Similar to the log-linear wind profile with a roughness parameter, such profiles and paunsolved in all these papers. Using the experimental data of the direct measure- rameters were also defined for scalars (Lours, 1979). ments of the molecular temperature sub-layer and an The near surface conditions were parametrized with analysis of the temperature profile in the buffer layer a so-called sublayer· Stanton number (WESELY and a simple parametrization of the dimensionless tempera- HICKS, 1977). Despite some inconsistencies of this ap-

-Km =

K

z+]

[z+ -zT+ ·tanh-

(3.3)

Meteorol. Z, 11, 2002

Th. Foken: Viscous sublayer aspects

proach (KRAMM et al., 1996), this is the current state of art for most of the models.

4

,)

Conclusions

The experimental investigations of the molecular sublayer were an interesting example to show how similar the atmosphere near the surface and hydrodynamical boundary layers over a plate are. Both experimental works have shown that even a complicated handling of experimental devices under natural conditions can realise not only qualitative but even quantitative and generalizable results. It was shown that the results are a useful tool for the parametrization of air-sea interaction processes and probably also for a general parametrization of the viscous sublayer. These investigations are restricted to the exchange process in the molecular sublayer and very important problems of air-sea exchange over a wavy surface such as the influence of pressure fluctuations and convective conditions could not be taken into consideration. The results did not really get practical use and the papers, even those in international journals, are not well-known. The reason may be that the development of three-layer models of air-sea exchange was not continued in the 1980s and the thickness of the molecular temperature boundary layer is only one puzzle in parametrizing such models. Recently for hurricane forecast models it was found that the LOUIS (1979) approach underestimates the energy exchange between the atmosphere and the ocean in the central parts of the hurricane. A replacement of the parameterizations by the MANGARELLA et al. (1973) hydrodynamical model, which is similar to the experimental investigations discussed above, gave more realistic results. It should be assumed that with a wider use of meso-scale models the parametrizations of molecular or laminar sublayers should be discussed again.

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