Calculating the Annual Cycle of the Vertical Eddy Viscosity in the North Sea With a Three-dimensional Baroclinic Shelf Sea Circulation Model

Pergamon Continental ShelfResearch, Vol. 16, No. 2, 147-161,1996 pp.

Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All tights reserved

0278-4343/96 $9.50+0.00

0278_4343(94)EOO37-M

Calculating the annual cycle of the vertical eddy viscosity in the North Sea with a three-dimensional haroclinic shelf sea

circulation model

THOMAS POHLMANN*

(Received 30 April 1993; in revised form 20 October 1993; accepted 17 March 1994)

Abstract-The vertical eddy viscosity (A,) is estimated using a three-dimensional baroclinic shelf sea model that treats the temperature as a prognostic quantity. A, is calculated by means of a turbulent closure approach proposed by Kochergin [(1987) Three-dimensional coastal ocean models, American Geophysical Union, pp. 201-2081 which is closely related to a Mellor and Yamada [(1974) Journal of Atmospheric Science, 31, pp. 1791-18061 level-2-model that has been used very successfully in a large number of applications.

The annual cycle of the vertical eddy viscosity is discussed by looking at horizontal and vertical A,,-distributions for the year 1988. These examples show that the vertical eddy viscosity is subject to a pronounced annual cycle which can be related to heating and cooling processes as well as to mixing induced by wind and bottom friction. A comparison of these results with A, -distributions calculated for the year 1987 additionally demonstrates a strong inter-annual variability.

INTRODUCTION

The vertical eddy viscosity (A,) in the North Sea has been investigated for the first time by Kraav (1969). He used a two-dimensional barotropic storm surge model in combination with a one-equation turbulent closure scheme. Due to this configuration only vertically averaged results could be obtained. They show a close relationship between A, and the maximum tidal current velocities. Thus, in the central and northeastern parts of the North Sea A, is smaller than 10 cm2 s-l, while along the British coast and in the Southern Bight values exceed 500 cm2 s-l . Here maximum A.-rates of more than 900 cm2 s-* are reached.

Later the employment of three-dimensional models made it necessary to determine realistic space- and time-dependent vertical eddy viscosities rates. This is of great importance in many respects because A, influences not only the structure of the current profiles but also the calculated tidal elevations significantly (Davies, 1991). Moreover it has been shown that the vertical eddy viscosity and diffusivity play a decisive role in the complex system of physical and biological processes, investigated in ecological models (see e.g. Radach and Mall, 1993).

* Zentrum fir Meeres- und Klimaforschung der Universitlt Hamburg, TroplowitzstraBe 7, D-22529 Hamburg, Germany.

147

148 T. Pohlmann

Until recently simple empirical relations between the vertical eddy viscosity and the how field were applied. These were supported by various measurements (e.g. Munk and Anderson, 1948; Bowden and Hamilton, 1975; Henderson-Sellers 1982). Being used in a wide range of different three-dimensional model applications (barotropic/baroclinic, steady-state/time dependent) these empirical approaches led to a number of very satisfac- tory results. For further details also concerning the model validation see for example: Davies and Furnes, 1980; Davies, 1986; Backhaus and Hainbucher, 1987; Hainbucher et al., 1987 or Backhaus, 1990.

Nevertheless, these empirical approaches cannot be expected to be applicable in general for all the hydrographic conditions and for all types of shelf sea models. Therefore it was desirable to develop approaches which are based on more theoretical assumptions. In particular the turbulent kinetic energy concept has been used very successfully to describe turbulent processes in meteorology (e.g. Mellor and Yamada, 1974) and in oceanography (e.g. Mellor and Durbin, 1975; Blumberg and Mellor, 1987).

For the northwestern European continental shelf this concept was applied for the first time by Davies and Jones (1990). Comparing the results calculated with the eddy viscosity and with the turbulent energy concept they concluded that differences are relatively small compared to other uncertainties. Nevertheless they only carried out purely barotropic simulations and it can be expected that for baroclinic simulations the empirical eddy viscosity approaches are no longer suitable--especially because it is known that under stratified conditions buoyancy can play a decisive role with respect to turbulent processes.

In the present paper a turbulent energy approach proposed by Kochergin (1987) is incorporated into a baroclinic circulation model treating temperature and salinity as prognostic quantities. Using this model configuration to simulate actual periods of the order of a few years is a significant step towards a more realistic description of shelf sea processes. In particular the newly introduced prognostic treatment of temperature allows to resolve processes connected with the existence of the summerly thermocline; one of the most pronounced summerly phenomena in the central and northern North Sea.

It has to be mentioned that the aim of this study, to simulate the hydrographic situation of the North Sea as realistically as possible, is to some degree contradictory to the demand to quantify the causes of mixing. Due to the complexity of the system under consideration here only rough estimates can be presented. On the other hand this investigation provides a much deeper insight into time- and space-scales of the physical phenomena in the North Sea than any case study would allow.

At the beginning a brief description of the model equations is presented, which mainly focuses on the determination of the vertical eddy viscosity. Then some details concerning the specific model experiment are given. This is followed by a presentation of the annual variability of the vertical eddy viscosity which, as an example, is demonstrated for the year 1988. Moreover an impression is given of the variability occurring on the adjacent frequency bands. For this reason the intra-monthly as well as the inter-annual variability will be investigated.

MODEL EQUATIONS AND TREATMENT OF A,

The model under consideration is a modified version of the shelf sea model developed by Backhaus (1985). A detailed description of these modifications is given in Pohlmann (1996). They mainly affect the treatment of the turbulent surface and bottom layers.

Vertical eddy viscosity in the North Sea 149

The present paper gives only a brief description of the theory of the model. It focuses on the determination of the vertical mixing coefficients.

The governing equations are the shallow water equations in combination with the hydrostatic assumption, the equation of continuity, the transport equations for temperature and salinity as well as the equation of state for sea water (Fofonoff and Millard, 1983). The vertical diffusion coefficients for momentum, temperature and salinity are deter- mined by simplifying the physically based k - e-equations brought into discussion by Rodi (1980).

In the equation for the turbulent kinetic energy k a local equilibrium is assumed. Thus advection and vertical diffusion can be neglected. The same assumptions have been made by Mellor and Yamada (1974) to obtain their level-2-model, that according to them gives reasonable results. Therefore the k-equation is simplified to:

where u and v are the velocity components pointing to the East and North, respectively and p denotes the density. AI, and A,,,,” represent the vertical viscosity and diffusivity coefficient, respectively.

E is the dissipation rate of the turbulent kinetic energy, whereas the s-equation is replaced by the generally accepted Kolmogorov-Prandtl expression

E = c,~~‘~/L. (2) where c, denotes an empirical constant and L represents the characteristic length scale of the turbulent motion.

With the help of these two equations the following expression for the vertical eddy viscosity can be deduced (for further details see Appendix):

where SM is the turbulent Schmidt-Prandtl number stating the relation between AI, and A f&L MV> the thickness of the mixed layer and cML an empirical constant that has been calibrated with the help of observational data (Kochergin, 1987).

Originally Kochergin proposed this approach to determine the vertical eddy viscosity and diffusion coefficients only in the surface mixed layer. Because of the strong influence of the bottom friction in shallow shelf seas like the North Sea in this study Kochergin’s approach is applied also to the bottom mixed layer. In general these two turbulent zones are divided by a region of laminar conditions. To determine the position of the interface between the turbulent and the laminar part of the water column the critical Richardson number is employed. In literature all the given values of this number are varying within a remarkably narrow interval between 0.21 and 0.25. In the present work a value of 0.22 derived by Mellor and Yamada (1974) is applied. In the turbulent zone A, is chosen according to equation (3) while in the laminar part of the water column only molecular diffusion (0.0134 cm2 s-l) is assumed. Knowing the position of the transition zone, it is also possible to calculate the thickness of both mixed layers hML, that appears in equation (3).

In order to allow for the simulation of unstratified conditions, which occur in summer in the southern North Sea and in winter in all over the North Sea, an overlapping of the surface and the bottom mixed layer is permitted.

150 T. Pohlmann

Due to the present lack of computing capacities, it is still necessary to parameterize the vertical gradients appearing directly at the thermocline. These are calculated by introduc- ing an artificially decreased vertical grid distance hrh. The value of hTh has been calibrated by comparing calculated and measured temperature profiles. A more detailed description of this procedure is given in Pohlmann (1996). Furthermore, the possible vertical spacing does not allow to simulate small scale boundary effects realistically, which occur in the near bed bottom boundary layer. For that reason the logarithmic boundary profile cannot be resolved adequately.

MODEL APPLICATION TO THE NORTH SEA

The model explained in the previous section has been applied for the calculation of the hydrodynamic parameters u, v, w, S, T, p and A, for a period from 1 January 1982 to 31 December 1992. The driving forces of these simulations comprise 3-hourly wind stress and atmospheric pressure distributions, daily sea surface temperatures; both derived from analysed ship observations (Luthardt, 1987). At open boundaries salinity and temperature data as well as the sea surface elevations are prescribed. The former two are obtained from a climatological monthly mean data set (Damm, 1989). In order to supply the three- dimensional prognostic baroclinic model (“North Sea Model”) with adequate sea surface elevations at the open boundary, another three-dimensional diagnostic baroclinic model covering the entire northwestern European continental shelf was employed (“Shelf Sea Model”). The latter provides all the major far field effects like the M,-tide and external surges. The horizontal resolution of both models is 12’ in meridional and 20’ in longitudinal direction. In the Shelf Sea Model the vertical scale is resolved by 12 layers, whereas in the North Sea Model a maximum number of 19 layers are existing. For the sake of an adequate resolution in that part of the water column where the thermocline occurs, the thickness of the upper 10 layers is only 5 m. The thickness of the lower layers gradually increases from 10 to 400 m with increasing depth. The time step is 10 min for the Shelf Sea Model and 20 mins for the North Sea Model, which turned out to be sufficient for a reasonable resolution of processes induced by the M,-tide.

At the closed lateral boundaries a no-flux and semi-slip condition is applied for momentum, while for the sea bottom the quadratic stress law is chosen. At the open boundaries the transport gradients normal to the open boundaries are set to zero, whereas for temperature a radiation condition is applied. Furthermore, at the sea surface observed temperature date (SST) are prescribed. This employment of a Dirichlet’s boundary condition is justified, because for the North Sea SST-data are available in a reasonable time-space resolution.

The output of the model consists of data averaged over two tidal cycles (roughly 1 day). In order to obtain A,-distributions, which are representative for a specific month, monthly means are calculated from these daily A,-rates. In all figures showing A,-distributions the 1 cm2 s-* contour line is displayed in order to indicate all areas exhibiting only molecular viscosity during the whole month.

ANNUAL CYCLE OF THE VERTICAL EDDY VISCOSITY

In the following, the annual cycle of the vertical eddy viscosity is described. As an example the evolution in the year 1988 is presented. Figures 2(a)-5 give the horizontal distribution of the monthly mean vertical eddy viscosity for the 15 m level, while


- i3

il

io

9

._

-

6

._

5

v

3 _

2

-

I _

_

0

9

B

W -15 -I” -13 -17 .I, -10 -9 -* -7 -6 -5 -” -3 -2 -I 0 I 2 3 ” 5 6 7 8 9 IO II

Fig. 1. Domain and topography (m) of the North Sea and Shelf Sea Model, mentioned locations and position of vertical section.

Figs 6-9(a) show the vertical distribution on a section from the Fair Isle Passage to the German Bight (cf Fig. 1).

In February 1988 [Fig. 2(a)] the maximum values of more than 275 cm2 s-l are reached in the off-coastal parts of the North Sea. This is mainly caused by the intensive cooling taking place in this month. Only the Dogger Bank region forms an exception. Here a minimum of less than 200 cm’ SC’ . 1s attained. Thus in late winter the horizontal A, distribution more or less reflects the topography. This results from the fact that the mixed layer depth hML used in equation (3) is chosen to be the total water depth if vertically homogeneous conditions are present. Therefore in winter the topography has a decisive influence on the vertical eddy viscosity. The only exceptions in this respect are found in the Southern Bight and in the Norwegian Trench region. Here the tidally induced mixing and

152 T. Pohlmann

the haline stratification, respectively, are predominating and thus are governing the vertical eddy viscosity. In April [Fig. 31 the warming has already started which in connection with decreasing storm activities explains the strong decrease of Au-rates. Only in the Southern Bight maximum values of 100 cm2 s-i are found. In August [Fig. 4(a)] this decrease has continued. The entire northern and central North Sea shows A,-rates of less than 25 cm2 s-l. However, on the other hand in nearly all North Sea regions values are considerably larger than the molecular viscosity (0.0134 cm2 s-l) indicating that a remainder of turbulence below the thermocline is still existing even in mid summer. In October [Fig. 51 the vertical eddy viscosity again increases as a result of the cooling through the sea surface combined with the intensification of wind stress. Thus maximum rates of more than 200 cm2 s-l are reached in the central North Sea.

The vertical distribution in February 1988 [Fig. 61 shows a maximum in the shallower parts where almost vertically homogeneous conditions are present. In the northwestern North Sea A, decreases nearly linearly from the surface (300 cm2 s-l) to the bottom (150 cm2 s-l). Only in the regions southeast of the Dogger Bank in the last few meters above the bottom is an increase of A, recognizable. The distribution in April [Fig. 71 shows a strong change in the A,-distribution indicating the transition between winterly and summerly conditions. In contrast to the previous months the constant decrease from surface to bottom no longer persists. Instead, an interior minimum with values lower than 25 cm2 s-i appears in the Fladen Ground area in the depth interval between 20 and 100 m. Between 50 and 60 m values are even smaller than 1 cm2 s-l ’ indicating that here laminar conditions are present during the whole month. Maximum A,-rates of more than 75 cm2 s-i occur at the surface and of more than 100 cm2 s-l at the bottom. In August [Fig. 8(a)] this general summerly structure in principle remains unchanged. The maximum has decreased at the sea surface in particular, whereas the smallest surface values of less than 50 cm2 s-l are found in the central and northern North Sea. Values of less than 1 cm2 s-i, which can be attributed to predominating non-turbulent conditions now cover a significant part of the water column. They are found north of the Dogger Bank in the central and northwestern North Sea in a range between 30 and 70 m. The upper level of this minimum agrees with the maximum of the vertical temperature gradient [Fig. 8(b)]. This correlation between vertical eddy viscosity and temperature even is observable in the shallower regions southeast of the Dogger Bank. In October [Fig. 9(a)] surface values have increased drastically up to more than 275 cm2 SC’. Likewise the extension of the interior minimum decreases considerably connected with a shift of the minimum towards the bottom. A -rates of less than 1 cm2 SC’ are now only reached in the deep Fladen Ground area in a dlpth range between 50 and 60 m. As in August also in October the upper level of the laminar zone coincides with the depth of the maximum vertical temperature gradient [Fig. 9(b)]. In contrast to the situation in August in all the shallower regions north of the Dogger Bank this interior minimum has disappeared. The latter can be explained by the absence of vertical thermal stratification on the Dogger Bank and southeast of it, indicating that the presence of a thermocline is a necessary condition for an interior A.-minimum.

INTER-ANNUAL AND INTRA-MONTHLY VARIABILITY

To get an impression of the occurring inter-annual variability horizontal distributions for winter and summer for the year 1987 [Fig. 2(b) and Fig. 4(b)] are compared with those for the year 1988 [Fig. 2(a) and Fig. 4(a)].


Fig. 2(a). Horizontal distribution of the monthly mean vertical eddy viscosity (cm’ s-l) in 15 m depth in February 1988. (b). Horizontal distribution of the standard deviation of the monthly mean vertical eddy viscosity (cm2 s-‘) in 15 m depth in February 1988. (c). Horizontal distribution of the

monthly mean of the vertical eddy viscosity (cm2 s-‘) in 15 m depth in February 1987.

Fig. 3. Horizontal distribution of the monthly mean of the vertical eddy viscosity (cni! s-’ ) in 15 m depth in April 1988.

154 T. Pohlmann

Fig. 4(a). Horizontal distribution of the monthly mean of the vertical eddy viscosity (cm’ ss’ ) in 15 m depth in August 1988. (b). Horizontal distribution of the standard deviation of the monthly mean vertical eddy viscosity (cm2 SC’) in 15 m depth in August 1988. (c). Horizontal distribution of

the monthly mean of the vertical eddy viscosity (cm2 s-r) in 15 m depth in August 1987.

Fig. 5. Horizontal distribution of the monthly mean of the vertical eddy viscosity (cn? s-’ ) in 15 m depth in October 1988.


FAIR ISLE 70 GERMAN BIGHT NH - 9E

NORTH-SEA FEBRUARY 1988 MONTHLY MEAN

SECTleN FRBH t 59’29.N. 2’01 ‘W 16 53 ‘vi ‘N, 7’YO.E DISTRNCE IN KH

0 100 200 300 400 500 600 700 800

Fig. 6. Vertical distribution of the monthly mean vertical eddy viscosity (cd s-‘) in February 1988 (location of section: cf Fig. 1).

In February the vertical eddy viscosity shows significant differences for both months [Fig. 2(b) and Fig. 2(a)]. Only in coastal regions the general structure remains more or less the same while in all the other North Sea regions marked differences occur. In most central parts of the North Sea in February 1987 A,-values only reach about 50% of those in 1988. In contrast to February 1988 no major cooling events have taken place in February 1987. These results stress the fast and direct reaction of the vertical eddy viscosity on the winterly cooling. In August this situation has changed drastically. Although the external forcing is significantly different for both months the general distribution of the vertical eddy viscosity is almost the same [Fig. 4(b) and Fig. 4(a)]. Only next to the frontal zone, separating stratified from vertically mixed conditions, differences in A,-rates are noticeable. Altogether this implies that in summer A, is mainly governed by topographic and tidally induced effects. Interannual changes in the depth and extension of the thermocline only result in locally limited variations of the vertical eddy viscosity. Contrarily in winter variations of the meteorological forcing dominate the inter-annual variability.

In order to demonstrate the intra-monthly variability for two selected months {i.e. February 1988 [Fig. 2(c)] and August 1988 [Fig. 4(c)]} besides the monthly mean distribution already presented in the previous section additionally the corresponding standard deviations are displayed. It has to be noted that due to the use of daily integrated model output data the intra-monthly variability discussed in the following covers the frequency spectrum from one day to one month.

Figure 2(c) indicates that in winter a considerable intra-monthly variability is present,

156 T. Pohlmann

FAIR ISLE 18 GERNRN BIGHT NH - SE

NMTH-SEA APRIL 1988 N6NTHLY RERN

SECTleN FRBR t 59’29.N. 2’01 ‘W 16 53 11 ‘N. 7’YO’E OISTRNCE tN KM

0 100 200 300 400 500 600 100 600

,C”r C” ) /ST)

.“Y

Fig. 7. Vertical distribution of the monthly mean vertical eddy viscosity (cm2 s-‘) in April 1988 (location of section: cf. Fig. 1).

reaching values higher than 100 crn’~-~ in most North Sea regions. Obviously, horizontal gradients of the standard deviation are smaller than those observed for the monthly mean d$tribution [Fig. 2(a)]. I n most coastal areas standard deviations are smaller than 100 cm2 s . This holds also for the Dogger Bank region, which exhibits a marked minimum standard deviation of less than 75 cm2 s-l. In most other North Sea regions standard deviations exceed 150 cm2 s-’ amounting to about 50% of the monthly mean. Most strikingly in August [Fig. 4(c)] a maximum of the standard deviation is located on the Dogger Bank. This is in contrast to the situation in February, when here a minimum standard deviation is attained. In most other North Sea regions standard deviations vary within the range between 1 and 50 cm2 s-l, showing roughly the same distribution as the monthly mean patterns [Fig. 4(a)]. Only the Southern Bight and the eastern part of the English Channel form an exception. Here, although mean values reach their maximum no significant increase of the A,-standard deviation is observable. In this region this phenomenon is also noticeable for February indicating that in this region high A,-rates are primarily induced by tidal stirring, which is not a subject to pronounced day-to-day changes and therefore does not cause a strong intra-monthly variability.

CONCLUSION AND OUTLOOK

The study of the vertical eddy viscosity with the help of a three-dimensional baroclinic shelf sea model demonstrates that A, is subject to a strong annual cycle. This annual cycle


0

10

20 30 40

50 60

15

FAIR ISLE TO GERRRN BIGHT NY - SE

NORlH-SER AUGUST 1988 RONTHLY MEAN

SECTION FROM t 59’29’N. 2’01% TO 53Yl‘N. 7’UO’E DISTANCE IN KM

0 100 200 300 voo 500 600 700 BOO I I I , I

50.0 -L ~__25.0

y-

FAIR ISLE TO GERMAN BIGHT NH - SE

NORTH-SER AUGUST 1998 MONTHLY MEAN

SECTION FRBN L 59’29’N. 2’01 ‘II TO 53 ‘I1 ‘N, 7’YO.E DISTANCE IN MI

0 100 200 300 1100 500 600 700 600 I I I I I I 0

10

20 30 UO 50 60

Fig. 8(a). Vertical distribution of the monthly mean vertical eddy viscosity (cn? s-l) in August 1988 (location of section: cf. Fig. 1). (b). Vertical distribution of the monthly mean temperature

(“C) in August 1988 (location of section: cf. Fig. 1).

158 T. Pohlmann

FAIR ISLE TB GERRRN BIGHT NH - SE

NBRTH-SEA BCTBBER 1988 HONTHLY HERN

SECTION FRBR t 59’29’N. 2’Ol‘U 78 53’41’N. 7’UO’E OISTRNCE IN KH

0 100 200 300 400 500 600 700 600 I I I I I

0

10

20

30

110

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60

FRIR 1SLE TB GERMAN BIGHT NW - SE

NBRTH-SEA BCTBBER 1998 NBNTHLY RERN

SECTlBN FRBH i 59’29.N. 2’Ol’W TB 53”ll’N, 7’UO’E DISTANCE IN KH

0 100 200 300 400 500 600 700 600 I I I I 4

/ I

Fig. 9(a). Vertical distribution of the monthly mean vertical eddy viscosity (cd s-‘) in October 1988 (location of section: cf. Fig. 1). (b). Vertical distribution of the monthly mean temperature

(“C) in October 1988 (location of section: cf. Fig. 1).


can clearly be related to the annual development of the atmospheric forcing conditions, i.e. the thermal forcing and the wind stress at the sea surface. Usually the deeper regions of the North Sea are characterized by an interior minimum of the vertical eddy viscosity. Towards the surface and the bottom A,-rates increase due to the intensification of vertical mixing caused by the wind stress and the bottom friction, respectively. Only in late winter, when the thermocline has totally disappeared and convection reaches down to the bottom does this general structure change and the vertical eddy viscosity also decreases in deepest North Sea regions more or less linearly from surface to bottom. In all the shallower parts of the North Sea this interior minimum is much less pronounced and only persists from early to mid summer.

Moreover, for the winter months a considerable inter-annual variability of A, is found, which can mainly be attributed to the thermal forcing. On the other hand in summer the inter-annual variability is significantly smaller. Hence in the summer season the vertical eddy viscosity seems to be more dependent on topographic and tidally induced effects. Inter-annual variations of the thermocline depth and extension are only responsible for a relatively small inter-annual variability of A,, which is primarily limited to the area of the frontal zone. A more detailed discussion concerning the development of the thermocline is presented in Pohlmann (1991).

The other frequency band under consideration in this study comprises all the A,,-fluctuations from one day to one month, i.e. the intra-monthly variability. Interest- ingly the latter, here presented with the help of the standard deviation of the monthly means, shows a general correlation with the monthly mean distributions themselves. Not only the horizontal structures but also the order of magnitude are comparable for the winter as for the summer season. This indicates that the high-frequency variability of the meteorological forcing has a strong impact on the actual A,-distribution and for this reason cannot be neglected, if a realistic representation of conditions in the North Sea is required.

As shown above, the vertical eddy viscosity depends on vertical velocity shear and thermal stratification. On the other hand, the circulation pattern and the temperature distribution are influenced by vertical mixing. The non-linear interaction of these three quantities produces a complex hydrodynamic system. For that reason it is difficult to clearly quantify the impact of each single component on the A,,-distribution actually present in a specific region and month. But as already pointed out in the introduction, this disadvantage is compensated by the possibility to obtain for the first time in a comprehen- sive form realistic space- and time-dependent distributions of the vertical eddy viscosity in the North Sea.

For the future it is planned to carry out a more detailed statistical analysis, which is feasible just lately, because the simulation period has recently been extended from 5 years (1987-1991) to the H-years-period mentioned above. This will allow a more general view of the climatological means and corresponding inter- and intra-annual variabilities of the vertical eddy viscosity. Furthermore it will be possible to deduce the coherences between A, and the respective meteorological forcing fields systematically. Another interesting aspect that has to be considered in this context is the timescale of the “hydrographic memory” of the North Sea as it was brought into discussion by Elliott and Clarke (1991).

Acknowledgements-The author is indebted to J. 0. Backhaus for helpful discussions and comments. The author would also like to thank his colleagues P. Damm, I. Harms, H. Langenberg, C. Schrum and N. Verch. This

160 T. Pohlmann

research was funded by the German Federal Ministry for Research and Technology [BMFT-MFU 0620/6, (PRISMA)] and by the Commission of the European Communities [MAST-0050-C, (PROFILE)].

REFERENCES

Backhaus J. 0. (1985) A three-dimensional model for the simulation of shelf sea dynamics. Deursche Hydrographische Zeitschriji, 38,165187.

Backhaus J. 0. and D. Hainbucher (1987) A finite difference general circulation model for shelf seas and its application to low frequency variability on the North European Shelf. In: Three-dimensional models of marine and estuarine dynamics, J. C. J. Nihoul and B. M. Jamart, editors, (Elsevier Oceanography Series 45) Elsevier Amsterdam, pp. 221-244.

Backhaus J. 0. (1990) On the atmospherically induced variability of the circulation of the Northwest European Shelf Sea and related phenomena-a model experiment. In: Modeling marine systems Vol.1, A. M. Davies, editor, CRC Press, Boca Raton, Florida, pp. 93-134.

Blumberg A. F. and G. L. Mellor (1987) A description of a three-dimensional coastal Ocean Circulation model. In: Three-dimensional coastal ocean models, N. S. Heaps, editor, American Geophysical Union, Washington D. C., Coastal and Estuarine Science 4, 1-16.

Bowden K. F. and P. Hamilton (1975) Some experiments with a numerical model of circulation and mixing in a tidal estuary. Estuarine and Coastal Marine Science, 3,281-301.

Damm P. (1989) Klimatologischer Atlas des Saltzgehaltes, der Tempera&r und der Dichte in der Nordsee, 1968 1985. Institut fur Meereskunde der Universitat Hamburg, Technical Report, 689, pp. l-81.

Davies A. M. and G. K. Furnes (1980) Observed and computed Mz tidal currents in the North Sea. Journal of Physical Oceanography, 14,237-257.

Davies A. M. (1986) A three-dimensional model of the north west European Continental Shelf, with application to the Mz tide. Journal of Physical Oceanography, 16,797-813.

Davies A. M. and J. E. Jones (1990) Application of a three-dimensional turbulence energy model to the determination of tidal currents on the Northwest European Shelf. Journal of Geophysical Research, 95, 18,14%18,162.

Davies A. M. (1991) On using turbulence energy models to develop spectral viscosity models. Continental Shelf Research, 11,1313-1353.

Elliott A. J. and T. Clarke (1991) Seasonal stratification in the northwest European shelf seas. Continental Shelf Research, 11,467-492.

Fofonoff N. P. and R. C. Millard Jr (1983) Algorithms for computation of fundamentalproperties of sea water. UNESCO Technical Papers in Marine Science, 44.

Hainbucher D., T. Pohlmann and J. 0. Backhaus (1987) Transport of passive tracers in the North Sea: First results of a circulation and transport model. Continental Shelf Research, 2, 1161-1171.

Henderson-Sellers B. (1982) A simple formula for vertical eddy diffusion coefficients under conditions of non- neutral stability. Journal of Geophysical Research, 87,5860-5864.

Kochergin V. P. (1987) Three-dimensional prognostic models. In: Three-dimensional coastal ocean models, N. S. Heaps, editor, American Geophysical Union, Washington D.C., Coastal and Estuarine Science, 4, 201- 208.

Kraav V. K. (1969) Computation of the semidiurnal tide and turbulence parameters in the North Sea. Oceanology, 9,332-341.

Luthardt H. (1987) Analyse der wassernahen Druck- und Windfelder iiber der Nordsee aus Routinebeobachtun- gen. Hamburger Geophysikalische Einzelschriften A, 83,1-115.

Mellor G. L. and T. Yamada (1974) A hierachy of turbulence closure models for planetary boundary layers. Journal of Atmospheric Science, 31, 1791-1806.

Mellor G. L. and P. A. Durbin (1975) The structure and dynamics of the ocean surface mixed layer. Journal of Physical Oceanography, 5,718-728.

Munk W. H. and E. R. Anderson (1948) Note on the theory of the thermocline. Journal of Marine Research, 7, 276295.

Pohlmann T. (1991) Untersuchung hydro- und thermodynamischer Prozesse in der Nordsee mit einem dreidimensionalen numerischen Modell. Berichte des Zentrums fiir Meeres- und Klimaforschung, 23, l- 116.

Pohlmann T. (1996) Predicting the thermocline in a circulation model of the North Sea-Part I: model description, calibration and verification. Continental ShelfResearch, 16, 131-146.


Radach G. and A. Moll (1993) Estimation of the variability of production by simulating annual cycles of phytoplankton in the central North Sea. Progress in Oceanography, 31, 339-419.

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APPENDIX

Derivation of equation (3): Using the generally accepted empirical relation

A, = c$%, (4)

together with equation (2) it follows that:

AI, = LciI%, with: CL = c,,lc,, (5)

where cfl denotes a further empirical constant. To substitute the third term on the left-hand side of equation (1) by the expression (5) it is necessary to have a

function F with: A,, = F(A,,). Here a linear relation between A,, and A,,is used:

1 A,, = --.A,,.

SM (6)

Because the possible range of the Schmidt-Prandtl number 5, is relatively small, in this investigation a simple algebraic dependency on the Richardson number Ri is assumed, which originally was proposed by Mellor and Durbin (1975).

Inserting equations (2) and (5) together with (6) and (1) results in

Solving equation (7) with respect to k followed by the employment of (5) yields

(7)

It is assumed that the characteristic length scale L is proportional to the thickness of the mixed layer hMur

L = CML’hlUL, (9)

where cML’ is the proportionality factor With the definition

cML = cML 4cmF .u E)

thus finally equation (8) can be written as equation (3).

(10)

Calculating the Annual Cycle of the Vertical Eddy Viscosity in the North Sea With a Three-dimensional Baroclinic Shelf Sea Circulation Model

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