Estuarine, Coastal and Shelf Science (1990) 30.111-129

Thermally Driven Circulation Within an Experimental Enclosure

Lars Andersson’ and Lars Rahmb “SMHI Oceanographic Laboratory, Box 2212, S-403 14 Gothenburg and

bSMHI, S-601 76 Norrktiping, Sweden

Received 3 May 1988 and in revised form 26 September I989

Keywords: enclosures; advection-diffusion models; boundary layers; circu- lation processes; temperature profiles; velocity-depth profiles; Baltic Sea

A diagnostic advection-diffusion model for the temperature field in an exper- imental enclosure is presented. The enclosed fluid domain is divided into a buoyancy layer at the nonhorizontal walls, and an horizontally homogeneous, stably stratified interior. Given thermal forcing through the top surface and the side walls, together with the interior temperature profile, the interior vertical velocityfield,as wellastheeddydiffusivityatthe boundarylayer,areestimatedfor anadvection-dominated case. This model has been applied to a pilot experiment, a limnocorral study in the coastal region of the Baltic Sea. The thermally driven interior vertical circulation, forced by the buoyancy layer, is found in the range O-10-* mss’, while the characteristic eddy diffusivity at the buoyancy layer typi- cally varies between 10-‘-10m5 m’ ss’. Hence, the distribution of biologically interesting parameters, as well as their sources and sinks, may be estimated using a similar advection-diffusion model for the substance in question, but where the velocity field and eddy diffusivity is calculated, by use of the present model.

Introduction

It has been found, for chemical and biological reasons, convenient to perform experiments in enclosures semisubmersed into lakes or oceans. The aim has been to simulate ‘ real ’

conditions within a controlled volume of sea water (for a review see Grice & Reeve, 1982), though already Verduin (1969) has raised some objections on the realization of these intentions.

In order to be able to estimate fluxes of biologically active substances such as oxygen, carbon dioxide or nutrients, the dynamics within the bag and especially its circulation must be determined. Some of the methods available have been discussed in a paper by Hesslein and Quay (1973). They found a reasonable agreement between some of the different methods used, and attributed the dissimilarities observed to different sampling processes. One of the methods discussed was the classic one of McEwan (1929), which is based on the hypothesis that the heat transfer through at least one part of the vertical water column (the ‘ clinolimnion ‘), is by vertical diffusion only. Some shortcomings of this method were discussed by Powell and Jassby (1974), who instead presented an alternative procedure to calculate the eddy diffusivity in a contained stratified fluid (Jassby &

0272-7714/90/020111+ 19 $03.00/O @ 1990 Academic Press Limited

112 L. Andersson & L. Rahm

Powell, 1975). Their model is, however, still based on a one-dimensional heat diffusion process in the vertical, which is used as a diagnostic tool upon measured temperature profiles. Nevertheless, as already pointed out by Boyce (1974), the assumption of horizon- tal homogeneity might be erroneous. He assumed that horizontal exchange processes through the sidewalls were the prime mechanism for keeping the interior temperature profile in reasonable agreement with the outer one, and that this was not a reflection upon the equivalence of dynamic processes within and outside of the enclosure, which some- times has been proposed.

The potential importance of the sidewall heating can readily be estimated by consider- ing Newton’s law of heat transfer. The heat flux q becomes,

q = RAT/d

where 6 represents the thermal conductivity of the wall, d its thickness, and AT the temperature difference across the same wall. In the present field experiment a typical temperature difference of order 2-3 K was observed, which yields q e 6 x 10’ W m-l. The incoming solar radiation, on the other hand, often reached 5 x lo2 W mP2. Hence, these two fluxes may be of the same order of magnitude, but as the effective area of the sidewall sometimes exceeded that of the top surface by as much as a factor of five, the sidewall heating may sometimes dominate the heating process. Hence, it seems that the effect of buoyancy layers at the sidewalls is an overlooked aspect of the limnocorral dynamics.

In this work, it is furthermore hypothesized that the dynamics of an artificial enclosure like a limnocorral may, under certain conditions, be well described by a theory for stably stratified fluids originally presented by Walin (1971), and thereafter scrutinized in some laboratory experiments by, for example, Davis and Walin (1977) and Rahm and Walin (1979). The horizontal inhomogeneities discussed by, for example, Boyce (1974) also play a crucial role in the present model. The almost inevitable temperature differences between the interior of the vessel and the surrounding water, due to different forcing, will cause a buoyancy layer flow at the nonhorizontal walls. This flow governs the stably stratified and horizontally homogeneous interior by determining the interior vertical flow. The interior dynamics is hence constituted by a vertical advection diffusion balance. The calculations carried out in this paper shows that this system sometimes may degenerate into an advection-dominated one. This degeneracy indicates a possible further idealization and, consequently, a drastic simplification of the leading order dynamics.

According to the previous paragraph, the vessel constitutes an aquatic domain of its own with its own dynamics. Though in some ways similar to its environment, the bag defines its own characteristic scales of length and time. Thus, the sometimes advection-dominated interior may represent a marine system quite different from the one sought by the experi- mentalists. In fact, the resulting circulation due to thermal forcing, may be the dominant flux mechanism of e.g. chemical substances. This is an aspect that, to the best knowledge of the authors, has not been considered in previous limnocorral experiments.

The present model is outlined in the following section. It is then applied to a pilot experiment. The observed meteorological forcing, and the field measurements done during this experiment, are presented in the third section. The results achieved and discussion of the problems, inherent in this type of model, encountered are given in the final sections.

Diagnostic model

The diagnostic model is based on the behaviour of an idealized limnocorral. Due to the superstructure of the bag (gangways, frame of bag, wind mast, spray shields etc.), the

Thermally driven circulations 113

effect of wind stress is neglected as is the possible occurrence of Langmuir circulation. Furthermore, any deformation of the vessel due to wind, wave action or drag is neglected. In the pilot case discussed in this study, the first assumption seems valid due to the relatively high spray shield around the vessel, whereas the validity of the other assump- tions are more uncertain. It is further assumed that the contained fluid is only thermally stratified since the vessel was initially filled with water from the well-mixed brackish surface layer and then artificially mixed in order to ensure homohaline conditions.

The theoretical background to the present work is found in Walin (1971), and hence, only the primary features are outlined below. The basic assumption of this theory is that the system is in a stratification dominated regime, where the density field is essentially horizontally homogeneous. If the eddy diffusivity is sufficiently weak to allow for a bound- ary layer approach, the interior dynamics are governed by a single one-dimensional heat equation. If further, the thermal conductivity of the wall is limited, the resulting buoyancy layer equations may be linearized.

Using a singular perturbation method, the variables of the equation system governing the dynamics of the bag is partitioned into an interior part, VI, and a boundary layer part, @. (The indices Iand B indicate the interior and boundary layer variables.) The buoyancy layer variables are such that c$+ 9’ represent the complete solution. The interior variables are defined in the whole domain while the boundary layer counterparts only stay finite in the buoyancy layer and, by definition, vanish in the interior. The assumption of a stratifi- cation dominated regime yields, to lowest order, an interior heat equation, which represents an advection-diffusion heat balance in the vertical with a source, (cf. Walin, 1971),

d T’ ,aT’ a --$+wz=-

dz (2)

where J, K’ and T are the source term, the vertical eddy diffusivity and the temperature, respectively, while W’denotes the vertical interior velocity. The bag was thermally forced both by heat exchange processes at its boundaries and short-wave radiation penetrating through the surface. The vessel was submersed into the well-mixed surface layer but sometimes the colder deep water reached the level of the enclosure when the thermocline rose due to upwelling phenomena. As a result of the assumed high turbulence intensity in the ambient fluid, the temperature distribution in a neighbourhood of the vessel is assumed representative for the temperature F(zJ), at the outer surface of the wall. A temperature difference across the sidewall results in a heat flux, governed by the Newtonian heat flux condition,

8T -=s^(T- T) K

c=o

where < is directed along the inward normal to the boundary and s ̂is defined locally as

R j=-

kd (3b)

where k is the thermal conductivity of the fluid. This heat flux causes lateral density variations close to the boundary at the inside of the bag and consequently a buoyancy layer is formed (see Figure 1). The interior temperature field is adjusted to the sidewall bound- ary conditions by divergences in the buoyancy layer flow and by vertical advection in the

114 L. Andersson t3 L. Rahm

\

Y Y

Figure 1. Schematic illustration of the physical model. T’ and P denotes the interior Figure 1. Schematic illustration of the physical model. T’ and P denotes the interior and ambient temperatures while J represents the source term. The buoyancy layer flow and ambient temperatures while J represents the source term. The buoyancy layer flow ma, and the interior vertical velocity W’ are indicated. l and ( are the buoyancy-layer ma, and the interior vertical velocity W’ are indicated. l and ( are the buoyancy-layer coordinates parallel and normal to the boundary. coordinates parallel and normal to the boundary.

r: .-. :nj’ /’ ‘of the 1 .’ .A. Location MASKS

Figure 2. Investigation area at Asko on the Swedish east coast, where the location of the bag is shown.

Thermally driven circulations 115

ipray shield

4.5 m

v Figure 3. Geometry and dimensions of the bag.

TABLE 1. List of parameters used in the calculations

Parameter Numerical value

0.2Wrn~lK I 2*10 Jm 0.6Wm ‘K I 1.2* lo-‘m* s-’ 2.25 m 2260kJkg ’ 1.23kgm 3 1.4* 10 ’ 1.01 kJ kg I K-’ 1.4* 10 3 0.96 5.67* 10-RWm-2K ’ 0.61 0.05 0.6 1.0 0.4m-’

,

!-5 m

interior. The heat conductance of the wall is limited, which makes the temperature vari- ations small in the buoyancy layer and thus makes possible a linearization of the boundary layer equations. The lowest-order quasi-steady buoyancy layer equations become (cf. Walin, 1971),

a2 uB kCgaTB = - vpO-

ai I a2TB

&$ = KB- ai2

(44

(4b)

116 L. Anderson &3 L. Rahm

60

-20

22

16/7 17/7 18/7

Time date

Figure 4. Energy fluxes through the sea surface during the investigation period. (a) Incoming solar radiation. (b) Long wave radiation (-), latent heat (- - -) and sensible heat (, .). Local time is used.

where a, v, g and pO represents the coefficient of thermal expansion, the diffusivity of momentum, the gravity and a reference density respectively, while k; is the component of the vertical unit vector parallel to the boundary. WB and UB are the velocities normal and parallel to the local boundary within the coordinate system (t&J, see Figure 1. It is found convenient to separate the eddy diffusivity K into two parts, a boundary layer part h-a, and an interior vertically dependent part 15 I. The momentum equation represents a balance between buoyancy forces and viscous dissipation while the heat equation expresses a corresponding balance between advective heat flow along the boundary and a cross-flow heat-diffusion. The lowest order volume transport is obtained by integrating UB in equation (4a) across the buoyancy layer, utilizing boundary condition equation (3a), where, as a consequence of the linearization mentioned above, T is approximated by T’.

The boundary layer transport per unit width, mB, turns out as (cf. Walin, 1971),

Thermally driven circulations 117

20

I L

, I’.. i !i i !i ! ! ' '.., , ! ! .,.. $ \

i 1 I , \ \

\ \ \ .’

1’

G I/

A

:, 1 ‘. i!

. . ,.. I

.t I ‘I I ! \ ! i

I I

I \ ; i \ 8

, \ ’ ’ \ ‘.. -’

\ I -’ -.\ ,

Darkened I I

I I I I I, I , I , / , , , ,

00 12 00 12 00 12 00 12 00 E/7 16/7 17/7 1817

Mixing Mixing Mixing Mixing Time date

Figure 5. Temperature evolution at four different depths (-, 1 m; - - -, 3 m; ,5 m; -. - ., 7 m) as function of time for the whole investigation period. The bag was covered during the period denoted ‘ darkened ‘.

!- I

;(T’- nir

p# = - KB ai

ar’

at

(5)

r=o

Since the vessel is closed and the interior is assumed stably stratified, volume continuity yields,

AW’++mBdl=O (6)

where $ dl represents the curvelinear integral around the vessel in a horizontal plane of cross-sectional area A. The interior vertical velocity may then be expressed within a Cartesian coordinate system and in interior variables only,

w’ = f

A aT’ -case

\ a2

\

dl - ;

I

(7)

where 0 is the angle between the c-axis and the horizontal plane. Elimination of WI in equation (2) makes it linear and it turns out as,

ar’ K* f(T’-f)

at +A4

2aA aTI a

cos e dZ----z=dz (8)

118 L. Andersson t3 L. Rahnz

2-

2 4- L

5

o 6-

a-

V

Temperature (“C)

IO 12 14 16 18 20 I, I, I, I ,\I ,>

(a) 00.00

I -0

/

Temperature (“C)

14 16 18 20

Temperature i”Ci

IO 12 14 I6 I8 20 I , , / I , I I , z-

- (e) 08.00

2-

g 4- I:

5 o 6-

8-

V

Figure 6 (a)-(f).

Temperature (“C)

IO I2 I4 16 18 20 I , 1 , / / I II,

- (0102.00

2-

6- ,-- 0

8-

V

Temperature (“Cl

IO 12 I4 16 18 20 I , I , I , I I , >

- (d.1 06.00

2-

Temperature (“Cl

‘0

: 4-

f

o 6-

8-

Temperature (“C)

IO I2 14 16 18 20 I , I / I ,

- (f ) IO.15 2 _ Mixing 10.00

2 4- c

5 o 6-

8-

V

This heat equation may be used diagnostically to determine the vertical interior velocity distribution, and hence the lowest-order vertical circulation within the bag. This requires, however, that ~‘and K~ are both specified. Since the main turbulence production probably occurs in the buoyancy-layer region, with its relatively strong shear and high stress, and

Thermally driven circulations 119

Temperoture (“C) Temperoture (“Cl

10 12 14 16 18 20 1 , I , 1 , I r , '

- (h) 14.00

2-

2 4- .c z - aa

D 6-

8-

Y

Temperoture (“Cl

IO 12 14 16 18 20 I’ I ’ I’>’ ”

i ) 16.00

Figure 6 (a j(i). Temperature profiles within and outside the bag during nine consecu- tive measurements on 16 July (i, inside; o, outside).

not in the horizontally homogeneous stratified interior (the sometimes unstable region at the top neglected), it seems reasonable to assume that the vertical eddy diffusivity K'

cannot exceed K~. Hence, it is first postulated that K'< < K~, which yields an advection- dominated regime. The interior will be nondissipative to lowest order while the buoyancy layer remains unaltered. The interior heat equation (8) then degenerates to,

dTr xB f(T’ - 0 t+A$ cos8 dl-gg=J (9)

Consequently there are two ways to determine W’ from the observed temperature fields, either by using the degenerated version of equation (2) which becomes,

or by using equation (7). Furthermore, k.B is determined by combining equations (7) and

(lo),

120 L. Anderson & L. Ruhm

KB(m’ I’l KB(m’ I’l

10-7 10-7 10-C 10-C 10-S 10-S I

8-

(0 i 0.00-02.00 (0 i 0.00-02.00 v

fcB (m2 s-‘)

1O-7 10-G IO- I >

8 -’

(c I 04.00-06.00 V

Figure 7 (a)-(d).

( b 1 02.00- 04.00

K im2 s-‘)

Id ) 06.00-08.00

!+ f(T’ - ?+,I t3A dT’

cos B dl-i)’ z ‘Z

The interior vertical velocity and the boundary layer eddy diffusivity are determined at each level independently of adjacent levels. The expressions (7), (10) and (11) depend on, at most, first order derivatives, something which makes these estimates rather robust in a diagnostic context.

The validity of the assumption of advection-dominated interior dynamics may be checked by making the same calculations as above but now for the nondegenerated case, say &of the same order as I?. Hence, the eddy diffusivity represents horizontally averaged values. Equation (8) yields

ad - = G,K’ + Gz a2 (124

Thermally driven circulations 121

Ic8 (m’ +I

lO-7 10-G 10-5

~8 Cm2 s-l) ~8 Cm2 s-l)

10-T 10-T 10-G 10-G 10-S 10-S --I I

-

g4 H

8 8 -’ (g) 12.00-14.00 (g) 12.00-14.00 (h ) 14.00-16.00

(Figure 7 (a j(h). Calculated tiB as a function of depth for the degenerated case, equation (1 l), based on the temperature profiles in Figure 6 on 16 July.

(12b)

$ $T’- n dA aT’ Z2T’

dl-zz-A- c, =

cos e a2

AZ

aT’ -- T J

c,z Ot

iiT’

az

(12c)

where I? is determined by only one boundary condition. When K’ has been calculated, W’ is obtainable by use of equation (7). However, the proper boundary condition on ti’ is not known, but as a sensitivity test, a series of numerical experiments with different boundary

122 L. Andersson & L. Rahm

(c)

0 0 \ \

v v

Depth Cm) Depth Cm)

Depth Cm)

Figure 8 (a)-(d).

V

Depth Cm)

W’ Cm s-l) W’ Cm s-l)

-,0-z -10-4 -,0-q ,0-e -,0-z -10-4 -,0-q L L k ,0-e 10.’ 10-z 10.’ 10-z I ’ I ’ IY $1 1 I ‘1 I ’

06.00-00.00

2

(d)

4 Depth (m) Depth (m)

conditions on ICI [(0-lOO)*n-,,,, where K,,, is the molecular diffusivity] was carried out. These calculations reveal that the results are not, at least in the range investigated, sensi- tive to the numerical value of the boundary condition, only a thin region very close to the bottom is influenced. Furthermore, a simple order of magnitude analysis yields that the length scale, on which the interior diffusive processes acts, is given by h=max(R/s^; K’AT/J)“~, where R is the radius of the bag, something which confirms the results above.

Measurements and calculations

Measurements

The experiment was carried out at Asko on the Swedish east coast, see Figure 2, during one week in July 1985. A report on this experiment, including results from the different

Thermally driven circulations 123

(e)

\1 Depth (m)

(g ) 12.00-14.00 2

Depth (ml

W’ Cm s-‘1 -(o-Z -10-a -,0-60 10-6

L L I 1 I ’ I* YI ’

10.15-12.00 2 -

6-

8 F c JI

Depth (m)

Wr (m s-‘)

(h 1 1400-16.00 2

‘0

:

Depth (m 1

Figure 8 (a)-(h). Calculated interior vertical velocity IV’, as a function of depth for the degenerated case, equation (7), based on the temperature profiles in Figure 6.

biochemical experiments carried out, is presented in Wulff & Koop (1990). The geometry and overall dimensions of the limnocorral are shown in Figure 3.

The observations were done at intervals of approximately 2 h during three periods from 15 July to 18 July. The water temperature was measured as a function of depth inside and outside the bag using a TD-probe (for details see Andersson & Rahm, 1986). To be able to determine the source term, i.e. the energy exchange through the sea surface and its distri- bution over depth, the following meteorological parameters were measured. Wind speed, recorded at a height of 3 mat the site of the bag (and then adjusted to represent the velocity at standard height, 10 m). Incoming solar radiation, in the interval 350-2800 nm, was continuously recorded by use of a pyrheliometer. Air temperature and dew-point tem- perature were measured on a raft close to the bag, and finally cloud cover was estimated at the same location.

In order to determine the latent and sensible heat fluxes, as well as the outgoing long- wave radiation, the following empirical relations have been used:

124 L. Anderson &L. Rahm

8

8 -.

V ( b) 02.00-04.00

Ks(m2 5’) I( id s-‘)

1-7 I o-6 10-5 10-T 10-6 10-S

‘, ( c) 04.00-06.00 (d 1 06.00-08.00

Figure 9 (a)-(d)

(i) The latest heat flux L, at the surface (Gill, 1982),

L, = LP, C.&Q, - Q,Ju, (13)

where L is the latent heat of evaporation, pA the density of air, C, the Dalton number. Q, and QA represent the specific humidity at the sea surface and at the standard height, respectively, while U, denotes the wind speed.

(ii) Sensible heat flux S, is also calculated by an empirical formula (Gill, 1982),

s=~,c,c,(T,- T,>u, (14)

where C, is the specific heat of air and C, the Stanton number. T, and T, are the sea surface and air temperatures, respectively.

(iii) The effective outgoing long-wave radiation I* from the sea surface, is given by the Brunt equation (Sellers, 1965),

I* = UJT A 4 (1 - k*n”) (1 - a - beli2) (1%

where E represents a correction coefficient for the sea surface not being a black body, and cr is the Stefan-Boltzman constant. The letters a, b, Fz* and m are empirical coefficients given

Thermally driven circulations 125

K ’ ( m2 5-l )

10-7 10-6 10-S I >

6 -.

8 -.

V (e 1 08.00-10.15

8 i 9 ) 12.00-14.00

K (m2 s-‘I K (m2 s-‘I

IO-’ IO-’ 10-6 10-G 10-5 10-5 I

I ’ I I > >

4 -.

L

’ : 6

8 --

V (f (f 1 10.15-12.00 1 10.15-12.00

K Cm2 s-‘i K Cm2 s-‘i

10-T 10-T 10-G 10-G 10-S 10-S

2

8

I

( h ) 14.00- 16.00

Figure 9 (a)-(h). Calculated I? as a function of depth for the undegenerated case, equation (12), based on the temperature profiles in Figure 6 on 16 July.

Time date

-10’0 L Darkened

Figure 10. Change in total thermal energy content within the bag (-), compared with the energy exchange through the side walls (- - -) and through the top surface (. .).

126 L. Anderson & L. Rahm

by Budyko (1963), while e and n represent the vapour pressure and the cloud cover, respectively. The numerical values of the coefficients are given in Table 1.

Note that the bulk formulae used are assumed valid for an open and exposed sea- surface, consequently their validity are questionable in this application where, for example, the spray shield of the bag may alter the heat transfer processes. However, the errors induced are assumed to be of minor importance.

The optical conditions within the bag, i.e. the attenuation of the incoming solar radiation, were measured during the experiment. The resulting distribution over depth indicated a rather constant attenuation coefficient c z 0.4 m-‘, and hence this value has been used throughout the calculations. The intensity I of the incoming light is then assumed to decay as,

Z(z,t) = Z,(t)exp( - cz) (1’5)

Calculations The temperature profiles were averaged for each time step at each depth interval (0.1 m), as were both the area of the vessel and the source function. Assuming piecewise linear gradients, the temperatures at the levels immediately above the top and below the bottom were estimated from the very top and bottom values. This was performed in order to facilitate calculations of temperature gradients at the very top and bottom. The errors induced are assumed negligible.

In determining the heat source J, the heat fluxes due to latent heat, sensible heat and long-wave radiation were included in the top level together with the calculated absorption of incoming short-wave radiation in this layer. The magnitude of these fluxes is shown in Figure 4. The source term in all other depth intervals was constituted by the absorption of short-wave radiation only. It was defined in all segments by integrating the absorbed fraction of the short-wave radiation over the actual timestep.

The governing equations were then solved using a finite-difference technique.

Results

The temperature profiles were taken in the centre of the bag, but at some occasions additional proliferations were done near the curved wall, measurements which confirmed the assumptions of horizontal homogeneity in the interior. The temperature evolution in the interior, at some different depths over the entire period, is shown in Figure 5. Since the measurements were done during an otherwise entirely biological experiment, the con- ditions for the stratification experiment were not always ideal, e.g. several artificial mixing events (indicated in Figure 5) were carried out. Moreover, during the latter part of the experiment, the upper part of the bag was covered with black plastic sheets reaching down to a depth of 1 m below the surface. The bag was covered in order to limit the photo- synthesis, but this of course also altered the heat exchange through the surface.

Typical temperature profiles taken during one day within and outside the enclosure are shown in Figure 6(a-i). The presence of the thermocline is evident in both Figures 5 and 6. It is also clear that the system always strives to approach the outside temperature distri- bution, but due to the time scales involved, in both the forcing process and in the system itself, a substantial lag is almost ever-present, which causes the sidewall thermal forcing. The estimated adjustment time of the system, based on equation (1 I), is of the order

r = min[AT/J; R/(2&)] (17)

Thermally driven circulations 12’7

Typical values fall within the range 102-lo4 s. [The ‘ diffusion time ’ (N R2/~) is about 105-lo7 s.] The observed temperature evolution in the bag supports the assumption of an advection-dominated interior.

A common feature in these measurements is a temperature decrease towards the sur- face, indicating a thin region of gravitationally unstable stratification and therefore defining a region invalid for the model. However, this phenomena is probably mainly due to the thermal mass of the embedded thermistor in the probe. The generally wet instru- ment was kept in the air for a short time before every lowering into the water, this to allow for an adjustment of its pressure gauge. This inevitably led to a rapid cooling of the thermistor, reminiscences of which are found at the top of the temperature profiles. The adjustment time of the thermistor is estimated to O-2 s, and since the lowering speed of the sond was roughly 0.3 m s-l, the temperature profiles will be displaced downwards ap- proximately 0.1 m. The main effect will be a slight underestimation of the strength of the sourceJ. Hence, the errors induced by this time-lag are generally insignificant except in a thin region at the top. No time-lag has been observed for the pressure gauge, probably because of the low lowering speed.

Some results typical for the experiment, and obtained from calculations of equations (10) and (1 l), under the assumption of advection dominated interior dynamics are pre- sented with regard for @ in Figure 7(a-h), and for IV’ in Figure 8(a-h). The eddy diffusivity characteristic for the buoyancy layer is found in the range 10-7-10P5 m2 s- ‘. It is evident that in the more stratified parts of the enclosure the eddy coefficients approach the molecular limit, while in the upper more homogeneous parts, higher values are fre- quently found. The norm of W’ generally fall within the range of O-10m2 m s-‘. The higher velocity values are sometimes associated with rapid movements of the exterior thermocline and the corresponding high heat fluxes through parts of the sidewall. High velocities may also occur in the more well-mixed upper parts of the vessel, since the buoyancy-layer transport is inversely proportional to the stratification. The model is, however, derived under the assumption of strong stratification, consequently its validity will only remain if the imposed temperature difference decreases in consort with the stratification (cf. Walin, 1971).

Solutions to the undegenerated equation (12), using the same set of forcing data, are shown in Figure 9(a-h). These results reveal a high degree of correspondence with the advection-dominated nondissipative case, Figure 7(a-h). Hence, to lowest order, dissi- pative processes only occurs in the boundary layers, the interior diffusion is insignificant. Note that this statement does not exclude horizontal advection-diffusion processes in the interior, which in fact cause the horizontally homogeneous conditions for T’ and W’/‘.

Discussion and conclusion

To summarize the above, the degenerate model yielded approximately the same results as the undegenerated one, and that despite the prescribed relation between xB and K’, which probably grossly overestimated the interior eddy diffusivity. Hence, in the stratification adjustment process, advection dominates over diffusion in the interior, while the heat diffusivity still plays a crucial role in the thin buoyancy layer. It is the boundary layer transport and its divergence that determines the interior stratification. This is a result in contrast to previous models, which have been based on turbulent diffusive interior pro- cesses only. The degeneracy is, however, partly due to the large vertical excursions of the

128 L. Andersson & L. Rahm

thermocline and the corresponding strong thermal forcing. Under more quiescent con- ditions this degeneracy should be less pronounced, but even under these circumstances, mixing at the sidewalls would probably generate buoyancy layer transports that would govern the interior dynamics.

Eddy diffusivity values below the molecular one were sometimes obtained even in this experiment, in fact, at some occasions even negative values were obtained. This is prob- ably due to uncertainties in the calculations of heat fluxes through the sea surface. A separation into direct and diffusive short-wave radiation has not been carried out, and the relations used for calculating latent and sensible heat fluxes are intended for large scale, well-exposed sea surface conditions. Other weak points are the rather sparse sampling in time and low resolution in the temperature data. Furthermore, the model is restricted to stably stratified conditions, something that was not always fulfilled during the experiment. Finally, the momentum flux through the sidewalls due to both wave action and motions of the superstructure itself, has not been considered, though these effects probably will be included in the calculated ‘ effective ’ eddy diffusivity coefficient xB.

An estimate of the change of total thermal energy content in the bag was carried out for two qualitatively different periods, see Figure 10. This in order to verify the assumptions about the importance of sidewall heating. The first period represents ‘ normal ’ con- ditions, while during the second period the top surface was covered. The change in energy content with time is compared with the corresponding integrated heat fluxes. At the beginning of the first period, the heat flux through the surface governs the thermal evolution, while during the rest of this period its role is superseded by the sidewall heating. The heat flux through the surface is almost eliminated by the covering during the second period, and consequently the sidewall heating completely controls the thermal evolution within the bag. Though the response of the bag to this forcing is not so distinct during the rest of the experimental period, the equal importance of the two forcing factors seems evident. Although the present model seems to describe the lowest order dynamics of the bag in a reasonable way, it would be of great interest to make a second experiment, where a comparison between the model predictions and those based on the distribution of a passive tracer could be carried out.

One conclusion that may be drawn from this work is that the indicated vertical circu- lation, caused by the buoyancy layer transport, must be considered in flux estimates of, for example, oxygen and nutrients as well as in estimates of the distribution of small plankton, calculations which are often carried out in this type of mesocosm studies. Since the circulation is determined by the temperature field, the sources and sinks of nonconserva- tive substances may be calculated by the use of measured concentration profiles, in an advection-diffusion equation analogous to equation (2).

Acknowledgements

The authors wish to thank Drs P. Williams and F. Wulff for initiating this study.

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