- Email: [email protected]
A laboratory model of a convectively driven ocean
Dynamics of Atmospheres and Oceans, 13 (1989) 77-93 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
77
A LABORATORY M O D E L OF A CONVECTIVELY DRIVEN O C E A N
SCOTT A. CONDIE
Research School of Earth Sciences, Australian National University, GPO Box 4, Canberra, A.C.T. 2601 (Australia) * (Received March 21, 1988; revised June 22, 1988; accepted June 24, 1988)
ABSTRACT Condie, S.A., 1989. A laboratory model of a convectively driven ocean. Dyn. Atmos. Oceans, 13: 77-93. Many large-scale flows in the ocean are driven by an imposed horizontal density gradient and the resulting circulation is strongly influenced by the Earth's rotation. Some of the essential features of such flows have been incorporated into a laboratory model by differentially heating and cooling the vertical end walls of a shallow rectangular cavity rotating about a vertical axis. Buoyancy driven boundary currents produced by the heating and cooling were unstable and the resulting eddy structures eventually dominated the system. A broad mean flow, perpendicular to the side walls, developed in the central region of the cavity. The resultant steady-state flow had a barotropic component consisting of two large-scale gyres of cyclonic and anticyclonic senses. In this paper, the evolution of the flow towards its final steady state is described. Measurements of the time-scales for the establishment of stratification in the cavity, point to the existence of two dynamical density adjustment modes.
1. INTRODUCTION
Hide (1958) reported the first experiments in a differentially heated rotating annulus (see review by Hide and Mason, 1975). Even today this configuration is yielding important results in the field of atmospheric dynamics and geostrophic turbulence (Buzyna et al., 1984; Hignett, 1985). While the similarities between atmospheric and ocean dynamics are well documented, there are two important features of the ocean which are absent from the annulus experiments. Firstly, the horizontal length-scales in the ocean are much larger than the vertical scales. Therefore, a laboratory experiment which models this system requires a small aspect ratio. Secondly, * Present address: Department of Oceanography, Old Dominion University, Norfolk, 23529, U.S.A.
0377-0265/89/$03.50
© 1989 Elsevier Science Publishers B.V.
VA
78 there are meridional boundaries or coastlines in the ocean which play a fundamental role in the circulation. Both of these features have been incorporated in a new and conceptually simple laboratory model of a thermally driven ocean basin on an f-plane. The model consists of a low-aspect-ratio rectangular cavity in which one end wall was heated and the other cooled, with all other boundaries insulated. This apparatus rotated about a vertical axis through its centre. Similar configurations have been examined previously without rotation and are reviewed in Bejan (1984). With the addition of rotation, the flow had features in common with the thermohaline circulation in the ocean and other large water bodies. Sugimoto and Whitehead (1983) considered a more complex configuration, in which a shallow rectangular sea of constant depth was bounded by a deep ocean through a uniformly sloping bottom region. The fluid was heated at the deep-ocean end wall and cooled at the surface, while the system rotated. The resulting flow included a number of gyres (dependent on the rotation rate), and a jet-like front along the shelf break. More recently, Speer and Whitehead (1988) conducted experiments in a rotating rectangular cavity with non-uniform heating from below. This system modelled convection in an ocean with non-uniform heating at the surface. Over much of the parameter range studied, the flow was dominated by a cyclonic gyre in the upper layer and much weaker anticyclonic motions near the bottom. The relative role of wind stress and buoyancy flux in the ocean circulation is still unclear. The commonly perceived dominance of wind stress has probably arisen owing to the simplicity of that formulation. Specifically, Luyten and Stommel (1986) suggested that the baroclinic circulation may be "driven by buoyancy and steered by wind". Our understanding of the deep circulation, where the direct effects of wind stress are less significant, is even more limited. This is largely owing to the difficulty in obtaining sufficient velocity data. Most descriptions of the deep circulation are inferred from measurements of property fields such as temperature, salinity and dissolved oxygen distributions (Warren, 1981). Under these circumstances, determining velocities requires assumptions of geostrophy and depths of no motion. Another important question relates to how the circulation depends on the surface boundary conditions. In particular, the response of the ocean to changes in atmospheric forcing is important to studies of global climate change. By examining the behaviour of simple systems over a wide parameter range, laboratory experiments can contribute to the understanding of the phenomena mentioned above. They have already made a significant contribution to the study of the dynamics of the large-scale circulation of the ocean. This is particularly true of homogeneous ocean models (for example
79 Stommel et al., 1958). It is hoped that the results reported here will assist our understanding of phenomena associated with stratification. Although the experiments were conducted on an f-plane, it is felt that an understanding of this relatively simple system is essential in order to understand the more complex dynamics of stratified flows on a r-plane. Section 2 begins with a detailed description of the experimental configuration. Appropriate parameters which govern the dynamics of the steady flow are then defined, followed by a description of the transient and steady-state flows. The time-scales for development of the stratification in the cavity are then examined in section 4. These last measurements may be relevant to the response of large-scale water bodies to seasonal and longer term climatic fluctuations. 2. DESCRIPTION OF EXPERIMENTS The experiments were conducted in a rectangular perspex cavity of height H = 15 cm, length L = 200 cm and width B - - 6 0 cm. Heat exchangers formed the two vertical end walls of the tank. These consisted of a 3.0-cm-thick aluminium block with 1.2-cm-square grooves through which heated or cooled water flowed. To ensure an even temperature distribution over the heat exchanger surface, a 1.0-cm-thick block of copper was in contact with the working fluid. Thermistors embedded in the copper and aluminium plates allowed both the temperature in contact with the fluid and the temperature gradient across the plates to be measured. When this system had been calibrated the latter quantity allowed the heat flux into the working fluid to be calculated. The base and sides of the cavity were insulated with 10-cm-thick expanded polystyrene. The lid was designed as a series of sealed transparent air cavities, and when visual access was not required a further 10 cm of polystyrene was fitted. This assembly was mounted on a 1.0-m-diameter rotating table. Fluid slip rings allowed the hot and cold baths supplying fluid to the end walls to be located off the table. There were also active themistor bridges on the turntable for amplification of thermistor probe signals. The amplified signals passed through electrical connections from the rotating table to the stationary reference frame where they were recorded. Before describing the experiments, suitable parameters which govern the dynamics of the flow will be defined. Batchelor (1954) pointed out that the non-rotating analogue of the present experiment can be described by three independent non-dimensional parameters. The first of these is the aspect ratio A = I-I/L
(1)
80 which depends only on the cavity geometry. For all the experiments reported here A = 0.075. The second is the Prandtl number, a property of the fluid and defined as the ratio of the kinematic viscosity to the thermal diffusivity, i.e. (2)
Pr = v / ~
The final parameter is the Rayleigh number which describes the forcing by the applied temperature difference AT between the end walls. This is defined by the relation gaATH 3
(3)
Ra PI(
where a is the coefficient of thermal expansion and g is the gravitational acceleration. Rotation introduces two additional independent variables and therefore requires two additional non-dimensional parameters. The first variable is the Coriolis parameter f (defined as twice the angular velocity) which has been included in an Ekman number Ek = v/(H2f)
(4)
The second additional variable is the width of the cavity, which is not required in the non-rotating case since the flow is then two dimensional. In our case the width is most usefully represented in terms of a rotational Froude number B2f 2 -
(5)
ge~A T H
which is the square of the ratio of the cavity width, B, to the Rossby deformation radius. For each experimental run the tank was filled with water and spun up to solid body rotation. Heating and cooling were then applied by pumping hot and cold fluid through the two end walls. The temperatures of the end walls reached their final steady values (to within 5%) within 100 s after 'start-up'. By this time the heat flux through the two end walls had also reached a value which was maintained (to within a few per cent) throughout the development of the flow. The end-wall boundary conditions throughout the duration of any given experiment were, therefore, constant temperatures and a constant heat flux into and out of the working fluid. These two heat fluxes were equal and the mean temperature of the water remained constant. Aspects of the transient flow in the rotating cavity configuration have been described by Condie and Ivey (1988). That study examined the flow of hot and cold boundary currents into an isothermal environment (t < 100
81 rotation periods). Their description began with the application of heating and cooling at the end walls. Buoyant fluid adjacent to the hot end wall rose vertically and was turned into the interior by the upper horizontal lid (with a symmetrical flow at the cold end). Coriolis forces pushed this current to the right (anti-clockwise rotation) to form a density current along the right-hand boundary, which was unstable to rotationally induced instabilities. When E k > 1 0 - 4 and Fr < 1, waves failed to break due to the damping effects of Ekman layer dissipation or the influence of the opposite side boundary. Conversely, for E k < 10 -4 and Fr > 1 the waves broke to form eddies and eddy pairs which penetrated into the lower layer and interacted with neighbouring eddies, thus broadening the current. This paper considers the longer term development of the flow and the establishment of the density field in the cavity. The steady-state flow will be described briefly, although a more detailed account of this aspect will be given in a later paper. 3. EVOLUTION OF THE FLOW FIELD Temperature sections were measured by vertically traversing a rack of 15 fast-response thermistors equally spaced across the tank. Cross-sectional profiles were recorded at regular intervals throughout the development of the flow at four positions along the length of the tank. Examples of these profiles in Fig. 1 show the development of the temperature field for a single set of conditions. This type of data was used along with observations of dye streaks released at the end walls (Fig. 2), to determine how the flow developed. The description which follows is for the conditions in Fig. 1; however, the basic character of the flow was similar provided Fr > 1. When this condition was not satisfied the flow approached that observed in a non-rotating cavity with differentially heated and cooled end walls (Patterson and Imberger, 1980). As the Froude number was increased, horizontal length-scales, such as the width of strong current regions and diameter of eddy structures, decreased. Column (a) in Fig. I shows the temperature structure at the four crosssections in the cavity (viewed from the heated end), 50 rotation periods after the heating and cooling were applied. Warm and cold intrusions with almost linear internal stratification, flowed with a side boundary on their right-hand side. The second profile in this column indicates that the warm current was particularly wide at that section, reflecting the presence of a large amplitude wave or eddy. The dimensions of the warm current were slightly larger than those of the cold bottom current. This resulted from the higher buoyancy flux associated with the warm flow, owing to the non-linear equation of state of water. The magnitude of this effect should not be underestimated, since
82
I
(a)
(b)
Fig. 1. C r o s s - s e c t i o n s o f the t e m p e r a t u r e s t r u c t u r e in the cavity w h e n Ra = 2.1 × 1 0 9 , Ek = 2.4 × 10 5 a n d Fr = 71.0. T h e c r o s s - s e c t i o n s are viewed f r o m the h o t e n d a n d i s o t h e r m s are at 1.0 o C intervals. T h e c o l u m n s c o r r e s p o n d to (a) t = 50, (b) t = 100, (c) t = 200 r o t a t i o n p e r i o d s and (d) s t e a d y state, while the rows c o r r e s p o n d to x / L = 0.28, 0.43, 0.65 a n d 0.80, respectively (where x is m e a s u r e d from the hot e n d wall). T h e d i r e c t i o n s of s t r o n g c u r r e n t s are also indicated.
83
(c)
Fig. 1 (continued).
(d)
84
Fig. 2. Plan views of the dyed hot intrusion when R a = 8 . 7 × 1 0 ~, E k = 2 . 4 × 1 0 5 and Fr = 71.0. The hot end wall is on the left-hand side and the cold end wall on the right. The clock shows the number of rotation periods since the heating and cooling were initiated. Rotation is counter-clockwise.
for the conditions described in Fig. 1, the coefficient of thermal expansion at the temperature of the hot end wall was double that at the cold end. This is also reflected by the stronger temperature gradients within the cold current. After 100 periods (column (b)) the growth and interactions of eddy structures had significantly broadened the currents. This is also evident in a photograph taken of the warm intrusion at 100 rotation periods (Fig. 2).
85 0.02
0.01
0.0(
0.0
012
014
0.'6
0'.8
.0
x/L Fig. 3. The cross-cavity mean velocity distribution for the hot intrusion as a function of distance along the cavity. Each datum point was determined by plotting the position of the warm front (as defined by the spreading of the dye cloud) at the appropriate value of x, as a function of time. For the non-dimensionalizationused in the figure, the data from runs with different conditions collapsed on to a single line. The best-fit slope of this line gave a non-dimensional cross-cavity velocity representative of all the parameter range except F r < 1.
Condie and Ivey (1988) noted that when the initial intrusions flowed into the cavity, they compressed columns of the stationary fluid. To conserve potential vorticity, a counter-flow developed beneath the hot intrusion that carried cold fluid away from the base of the cold end wall (i.e., along the bottom right-hand corner of the cross-section). It is evident in the fourth cross-section in column (b), that additional fluid was supplied to this flow through spreading of the original cold current away from the opposite side wall. Much of the buoyancy in the hot boundary current was transported across the cavity. In fact, Condie and Ivey (1988) pointed out that at the higher rotation rates ( f > 1.0 s -1) the nose of the current almost stagnated before reaching the end of the cavity, indicating that the cross-cavity flux exceeded the along-shore flux. The dynamics of the cross flow were of particular interest, since they appeared to influence the flow throughout the cavity. While eddies were extremely prominent in the region, the mean cross-stream velocities were essentially constant, suggesting an advective rather than a turbulent diffusive transport process. It is, however, quite feasible that the Reynolds stresses associated with the eddy field significantly contributed to the advection. The mean cross-cavity velocity at the surface, as determined by the spreading of dye clouds, is plotted in Fig. 3 as a function of the position along the cavity. There was relatively little transport near the hot end wall where waves had not yet broken. However, as Fig. 3 shows, the cross-stream component of the hot flow was substantial in the central region of the tank. Further downstream, where less buoyant fluid was available, the transport again dropped to small values. By 200 rotation periods (column (c) of Fig. 1 and the second photograph
86 in Fig. 2) the cross-cavity flow had considerably altered the nature of the flow field. When the cross-flow impinged near the centre of the opposite side wall, it produced two inertial jets along this boundary. The weaker jet went to the left and returned a small amount of warm fluid to the hot end. The stronger jet carried the remainder of the warm fluid to the right towards the cold end wall (this feature is quite distinct in the last two profiles in column (c)). This region of the flow may be analogous to that analysed by Whitehead (1985), in which a baroclinic jet impinged on a wall in a rotating fluid. In the present experiment, the cross-cavity flow struck the wall almost normally and active mixing by meso-scale eddies suggests that the potential vorticity distribution would have been relatively uniform. For a constant potential vorticity jet normal to the wall, Whitehead's model indicates that approximately 70% of the fluid goes to the right and 30% to the left. This result is consistent with qualitative observations of dye transports in the cavity, even though the turbulent nature of the cross-cavity flow and associated non-linear effects complicate the exact description of the flow in our experiments. As stratification in the cavity developed towards the steady-state profiles (column (d) of Fig. 1), the horizontal density gradients were reduced. This process acted to weaken the buoyancy driven components of the flow. Dye streaks showed that the larger scales of motion became more barotropic, while the inertial jets became more significant. The weakening of baroclinic components of the flow was accompanied by a decay in the cold flux from the base of the left side (viewed from the hot end) of the cold end wall. A somewhat surprising observation was that the flow from the right-hand side o f the cold end, thereafter, carried all of the cold fluid. This cold intrusion moved towards the centre of the cavity (with the side wall to its left when looking in the direction of flow), then meandered across to the opposite boundary (the left-hand side wall when viewed from the hot end). After the cold water reached the left-hand wall, most of it continued on to the hot end beneath the weak inertial warm jet flowing in the same direction.
i
coLD
HOT
m
Hot Fluid
r'--]Cold Fluid
Fig. 4. A schematic representation of the mean flow observed in the cavity at steady state when Fr > 1. The wide arrows represent the slow turbulent cross-cavity flow.
87 The barotropic component of the steady-state mean flow can therefore be described as two large-scale gyres. This is represented schematically in Fig. 4. A cyclonic gyre filled approximately half of the cavity closest to the hot wall and an anticyclonic gyre occupied the other half. This aspect of the steady-state flow is not yet fully understood and is the subject of continuing investigation. 4. DEVELOPMENT OF THE STRATIFICATION The aim of this part of the study was to determine the time-scale for development of the density structure in the cavity and investigate its dependence on the Ekman, Froude and Rayleigh numbers. A scaled estimate for this quantity in a non-rotating cavity was derived by Patterson and Imberger (1980) based on a simple 'filling box' process in which the interior region was assumed to fill with heated fluid by horizontal layering. This process has been studied experimentally in a cavity with only one differentially heated end wall (Worster and Leitch, 1985). Such a configuration more closely models the idealized filling box, because of the absence of a sink for the buoyant fluid. However, experiments by Yewell et al. (1982), along with temperature sections such as those in Fig. 1, suggest that a similar type of process may be operating in the case of two differentially heated end walls. Patterson and Imberger showed that the volume flux p u m p e d through the thermal boundary layers of the end walls of a cavity, scales as BxRa a/4. The time-scale to p u m p all the fluid in the cavity through the boundaries is therefore 1"
HL xRa]/4
(6)
This quantity will be referred to as the 'heat-up' time. Cross-sectional profiles, such as those in Fig. 1, were used to determine the adjustment time-scales. Ten profiles recorded after the system had been allowed to reach steady state, were averaged to determine the mean steadystate temperature structure. The standard deviation of the instantaneous temperature values from the steady-state values was then calculated for each cross-sectional profile taken during the heat-up process. More precisely, if the two-dimensional matrix of temperature values for a given time and cross-section has dements T~j and the steady-state values at the same positions are then the standard deviation between the profiles is defined as
Sis, ( Y', E [Tij-Sij[2)a/2 n
¢$T=
m
i=lj=l
(7) nm
88
10 0 w *
u
8T 10-1
• a • • • + • x g
=,
i7 @
I.
•
i..i
I
I
0.0
Fr 0.33 0.33 0.33 0.33 8.28 8.28 8.28 8.28 32.6 32.6 32.6 32.6 71.0 71.0 71.0
Ek 3.6x104 3.6x104 3.6x104 3.6xl 0 .4 7.1x10 ~ 71x105 7.1x10 5 7.1xt0 ~ 3.6x10 ~ 3.6x105 3.6x10 ~ 3.6x10S 2.4x10 ~ 2.4x105 2.4x10 S
•
!! :• I"
1 0 -2
x/I. 0.28 0.43 0.65 0.80 0.28 0.43 0.65 0.80 0.28 0.43 0.65 0.80 0.28 0.43 0.80
1.0
"
•
.
°
I
o
,
.
20
t/'c Fig. 5. The standard deviation of the developing temperature structure from the mean steady-state temperature structure (as defined by eqn. (7)) as a function of time (non-dimensionalized by the applied temperature difference and the heat-up time, respectively). Runs cover a range of Ekman numbers, Froude numbers and positions along the cavity.
where n was the number of thermistors (n = 15) and m was the number of readings taken by each thermistor in a single profile (m = 100). Parameter 8T was calculated at the four cross-sections in the tank throughout the development of the flow. This procedure was carried out over the entire experimental parameter range. Values of ST, normalized by the end-wall temperature difference AT, are plotted as a function of time (non-dimensionalized by the heat-up time) for a range of Ekman and Froude numbers in Fig. 5 and for two Rayleigh numbers in Fig. 6. Although there is not a simple exponential behaviour, the data for any given run can be fitted satisfactorily by two exponential curves, corresponding to the early and later development of the stratification. In other words, the adjustment process can be divided into two regimes, each of which is described by an exponential time-scale. Attempts to fit the data to a
89
10
fiT
[] ,t, a *
x/L 0.28 0.43 0.65 0.28
Ra 5.8x10 ~ 5.8x10 B 5.8x10 e 2.1x109
[] a •
0.43 0.65 0.80
2.1x10 Q 2.1x109 2.1x109
10
AT
,..t
IBm
[] []
a 2
[]
[]a a
[]
•
II
a 4~ Ju
,
0.0
I
,
1.0
.
•
,
~2.0
t/~ Fig. 6. Standard deviation from the mean steady-state temperature structure as a function of time, for a range of Rayleigh numbers and positions along the cavity.
power law yielded unsatisfactory results. The trends for small times (t < 0.5"r) will be discussed first. The initial adjustment is well described by the exponential law ~ T = C A T e -kt/'~
(8)
where C and k are constants. Figure 5 indicates that the time-scales are independent of rotation rate, while Fig. 6 shows that the non-dimensionalization accurately describes the dependence on the end-wall forcing for the range of AT investigated. There is also no distinguishable dependence of any of the data on the position along the tank. Measurements of the slopes and intercepts in Figs. 5 and 6 give k = 1.23 + 0.17 and C = 0.15 + 0.01 for the early development of the temperature structure. This value of k corresponds to a time-scale of (0.81 _ 0.11)'r. Since the time-scale for the early development of the stratification scales with the heat-up time, it is reasonable to assume that the adjustment process is adequately approximated by the filling-box model. The transition point
90 1.0i
, , , - "
0,8 Z 0.6 H 0.4
0.2 0.0-0.5 -0.3 -0.1 0.1
0.3
0.5
2(T-To)/AT Fig. 7. A vertical temperature profile within the cavity at the transition time between the two adjustment regimes (t = 0.54T). The broken line represents the steady-state structure. With the temperature non-dimensionalized in this way, the profile is representative of the entire parameter range investigated. T0 is the average of the two end-wall temperatures.
f r o m the filling b o x to a s e c o n d a d j u s t m e n t process, as given b y the i n t e r s e c t i o n of fitted curves for the two regimes, o c c u r r e d at t = (0.54 + 0.10)~-. F o r c o m p a r i s o n , the sections in Fig. 1 c o r r e s p o n d to (a)t = 0.045~', ( b ) t = 0 . 0 9 ~ - a n d ( c ) t = 0 . 1 8 " r . F i g u r e 7 shows the level of s t r a t i f i c a t i o n outside the b o u n d a r y c u r r e n t s at t = 0.54~-, d e m o n s t r a t i n g t h a t the filling-box p r o c e s s e s t a b l i s h e d m u c h of the final d e n s i t y s t r u c t u r e within the cavity. T h e later a d j u s t m e n t p r o c e s s c a n t h e r e f o r e b e r e g a r d e d as a ' f i n e t u n i n g ' of the density field. T h e s e c o n d a d j u s t m e n t r e g i m e was significantly slower t h a n the first, b u t a g a i n c o u l d r e a s o n a b l y b e d e s c r i b e d b y an e x p o n e n t i a l law. T h e r a p i d i t y of the transition b e t w e e n the two r e g i m e s v a r i e d significantly, l e a d i n g to a large
50000
Timescale (seconds)
.1
1
10
100
Fr Fig. 8. The exponential time-scales for the second density adjustment process (t > 0.5~-), as a function of rotating Froude number. The error bars represent one standard deviation, calculated from the values obtained at different cross-sections in the cavity. The Rayleigh number was constant for these runs at Ra = 2.1 × 10 9.
91 degree of scatter in 8T in the second regime. However, as noted in the previous paragraph, the transition point was relatively constant. After the transition, the rate of change of 8T with time continued to be independent of rotation rate. Since this result is not clear from inspection of Fig. 5, the data has been replotted in Fig. 8, on which the time-scales are shown for the second adjustment regime as a function of rotating Froude number. Taking into account the scatter of the data, there is no discernible trend, confirming that within the parameter range studied adjustment times were independent of rotation. The data in Fig. 6 indicate that the second adjustment process had a relatively strong dependence on Rayleigh number. As noted by Patterson and Imberger (1980), the motion in the core region of the cavity during the approach to steady state is extremely complex, making detailed scaling of behaviour very difficult. However, in dimensional terms, the exponential time-scales were (6.8 ___1.5) x 104 S for Ra = 5.8 x 108 and (2.33 + 0.55) x 104 s for Ra = 2.1 x 109. Although the parameter range was insufficient to determine empirically the Rayleigh number dependence, these results suggest that the second stage adjustment time-scales may have a close to linear dependence on Ra. This contrasts with the weaker Ra 1/4 dependence of the first regime. A separate experiment was conducted in which the applied temperature difference, AT, was increased after the flow had been allowed to reach steady state. The transition was, therefore, from a steady-state flow with Ra = 5.8 x 108 to a second steady state with Ra = 2.1 × 109. This experiment yielded a single exponential time-scale of 1.9 × 104 s for the duration of the adjustment process. This time-scale agrees with the value of (2.33 + 0.55) X 104 S, given in the previous paragraph for the second adjustment when Ra = 2.1 x 109, indicating that the second, much longer, adjustment regime is relevant to the problem of modification of existing stratification by external forcing. Perhaps not a surprising result since strong stratification was present throughout the process. Clearly, if the initial stratification had been sufficiently weak, the filling-box mechanism would have operated at early times. Therefore, in summary, the initial conditions (strength of the initial stratification) determine the type of adjustment process, while the new boundary conditions determine the actual value of the adjustment time-scale. 5. DISCUSSION This paper extends the work of Condie and Ivey (1988), by considering the long-term development of the convection experiment to a steady-state situation. It has therefore incorporated the processes associated with the development of stratification, as well as some of the effects of a stratified
92
environment on boundary currents. These are important matters for the thermohaline component of the large-scale circulation of the ocean. The most restrictive feature of the study with respect to oceanic applications is that the experiments were conducted on an f-plane rather than a B-plane (simply introducing a sloping bottom in the presence of stratification does not simulate a B-plane). It is clear, however, that a knowledge of f-plane dynamics is an essential component of any understanding of large-scale ocean circulation. The study deals particularly with the heat-up time-scales of the convective flow, both from an isothermal state and as a perturbation of a steady state. These processes can be described in terms of two adjustment modes. While the fluid is only weakly stratified (relative to the gradients associated with the forcing), a filling-box process operates on an exponential time-scale of (0.81 +0.11)'r. The second mode then takes over. This is essentially a fine-tuning of the density field and the relevant time-scale is dependent on the Rayleigh number. Perhaps the most significant finding is that both adjustment modes are independent of rotation. The results summarized in the previous paragraph may be relevant to a range of geophysical phenomena. Common examples are the adjustment of systems such as large lakes, seas, gulfs and oceans to seasonal variations in conditions of atmospheric temperature, evaporation rate, precipitation and fresh-water run-off. While the response of the oceans to long-term climatic change is a very complex problem (see for example Broeker, 1987), some of the experimental results may be relevant to this problem. The formation of two large-scale gyres at steady state is particularly interesting, but not yet well understood. This aspect does, however, demonstrate, that a complex density flow can be produced by simple boundary conditions. An observation which may have significant implications for the role of buoyancy in the large-scale circulation of the ocean. ACKNOWLEDGEMENTS
I would like to thank Dr Ross Griffiths, Dr Gregory Ivey and Professor Stuart Turner for advice throughout this study, as well as their comments on the manuscript. Thanks also to Dr Roger Nokes who wrote the data logging and plotting routines. Mr Pat Travers constructed the convection tank, Mr Ross Wylde-Browne processed the photographs and Mr Derek Corrigan gave other valuable technical aid and advice.
93 REFERENCES Batchelor, G.K., 1954. Heat transfer by free convection across a closed cavity between vertical boundaries of different temperatures. Q. J. Appl. Maths., 12: 209. Bejan, A., 1984. Convective Heat Transfer. Wiley, New York. Broeker, W.S., 1987. Unpleasant surprises in the greenhouse. Nature, 328: 123-126. Buzyna, G., Pfeffer, R.L. and Kung, R., 1984. Transition to geostrophic turbulence in a rotating differentially heated annulus of fluid. J. Fluid Mech., 145: 377-403. Condie, S.A. and Ivey, G.N., 1988. Convectively driven coastal currents in a rotating basin. J. Mar. Res., 46: 473-494. Hide, R., 1958. An experimental study of thermal convection in a rotating liquid. Philos. Trans. R. Soc., Ser. A, 250: 441-478. Hide, R. and Mason, P.J., 1975. Sloping convection in a rotating annulus. Adv. Phys., 24: 47-100. Hignett, P., 1985. Characteristics of amplitude vacillation in a differentially heated rotating fluid annulus. Geophys. Astrophys. Fluid Dyn., 31: 247-281. Luyten, J. and Stommel, H., 1986. Gyres driven by combined wind and buoyancy flux. J. Phys. Oceanogr., 16: 1551-1560. Patterson, J.C. and Imberger, J., 1980. Unsteady natural convection in a rectangular cavity. J. Fluid Mech., 100: 65-86. Speer, K.G. and Whitehead, J.A., 1988. A gyre in a non-uniformly heated rotating fluid. Deep-Sea Res., in press. Stommel, H., Arons, A.B. and Failer, A.J., 1958. Some examples of stationary flow patterns in bounded basins. Tellus, 10: 179-187. Sugimoto, T. and Whitehead, J.A., 1983. Laboratory models of bay-type continental shelves in the winter. J. Phys. Oceanogr., 13: 1819-1828. Warren, B.A., 1981. Deep circulation in the world ocean. In: B.A. Warren and C. Wunsch (Editors), Evolution of Physical Oceanography. MIT Press, Cambridge, MA, pp. 6-41. Whitehead, J.A., 1985. The deflection of a baroclinic jet by a wall in a rotating fluid. J. Fluid Mech., 157: 79-93. Worster, M.G. and Leitch, A.M., 1985. Laminar free convection in confined regions. J. Fluid Mech., 156: 301-319. Yewell, P., Poulikakos, D. and Bejan, A., 1982. Transient natural convection experiments in shallow enclosures. Trans. ASME, J. Heat Transfer, 104: 533-538.
77
A LABORATORY M O D E L OF A CONVECTIVELY DRIVEN O C E A N
SCOTT A. CONDIE
Research School of Earth Sciences, Australian National University, GPO Box 4, Canberra, A.C.T. 2601 (Australia) * (Received March 21, 1988; revised June 22, 1988; accepted June 24, 1988)
ABSTRACT Condie, S.A., 1989. A laboratory model of a convectively driven ocean. Dyn. Atmos. Oceans, 13: 77-93. Many large-scale flows in the ocean are driven by an imposed horizontal density gradient and the resulting circulation is strongly influenced by the Earth's rotation. Some of the essential features of such flows have been incorporated into a laboratory model by differentially heating and cooling the vertical end walls of a shallow rectangular cavity rotating about a vertical axis. Buoyancy driven boundary currents produced by the heating and cooling were unstable and the resulting eddy structures eventually dominated the system. A broad mean flow, perpendicular to the side walls, developed in the central region of the cavity. The resultant steady-state flow had a barotropic component consisting of two large-scale gyres of cyclonic and anticyclonic senses. In this paper, the evolution of the flow towards its final steady state is described. Measurements of the time-scales for the establishment of stratification in the cavity, point to the existence of two dynamical density adjustment modes.
1. INTRODUCTION
Hide (1958) reported the first experiments in a differentially heated rotating annulus (see review by Hide and Mason, 1975). Even today this configuration is yielding important results in the field of atmospheric dynamics and geostrophic turbulence (Buzyna et al., 1984; Hignett, 1985). While the similarities between atmospheric and ocean dynamics are well documented, there are two important features of the ocean which are absent from the annulus experiments. Firstly, the horizontal length-scales in the ocean are much larger than the vertical scales. Therefore, a laboratory experiment which models this system requires a small aspect ratio. Secondly, * Present address: Department of Oceanography, Old Dominion University, Norfolk, 23529, U.S.A.
0377-0265/89/$03.50
© 1989 Elsevier Science Publishers B.V.
VA
78 there are meridional boundaries or coastlines in the ocean which play a fundamental role in the circulation. Both of these features have been incorporated in a new and conceptually simple laboratory model of a thermally driven ocean basin on an f-plane. The model consists of a low-aspect-ratio rectangular cavity in which one end wall was heated and the other cooled, with all other boundaries insulated. This apparatus rotated about a vertical axis through its centre. Similar configurations have been examined previously without rotation and are reviewed in Bejan (1984). With the addition of rotation, the flow had features in common with the thermohaline circulation in the ocean and other large water bodies. Sugimoto and Whitehead (1983) considered a more complex configuration, in which a shallow rectangular sea of constant depth was bounded by a deep ocean through a uniformly sloping bottom region. The fluid was heated at the deep-ocean end wall and cooled at the surface, while the system rotated. The resulting flow included a number of gyres (dependent on the rotation rate), and a jet-like front along the shelf break. More recently, Speer and Whitehead (1988) conducted experiments in a rotating rectangular cavity with non-uniform heating from below. This system modelled convection in an ocean with non-uniform heating at the surface. Over much of the parameter range studied, the flow was dominated by a cyclonic gyre in the upper layer and much weaker anticyclonic motions near the bottom. The relative role of wind stress and buoyancy flux in the ocean circulation is still unclear. The commonly perceived dominance of wind stress has probably arisen owing to the simplicity of that formulation. Specifically, Luyten and Stommel (1986) suggested that the baroclinic circulation may be "driven by buoyancy and steered by wind". Our understanding of the deep circulation, where the direct effects of wind stress are less significant, is even more limited. This is largely owing to the difficulty in obtaining sufficient velocity data. Most descriptions of the deep circulation are inferred from measurements of property fields such as temperature, salinity and dissolved oxygen distributions (Warren, 1981). Under these circumstances, determining velocities requires assumptions of geostrophy and depths of no motion. Another important question relates to how the circulation depends on the surface boundary conditions. In particular, the response of the ocean to changes in atmospheric forcing is important to studies of global climate change. By examining the behaviour of simple systems over a wide parameter range, laboratory experiments can contribute to the understanding of the phenomena mentioned above. They have already made a significant contribution to the study of the dynamics of the large-scale circulation of the ocean. This is particularly true of homogeneous ocean models (for example
79 Stommel et al., 1958). It is hoped that the results reported here will assist our understanding of phenomena associated with stratification. Although the experiments were conducted on an f-plane, it is felt that an understanding of this relatively simple system is essential in order to understand the more complex dynamics of stratified flows on a r-plane. Section 2 begins with a detailed description of the experimental configuration. Appropriate parameters which govern the dynamics of the steady flow are then defined, followed by a description of the transient and steady-state flows. The time-scales for development of the stratification in the cavity are then examined in section 4. These last measurements may be relevant to the response of large-scale water bodies to seasonal and longer term climatic fluctuations. 2. DESCRIPTION OF EXPERIMENTS The experiments were conducted in a rectangular perspex cavity of height H = 15 cm, length L = 200 cm and width B - - 6 0 cm. Heat exchangers formed the two vertical end walls of the tank. These consisted of a 3.0-cm-thick aluminium block with 1.2-cm-square grooves through which heated or cooled water flowed. To ensure an even temperature distribution over the heat exchanger surface, a 1.0-cm-thick block of copper was in contact with the working fluid. Thermistors embedded in the copper and aluminium plates allowed both the temperature in contact with the fluid and the temperature gradient across the plates to be measured. When this system had been calibrated the latter quantity allowed the heat flux into the working fluid to be calculated. The base and sides of the cavity were insulated with 10-cm-thick expanded polystyrene. The lid was designed as a series of sealed transparent air cavities, and when visual access was not required a further 10 cm of polystyrene was fitted. This assembly was mounted on a 1.0-m-diameter rotating table. Fluid slip rings allowed the hot and cold baths supplying fluid to the end walls to be located off the table. There were also active themistor bridges on the turntable for amplification of thermistor probe signals. The amplified signals passed through electrical connections from the rotating table to the stationary reference frame where they were recorded. Before describing the experiments, suitable parameters which govern the dynamics of the flow will be defined. Batchelor (1954) pointed out that the non-rotating analogue of the present experiment can be described by three independent non-dimensional parameters. The first of these is the aspect ratio A = I-I/L
(1)
80 which depends only on the cavity geometry. For all the experiments reported here A = 0.075. The second is the Prandtl number, a property of the fluid and defined as the ratio of the kinematic viscosity to the thermal diffusivity, i.e. (2)
Pr = v / ~
The final parameter is the Rayleigh number which describes the forcing by the applied temperature difference AT between the end walls. This is defined by the relation gaATH 3
(3)
Ra PI(
where a is the coefficient of thermal expansion and g is the gravitational acceleration. Rotation introduces two additional independent variables and therefore requires two additional non-dimensional parameters. The first variable is the Coriolis parameter f (defined as twice the angular velocity) which has been included in an Ekman number Ek = v/(H2f)
(4)
The second additional variable is the width of the cavity, which is not required in the non-rotating case since the flow is then two dimensional. In our case the width is most usefully represented in terms of a rotational Froude number B2f 2 -
(5)
ge~A T H
which is the square of the ratio of the cavity width, B, to the Rossby deformation radius. For each experimental run the tank was filled with water and spun up to solid body rotation. Heating and cooling were then applied by pumping hot and cold fluid through the two end walls. The temperatures of the end walls reached their final steady values (to within 5%) within 100 s after 'start-up'. By this time the heat flux through the two end walls had also reached a value which was maintained (to within a few per cent) throughout the development of the flow. The end-wall boundary conditions throughout the duration of any given experiment were, therefore, constant temperatures and a constant heat flux into and out of the working fluid. These two heat fluxes were equal and the mean temperature of the water remained constant. Aspects of the transient flow in the rotating cavity configuration have been described by Condie and Ivey (1988). That study examined the flow of hot and cold boundary currents into an isothermal environment (t < 100
81 rotation periods). Their description began with the application of heating and cooling at the end walls. Buoyant fluid adjacent to the hot end wall rose vertically and was turned into the interior by the upper horizontal lid (with a symmetrical flow at the cold end). Coriolis forces pushed this current to the right (anti-clockwise rotation) to form a density current along the right-hand boundary, which was unstable to rotationally induced instabilities. When E k > 1 0 - 4 and Fr < 1, waves failed to break due to the damping effects of Ekman layer dissipation or the influence of the opposite side boundary. Conversely, for E k < 10 -4 and Fr > 1 the waves broke to form eddies and eddy pairs which penetrated into the lower layer and interacted with neighbouring eddies, thus broadening the current. This paper considers the longer term development of the flow and the establishment of the density field in the cavity. The steady-state flow will be described briefly, although a more detailed account of this aspect will be given in a later paper. 3. EVOLUTION OF THE FLOW FIELD Temperature sections were measured by vertically traversing a rack of 15 fast-response thermistors equally spaced across the tank. Cross-sectional profiles were recorded at regular intervals throughout the development of the flow at four positions along the length of the tank. Examples of these profiles in Fig. 1 show the development of the temperature field for a single set of conditions. This type of data was used along with observations of dye streaks released at the end walls (Fig. 2), to determine how the flow developed. The description which follows is for the conditions in Fig. 1; however, the basic character of the flow was similar provided Fr > 1. When this condition was not satisfied the flow approached that observed in a non-rotating cavity with differentially heated and cooled end walls (Patterson and Imberger, 1980). As the Froude number was increased, horizontal length-scales, such as the width of strong current regions and diameter of eddy structures, decreased. Column (a) in Fig. I shows the temperature structure at the four crosssections in the cavity (viewed from the heated end), 50 rotation periods after the heating and cooling were applied. Warm and cold intrusions with almost linear internal stratification, flowed with a side boundary on their right-hand side. The second profile in this column indicates that the warm current was particularly wide at that section, reflecting the presence of a large amplitude wave or eddy. The dimensions of the warm current were slightly larger than those of the cold bottom current. This resulted from the higher buoyancy flux associated with the warm flow, owing to the non-linear equation of state of water. The magnitude of this effect should not be underestimated, since
82
I
(a)
(b)
Fig. 1. C r o s s - s e c t i o n s o f the t e m p e r a t u r e s t r u c t u r e in the cavity w h e n Ra = 2.1 × 1 0 9 , Ek = 2.4 × 10 5 a n d Fr = 71.0. T h e c r o s s - s e c t i o n s are viewed f r o m the h o t e n d a n d i s o t h e r m s are at 1.0 o C intervals. T h e c o l u m n s c o r r e s p o n d to (a) t = 50, (b) t = 100, (c) t = 200 r o t a t i o n p e r i o d s and (d) s t e a d y state, while the rows c o r r e s p o n d to x / L = 0.28, 0.43, 0.65 a n d 0.80, respectively (where x is m e a s u r e d from the hot e n d wall). T h e d i r e c t i o n s of s t r o n g c u r r e n t s are also indicated.
83
(c)
Fig. 1 (continued).
(d)
84
Fig. 2. Plan views of the dyed hot intrusion when R a = 8 . 7 × 1 0 ~, E k = 2 . 4 × 1 0 5 and Fr = 71.0. The hot end wall is on the left-hand side and the cold end wall on the right. The clock shows the number of rotation periods since the heating and cooling were initiated. Rotation is counter-clockwise.
for the conditions described in Fig. 1, the coefficient of thermal expansion at the temperature of the hot end wall was double that at the cold end. This is also reflected by the stronger temperature gradients within the cold current. After 100 periods (column (b)) the growth and interactions of eddy structures had significantly broadened the currents. This is also evident in a photograph taken of the warm intrusion at 100 rotation periods (Fig. 2).
85 0.02
0.01
0.0(
0.0
012
014
0.'6
0'.8
.0
x/L Fig. 3. The cross-cavity mean velocity distribution for the hot intrusion as a function of distance along the cavity. Each datum point was determined by plotting the position of the warm front (as defined by the spreading of the dye cloud) at the appropriate value of x, as a function of time. For the non-dimensionalizationused in the figure, the data from runs with different conditions collapsed on to a single line. The best-fit slope of this line gave a non-dimensional cross-cavity velocity representative of all the parameter range except F r < 1.
Condie and Ivey (1988) noted that when the initial intrusions flowed into the cavity, they compressed columns of the stationary fluid. To conserve potential vorticity, a counter-flow developed beneath the hot intrusion that carried cold fluid away from the base of the cold end wall (i.e., along the bottom right-hand corner of the cross-section). It is evident in the fourth cross-section in column (b), that additional fluid was supplied to this flow through spreading of the original cold current away from the opposite side wall. Much of the buoyancy in the hot boundary current was transported across the cavity. In fact, Condie and Ivey (1988) pointed out that at the higher rotation rates ( f > 1.0 s -1) the nose of the current almost stagnated before reaching the end of the cavity, indicating that the cross-cavity flux exceeded the along-shore flux. The dynamics of the cross flow were of particular interest, since they appeared to influence the flow throughout the cavity. While eddies were extremely prominent in the region, the mean cross-stream velocities were essentially constant, suggesting an advective rather than a turbulent diffusive transport process. It is, however, quite feasible that the Reynolds stresses associated with the eddy field significantly contributed to the advection. The mean cross-cavity velocity at the surface, as determined by the spreading of dye clouds, is plotted in Fig. 3 as a function of the position along the cavity. There was relatively little transport near the hot end wall where waves had not yet broken. However, as Fig. 3 shows, the cross-stream component of the hot flow was substantial in the central region of the tank. Further downstream, where less buoyant fluid was available, the transport again dropped to small values. By 200 rotation periods (column (c) of Fig. 1 and the second photograph
86 in Fig. 2) the cross-cavity flow had considerably altered the nature of the flow field. When the cross-flow impinged near the centre of the opposite side wall, it produced two inertial jets along this boundary. The weaker jet went to the left and returned a small amount of warm fluid to the hot end. The stronger jet carried the remainder of the warm fluid to the right towards the cold end wall (this feature is quite distinct in the last two profiles in column (c)). This region of the flow may be analogous to that analysed by Whitehead (1985), in which a baroclinic jet impinged on a wall in a rotating fluid. In the present experiment, the cross-cavity flow struck the wall almost normally and active mixing by meso-scale eddies suggests that the potential vorticity distribution would have been relatively uniform. For a constant potential vorticity jet normal to the wall, Whitehead's model indicates that approximately 70% of the fluid goes to the right and 30% to the left. This result is consistent with qualitative observations of dye transports in the cavity, even though the turbulent nature of the cross-cavity flow and associated non-linear effects complicate the exact description of the flow in our experiments. As stratification in the cavity developed towards the steady-state profiles (column (d) of Fig. 1), the horizontal density gradients were reduced. This process acted to weaken the buoyancy driven components of the flow. Dye streaks showed that the larger scales of motion became more barotropic, while the inertial jets became more significant. The weakening of baroclinic components of the flow was accompanied by a decay in the cold flux from the base of the left side (viewed from the hot end) of the cold end wall. A somewhat surprising observation was that the flow from the right-hand side o f the cold end, thereafter, carried all of the cold fluid. This cold intrusion moved towards the centre of the cavity (with the side wall to its left when looking in the direction of flow), then meandered across to the opposite boundary (the left-hand side wall when viewed from the hot end). After the cold water reached the left-hand wall, most of it continued on to the hot end beneath the weak inertial warm jet flowing in the same direction.
i
coLD
HOT
m
Hot Fluid
r'--]Cold Fluid
Fig. 4. A schematic representation of the mean flow observed in the cavity at steady state when Fr > 1. The wide arrows represent the slow turbulent cross-cavity flow.
87 The barotropic component of the steady-state mean flow can therefore be described as two large-scale gyres. This is represented schematically in Fig. 4. A cyclonic gyre filled approximately half of the cavity closest to the hot wall and an anticyclonic gyre occupied the other half. This aspect of the steady-state flow is not yet fully understood and is the subject of continuing investigation. 4. DEVELOPMENT OF THE STRATIFICATION The aim of this part of the study was to determine the time-scale for development of the density structure in the cavity and investigate its dependence on the Ekman, Froude and Rayleigh numbers. A scaled estimate for this quantity in a non-rotating cavity was derived by Patterson and Imberger (1980) based on a simple 'filling box' process in which the interior region was assumed to fill with heated fluid by horizontal layering. This process has been studied experimentally in a cavity with only one differentially heated end wall (Worster and Leitch, 1985). Such a configuration more closely models the idealized filling box, because of the absence of a sink for the buoyant fluid. However, experiments by Yewell et al. (1982), along with temperature sections such as those in Fig. 1, suggest that a similar type of process may be operating in the case of two differentially heated end walls. Patterson and Imberger showed that the volume flux p u m p e d through the thermal boundary layers of the end walls of a cavity, scales as BxRa a/4. The time-scale to p u m p all the fluid in the cavity through the boundaries is therefore 1"
HL xRa]/4
(6)
This quantity will be referred to as the 'heat-up' time. Cross-sectional profiles, such as those in Fig. 1, were used to determine the adjustment time-scales. Ten profiles recorded after the system had been allowed to reach steady state, were averaged to determine the mean steadystate temperature structure. The standard deviation of the instantaneous temperature values from the steady-state values was then calculated for each cross-sectional profile taken during the heat-up process. More precisely, if the two-dimensional matrix of temperature values for a given time and cross-section has dements T~j and the steady-state values at the same positions are then the standard deviation between the profiles is defined as
Sis, ( Y', E [Tij-Sij[2)a/2 n
¢$T=
m
i=lj=l
(7) nm
88
10 0 w *
u
8T 10-1
• a • • • + • x g
=,
i7 @
I.
•
i..i
I
I
0.0
Fr 0.33 0.33 0.33 0.33 8.28 8.28 8.28 8.28 32.6 32.6 32.6 32.6 71.0 71.0 71.0
Ek 3.6x104 3.6x104 3.6x104 3.6xl 0 .4 7.1x10 ~ 71x105 7.1x10 5 7.1xt0 ~ 3.6x10 ~ 3.6x105 3.6x10 ~ 3.6x10S 2.4x10 ~ 2.4x105 2.4x10 S
•
!! :• I"
1 0 -2
x/I. 0.28 0.43 0.65 0.80 0.28 0.43 0.65 0.80 0.28 0.43 0.65 0.80 0.28 0.43 0.80
1.0
"
•
.
°
I
o
,
.
20
t/'c Fig. 5. The standard deviation of the developing temperature structure from the mean steady-state temperature structure (as defined by eqn. (7)) as a function of time (non-dimensionalized by the applied temperature difference and the heat-up time, respectively). Runs cover a range of Ekman numbers, Froude numbers and positions along the cavity.
where n was the number of thermistors (n = 15) and m was the number of readings taken by each thermistor in a single profile (m = 100). Parameter 8T was calculated at the four cross-sections in the tank throughout the development of the flow. This procedure was carried out over the entire experimental parameter range. Values of ST, normalized by the end-wall temperature difference AT, are plotted as a function of time (non-dimensionalized by the heat-up time) for a range of Ekman and Froude numbers in Fig. 5 and for two Rayleigh numbers in Fig. 6. Although there is not a simple exponential behaviour, the data for any given run can be fitted satisfactorily by two exponential curves, corresponding to the early and later development of the stratification. In other words, the adjustment process can be divided into two regimes, each of which is described by an exponential time-scale. Attempts to fit the data to a
89
10
fiT
[] ,t, a *
x/L 0.28 0.43 0.65 0.28
Ra 5.8x10 ~ 5.8x10 B 5.8x10 e 2.1x109
[] a •
0.43 0.65 0.80
2.1x10 Q 2.1x109 2.1x109
10
AT
,..t
IBm
[] []
a 2
[]
[]a a
[]
•
II
a 4~ Ju
,
0.0
I
,
1.0
.
•
,
~2.0
t/~ Fig. 6. Standard deviation from the mean steady-state temperature structure as a function of time, for a range of Rayleigh numbers and positions along the cavity.
power law yielded unsatisfactory results. The trends for small times (t < 0.5"r) will be discussed first. The initial adjustment is well described by the exponential law ~ T = C A T e -kt/'~
(8)
where C and k are constants. Figure 5 indicates that the time-scales are independent of rotation rate, while Fig. 6 shows that the non-dimensionalization accurately describes the dependence on the end-wall forcing for the range of AT investigated. There is also no distinguishable dependence of any of the data on the position along the tank. Measurements of the slopes and intercepts in Figs. 5 and 6 give k = 1.23 + 0.17 and C = 0.15 + 0.01 for the early development of the temperature structure. This value of k corresponds to a time-scale of (0.81 _ 0.11)'r. Since the time-scale for the early development of the stratification scales with the heat-up time, it is reasonable to assume that the adjustment process is adequately approximated by the filling-box model. The transition point
90 1.0i
, , , - "
0,8 Z 0.6 H 0.4
0.2 0.0-0.5 -0.3 -0.1 0.1
0.3
0.5
2(T-To)/AT Fig. 7. A vertical temperature profile within the cavity at the transition time between the two adjustment regimes (t = 0.54T). The broken line represents the steady-state structure. With the temperature non-dimensionalized in this way, the profile is representative of the entire parameter range investigated. T0 is the average of the two end-wall temperatures.
f r o m the filling b o x to a s e c o n d a d j u s t m e n t process, as given b y the i n t e r s e c t i o n of fitted curves for the two regimes, o c c u r r e d at t = (0.54 + 0.10)~-. F o r c o m p a r i s o n , the sections in Fig. 1 c o r r e s p o n d to (a)t = 0.045~', ( b ) t = 0 . 0 9 ~ - a n d ( c ) t = 0 . 1 8 " r . F i g u r e 7 shows the level of s t r a t i f i c a t i o n outside the b o u n d a r y c u r r e n t s at t = 0.54~-, d e m o n s t r a t i n g t h a t the filling-box p r o c e s s e s t a b l i s h e d m u c h of the final d e n s i t y s t r u c t u r e within the cavity. T h e later a d j u s t m e n t p r o c e s s c a n t h e r e f o r e b e r e g a r d e d as a ' f i n e t u n i n g ' of the density field. T h e s e c o n d a d j u s t m e n t r e g i m e was significantly slower t h a n the first, b u t a g a i n c o u l d r e a s o n a b l y b e d e s c r i b e d b y an e x p o n e n t i a l law. T h e r a p i d i t y of the transition b e t w e e n the two r e g i m e s v a r i e d significantly, l e a d i n g to a large
50000
Timescale (seconds)
.1
1
10
100
Fr Fig. 8. The exponential time-scales for the second density adjustment process (t > 0.5~-), as a function of rotating Froude number. The error bars represent one standard deviation, calculated from the values obtained at different cross-sections in the cavity. The Rayleigh number was constant for these runs at Ra = 2.1 × 10 9.
91 degree of scatter in 8T in the second regime. However, as noted in the previous paragraph, the transition point was relatively constant. After the transition, the rate of change of 8T with time continued to be independent of rotation rate. Since this result is not clear from inspection of Fig. 5, the data has been replotted in Fig. 8, on which the time-scales are shown for the second adjustment regime as a function of rotating Froude number. Taking into account the scatter of the data, there is no discernible trend, confirming that within the parameter range studied adjustment times were independent of rotation. The data in Fig. 6 indicate that the second adjustment process had a relatively strong dependence on Rayleigh number. As noted by Patterson and Imberger (1980), the motion in the core region of the cavity during the approach to steady state is extremely complex, making detailed scaling of behaviour very difficult. However, in dimensional terms, the exponential time-scales were (6.8 ___1.5) x 104 S for Ra = 5.8 x 108 and (2.33 + 0.55) x 104 s for Ra = 2.1 x 109. Although the parameter range was insufficient to determine empirically the Rayleigh number dependence, these results suggest that the second stage adjustment time-scales may have a close to linear dependence on Ra. This contrasts with the weaker Ra 1/4 dependence of the first regime. A separate experiment was conducted in which the applied temperature difference, AT, was increased after the flow had been allowed to reach steady state. The transition was, therefore, from a steady-state flow with Ra = 5.8 x 108 to a second steady state with Ra = 2.1 × 109. This experiment yielded a single exponential time-scale of 1.9 × 104 s for the duration of the adjustment process. This time-scale agrees with the value of (2.33 + 0.55) X 104 S, given in the previous paragraph for the second adjustment when Ra = 2.1 x 109, indicating that the second, much longer, adjustment regime is relevant to the problem of modification of existing stratification by external forcing. Perhaps not a surprising result since strong stratification was present throughout the process. Clearly, if the initial stratification had been sufficiently weak, the filling-box mechanism would have operated at early times. Therefore, in summary, the initial conditions (strength of the initial stratification) determine the type of adjustment process, while the new boundary conditions determine the actual value of the adjustment time-scale. 5. DISCUSSION This paper extends the work of Condie and Ivey (1988), by considering the long-term development of the convection experiment to a steady-state situation. It has therefore incorporated the processes associated with the development of stratification, as well as some of the effects of a stratified
92
environment on boundary currents. These are important matters for the thermohaline component of the large-scale circulation of the ocean. The most restrictive feature of the study with respect to oceanic applications is that the experiments were conducted on an f-plane rather than a B-plane (simply introducing a sloping bottom in the presence of stratification does not simulate a B-plane). It is clear, however, that a knowledge of f-plane dynamics is an essential component of any understanding of large-scale ocean circulation. The study deals particularly with the heat-up time-scales of the convective flow, both from an isothermal state and as a perturbation of a steady state. These processes can be described in terms of two adjustment modes. While the fluid is only weakly stratified (relative to the gradients associated with the forcing), a filling-box process operates on an exponential time-scale of (0.81 +0.11)'r. The second mode then takes over. This is essentially a fine-tuning of the density field and the relevant time-scale is dependent on the Rayleigh number. Perhaps the most significant finding is that both adjustment modes are independent of rotation. The results summarized in the previous paragraph may be relevant to a range of geophysical phenomena. Common examples are the adjustment of systems such as large lakes, seas, gulfs and oceans to seasonal variations in conditions of atmospheric temperature, evaporation rate, precipitation and fresh-water run-off. While the response of the oceans to long-term climatic change is a very complex problem (see for example Broeker, 1987), some of the experimental results may be relevant to this problem. The formation of two large-scale gyres at steady state is particularly interesting, but not yet well understood. This aspect does, however, demonstrate, that a complex density flow can be produced by simple boundary conditions. An observation which may have significant implications for the role of buoyancy in the large-scale circulation of the ocean. ACKNOWLEDGEMENTS
I would like to thank Dr Ross Griffiths, Dr Gregory Ivey and Professor Stuart Turner for advice throughout this study, as well as their comments on the manuscript. Thanks also to Dr Roger Nokes who wrote the data logging and plotting routines. Mr Pat Travers constructed the convection tank, Mr Ross Wylde-Browne processed the photographs and Mr Derek Corrigan gave other valuable technical aid and advice.
93 REFERENCES Batchelor, G.K., 1954. Heat transfer by free convection across a closed cavity between vertical boundaries of different temperatures. Q. J. Appl. Maths., 12: 209. Bejan, A., 1984. Convective Heat Transfer. Wiley, New York. Broeker, W.S., 1987. Unpleasant surprises in the greenhouse. Nature, 328: 123-126. Buzyna, G., Pfeffer, R.L. and Kung, R., 1984. Transition to geostrophic turbulence in a rotating differentially heated annulus of fluid. J. Fluid Mech., 145: 377-403. Condie, S.A. and Ivey, G.N., 1988. Convectively driven coastal currents in a rotating basin. J. Mar. Res., 46: 473-494. Hide, R., 1958. An experimental study of thermal convection in a rotating liquid. Philos. Trans. R. Soc., Ser. A, 250: 441-478. Hide, R. and Mason, P.J., 1975. Sloping convection in a rotating annulus. Adv. Phys., 24: 47-100. Hignett, P., 1985. Characteristics of amplitude vacillation in a differentially heated rotating fluid annulus. Geophys. Astrophys. Fluid Dyn., 31: 247-281. Luyten, J. and Stommel, H., 1986. Gyres driven by combined wind and buoyancy flux. J. Phys. Oceanogr., 16: 1551-1560. Patterson, J.C. and Imberger, J., 1980. Unsteady natural convection in a rectangular cavity. J. Fluid Mech., 100: 65-86. Speer, K.G. and Whitehead, J.A., 1988. A gyre in a non-uniformly heated rotating fluid. Deep-Sea Res., in press. Stommel, H., Arons, A.B. and Failer, A.J., 1958. Some examples of stationary flow patterns in bounded basins. Tellus, 10: 179-187. Sugimoto, T. and Whitehead, J.A., 1983. Laboratory models of bay-type continental shelves in the winter. J. Phys. Oceanogr., 13: 1819-1828. Warren, B.A., 1981. Deep circulation in the world ocean. In: B.A. Warren and C. Wunsch (Editors), Evolution of Physical Oceanography. MIT Press, Cambridge, MA, pp. 6-41. Whitehead, J.A., 1985. The deflection of a baroclinic jet by a wall in a rotating fluid. J. Fluid Mech., 157: 79-93. Worster, M.G. and Leitch, A.M., 1985. Laminar free convection in confined regions. J. Fluid Mech., 156: 301-319. Yewell, P., Poulikakos, D. and Bejan, A., 1982. Transient natural convection experiments in shallow enclosures. Trans. ASME, J. Heat Transfer, 104: 533-538.