- Email: [email protected]
Laboratory Experiments on Dipole Structures in a Stratified Fluid
591
LABORATORY EXPERIMENTS ON DIPOLE STRUCTURES IN A STRATIFIED FLUID G.J.F. VAN HEIJST and J.B. FLOR Institute of Meteorology and Oceanography University of Utrecht Princetonplein 5 3584 CC Utrecht, The Netherlands
ABSTRACT Coherent flow structures (vortices, dipoles) as occurring in geophysical fluid systems have been investigated in the laboratory. This paper concentrates on dipole experiments, in which locally 3D turbulence is generated in a continuously stratified fluid, viz. by horizontal injection of a fixed volume of fluid during a short period of time. The cone-shaped region of 3D turbulent fluid collapses under gravity (due to the stratified ambient fluid), resulting in a nearly 2D flow confined within a thin layer of fluid. A s can be expected from 2Pturbulence theory, one observes a spectral flux of energy towards larger scales, and the collapsed fluid eventually gets organized in a dipole-shaped structure. The formation process leading to these dipoles as well as their dynamics will be discussed. Two experiments were carried out in order to study interaction properties of dipoles, viz. a head-on collision between two dipoles, and a collision of a single dipole against a solid wall. Some preliminary results are presented. 1.
INTRODUCTION Coherent flow structures are typical features of geostrophic or quasi two-
dimensional turbulence, and these structures play an important part in largescale geophysical flows. Essential elements in such flows are rotation and/or stratification, which tend to make the motion two-dimensional. Flow coherency has been observed in the atmosphere as well as in the oceans, and has been studied theoretically (both analytically and numerically) and by laboratory experiments. In previous studies it was found that coherent structures in quasi two-dimensional flows can take the appearance of 'monopoles' (i.e. single vortices) and 'dipoles' (i.e.
vortex pairs), and recently evidence was obtained
for the existence of tripolar flow structures, see e.g. Legras, Santangelo and Benzi (1988) and Kloosterziel and Van Heijst (1989a,b).
The present study
concentrates on a laboratory experiment in which a dipole structure is generated by turbulent horizontal injection of a small fluid volume i n a stratified, nonrotating ambient fluid. The set-up of this experiment is relatively simple, and it provides valuable information about the dipole formation as well as the flow dynamics of the ultimate dipole structure.
592
Observations by satellites have revealed the occurrence of dipolar flow patterns in the ocean. Fedorov and Ginsburg (1986) describe some examples of oceanic dipoles in the Black Sea and the Sea of Okhotsk, and the satellite observations presented by Ahlniis, Royer and George (1987) reveal an abundancy of dipolar structures at the edges of the Alaska Coastal Current where it separates from Kayak Island, These patterns have horizontal dimensions of typically 20 km. Other examples of oceanic dipoles can be found in a recent paper by Stockton and Lutjeharms (1988), which describes observations of dipolar vortex structures on the Benguela upwelling front. Blocking events in the atmosphere are usually characterized by dipolar patterns in the isobars. Because of the large horizonal scale of such dipolar systems, the B-effect plays a non-negligible role in their dynamics. An analytical solution for a coherent dipole structure on a +plane was given by Stern (1975), who named it 'modon'. Motivated by geophysical applications, a number of laboratory experiments on rotating fluids were performed, in which dipolar flow structures were
observed to occur. For example, Griffiths and Linden (1981) found evidence of dipoles in a laboratory study of an unstable density front in a rotating fluid, whereas Kloosterziel and Van Heijst (1989a) observed dipole formation in rotating fluid experiments on unstable anticyclonic barotropic vortices, These experiments demonstrate that dipoles can be generated by a rather non-descript forcing. This was also observed in the experiment described by Flierl, Stern and Whitehead (1983):
they found that the turbulent flow induced by a pulsed
horizontal jet in a homogeneous rotating fluid becomes two-dimensional (due to the rotation of the system) within one or two rotation periods, and a subsequent organization into a single dipole structure was observed. Two-dimensionality of the flow can also be established by other forces than those associated with rotation or stratification. For example, magnetic forces have similar effects on electrically conducting fluids, and the occurrence of coherent structures has indeed been observed in such fluids. Solitary vortices are common features in plasma flows and their dynamics have been studied by numerous investigators (see e.g. Nycander (1988)).
Makino, Kamimura and Taniuti (1981) and
Interesting laboratory experiments on dipole structures in
magnetohydrodynamic flows were recently performed by Nguyen h c and Sommeria (1988).
The basic set-up of their experiment was a thin layer of mercury
subjected to a homogeneous magnetic field perpendicular to the fluid layer; the dipoles were generated by an electric current pulse between two electrodes placed in the fluid at some distance apart. The dynamics of the dipole structures thus created were investigated by visualization of the flow with small tracer particles, which allowed accurate measurement of the flow field and its vorticity. Coherent flow structures were also observed in the soap film
593
experiments by Couder and Basdevant (1986).
As a consequence of the geometry,
turbulence generated in a soap film is essentially two-dimensional, and the formation of single and dipolar vortices in a turbulent wake behind an obstacle moving through the film was indeed observed. The present study concentrates on an experiment with a turbulent jet in a stratified, non-rotating fluid. By horizontally injecting a small volume of fluid a region of three-dimensionally turbulent flow is created, which collapses due to gravity forces associated with the ambient stratification. The resulting flow quickly becomes two-dimensional, and the subsequent formation of coherent structures can be clearly observed, eventually leading to a single dipole structure. In these experiments the flow was visualized by adding dye to the injected fluid or by putting neutrally buoyant tracer particles in the stratified fluid. The set-up of this laboratory experiment is extremely simple, and allows one to study in detail (1) the transition process from threedimensional turbulence to two-dimensional flow, ( 2 ) the subsequent emergence of coherent flow structures, and ( 3 ) the dynamics of the eventually arising dipole structures. It has been found by many investigators that dipolar flow structures are very robust and that they behave in a soliton-like manner in interactions with other dipoles (e.g.
Makino et al. (1981), McWilliams and Zabusky (1982),
Couder and Basdevant (1986)),
and experiments on colliding dipoles to be
discussed in this paper confirm these results. Dipoles were also studied in a similar experiment by Voropayev and Fillipov (1985).
However, their experiment
is essentially different from the one discussed in the present paper, in that in their case fluid was injected continuously rather than in a short pulse, and the injection produced a smooth laminar flow rather than a turbulent jet. The paper is organized as follows: the set-up of the laboratory experiment as well as the observed flow phenomena are described in Section 2. Quantitative observations of the flow before, during and after the gravitational collapse of the turbulent region are presented in Section 3. Interaction properties of dipoles are considered in Section 4 , attention being focussed on two particular cases, viz. a head-on collision between two dipoles, and a collision of a single dipole against a solid wall. Finally, a summary of the results is given in Section 5.
2. LABORATORY EXPERIMENT
In order to study the transition from three-dimensional turbulence to twodimensional turbulence and the subsequent emergence of coherent structures, experiments were carried out with a horizontal turbulent jet in a stratified
594
fluid, The basic set-up of this laboratory experiment is depicted schematically in Fig. 1. A square perspex tank of dimensions 1 m
x
1 m x 0.3 m i s filled with
a linearly stratified salt solution. During a short period of time (At) some fixed volume (Q) of fluid is injected horizontally through a nozzle of diameter d. The density of the injected fluid exactly matches the density of the ambient stratified salt solution at the level of the nozzle, so that a horizontal jet flow is created. In the experiments the injection velocity U is taken large enough to ensure that the jet is turbulent. Typical experimental conditions are: U
FI
1 m/s
,d
a
m, and the kinematic fluid viscosity V =
the Reynolds number Red
=
m2/s, so that
Ud/v based o n the nozzle diameter has a value of
typically lo3. The flow was visualized by adding dye to the injected fluid, and the subsequent deformation of the injected dye cloud was recorded both from above and from the side by remotely controlled photocameras. Because of the short duration of the injection (At of a ' b l o b '
<
1
s),
the induced flow has the appearance
of turbulent flow in an otherwise motionless stratified environment. QAt
Fig.
1. The basic set-up of the laboratory experiment.
A s in a non-stratified environment, this jet
-
or I b l o b '
-
initially has a
conical shape with a circular cross-section, and turbulent entrainment of ambient fluid is observed to occur, both horizontally and vertically. In the next stage, this three-dimensionally turbulent region is observed to collapse under gravity, as a result of the ambient stratification: in this stage the vertical velocity component is suppressed, and the flow shows a transition from 3D to 2D turbulence. A well known property of two-dimensional flow is the so-
called 'inverse energy cascade': in contrast to three-dimensional turbulence, kinematic energy is in two-dimensional turbulence transferred from smaller to larger scales ( s e e e.g. Batchelor (1969) and Salmon (1982)),
leading to an
apparent organization of the flow into larger coherent structures. In this experiment the two-dimensional flow is observed to get organized in a welldefined flat dipole structure, as illustrated by the sequence of photographs presented in Fig. 2. Once this dipole is formed, it moves slowly forward without
595
Fig. 2. A sequence of plan-view photographs showing the evolution of the turbulent volume of h o r i z o n t a l l y i n j e c t e d f l u i d . The p i c t u r e s were taken ( a ) 3 s , (b) 12 s , ( c ) 50 s, and (d) 180 s a f t e r the i n j e c t i o n was stopped. Experimental parameters are: Q = 2.44 ml, A t = .I9 s , N = 2 . 2 7 s.
596
any d r a s t i c changes i n i t s shape. T y p i c a l v a l u e s of i t s speed of advance V and
i t s d i a m e t e r D are: V
m / s and D
10-l m y so t h a t t h e Reynolds number
based on D t a k e s a t y p i c a l v a l u e of l o 2 . The d i p o l e f o r m a t i o n was s t u d i e d i n some 1 5 e x p e r i m e n t s , i n which t h r e e e x p e r i m e n t a l parameters were v a r i e d : t h e i n j e c t e d volume Q ranged from 2.44 m l t o 14.6 m l ,
t h e speed of i n j e c t i o n U through t h e n o z z l e was v a r i e d between 2.1
m / s and 4.4 mfs, and t h e v a l u e of t h e buoyancy f r e q u e n c y N of t h e s t r a t i f i c a t i o n ranged from 1.2 /s t o 2.52 /s. exactly a t half-depth,
I n a l l t h e e x p e r i m e n t s t h e n o z z l e was p l a c e d
and i t s d i a m e t e r measured 1.9 mm.
During and a f t e r t h e c o l l a p s e of t h e t u r b u l e n t r e g i o n , i n t e r n a l waves a r e observed t o propagate away from t h e c o l l a p s i n g f l o w r e g i o n , t h u s i n d u c i n g a p p r e c i a b l e motions in t h e ambient f l u i d . Although t h e s e waves are r e f l e c t e d by t h e s o l i d t a n k walls, t h e y a p p e a r t o decay r a t h e r q u i c k l y , w i t h o u t s e r i o u s l y a f f e c t i n g t h e dipole s t r u c t u r e t h a t a r i s e s i n t h e collapsed region. N e v e r t h e l e s s , t h e i n t e r n a l waves p l a y a n i m p o r t a n t r o l e i n t h e f o r m a t i o n p r o c e s s of t h e d i p o l e , as w i l l be p o i n t e d o u t i n t h e n e x t s e c t i o n ,
3. LABORATORY OBSERVATIONS AND RESULTS An i n t r i g u i n g q u e s t i o n concerns t h e f o r m a t i o n of c o h e r e n t s t r u c t u r e s a f t e r t h e c o l l a p s e of t h e i n j e c t e d t u r b u l e n t f l u i d r e g i o n : i t i s obvious t h a t i n i t i a l l y t h e f l o w is e s s e n t i a l l y three-dimensional,
i t s appearance b e i n g
i r r e g u l a r , w h i l e t h e u l t i m a t e two-dimensional f l o w h a s a r e g u l a r , well-organized s t r u c t u r e , I n o r d e r t o g a i n i n s i g h t i n t o t h e f o r m a t i o n of t h e u l t i m a t e d i p o l e s t r u c t u r e , c a r e f u l o b s e r v a t i o n s were c a r r i e d o u t by r e c o r d i n g t h e dyed c l o u d both from above and from t h e s i d e by photocameras and by a v i d e o camera. The s c e n a r i o of e v e n t s l e a d i n g t o t h e e v e n t u a l d i p o l a r f l o w p a t t e r n is s k e t c h e d i n Fig. 3, showing 5 s t a g e s i n t h e f o r m a t i o n p r o c e s s as observed from above. I n i t i a l l y , t h e t u r b u l e n t j e t v e r y much r e s e m b l e s a h o r i z o n t a l j e t i n a nons t r a t i f i e d ambient f l u i d : b e f o r e t h e c o l l a p s e t h e j e t o c c u p i e s a c o n i c a l volume and t h e t u r b u l e n t motion i n s i d e i t is e s s e n t i a l l y three-dimensional
(Fig. 3a).
I n t h e n e x t s t a g e , t h e t u r b u l e n t c l o u d c o l l a p s e s under g r a v i t y , and i n t e r n a l waves are observed t o p r o p a g a t e away from a n a p p a r e n t c e n t r e l o c a t e d somewhere i n s i d e t h e i n j e c t e d f l u i d volume. T h i s is s k e t c h e d i n Fig.
3b, where t h e
i n t e r n a l wave crests are i n d i c a t e d by t h e arcs. The dominant mode of t h e i n t e r n a l wave motion is t h e m = 2 mode, w i t h a h o r i z o n t a l p l a n e of symmetry a p p r o x i m a t e l y a t t h e i n j e c t i o n l e v e l . The i n t e r n a l waves p r o p a g a t e away from t h e c o l l a p s i n g r e g i o n and t h i s i m p l i e s t h a t a l a r g e p o r t i o n of t h e k i n e t i c energy a s s o c i a t e d w i t h t h e i n i t i a l j e t f l o w is r a d i a t e d away from t h e mixed r e g i o n . Due
597 t o r e f l e c t i o n a t t h e t a n k walls, complicated wave motions can be observed throughout t h e tank, b u t t h e s e appear t o decay r e l a t i v e l y q u i c k l y , a p p a r e n t l y w i t h o u t a f f e c t i n g t h e formation of c o h e r e n t flow s t r u c t u r e s i n t h e c o l l a p s e d region.
before collapse:
apparent centre of radiation
during collapse:
+
-
(b)
internal wave crests
after collapse:
Fig. 3. Schematic plan-view drawing of t h e p r i n c i p a l s t a g e s in t h e d i p o l e formation: ( a ) t h e 3D t u r b u l e n t jet b e f o r e c o l l a p s e , ( b ) t h e o c c u r r e n c e of i n t e r n a l waves d u r i n g (and a f t e r ) t h e c o l l a p s e , ( c ) t h e emergence of two secondary d i p o l e s , which move i n l o o p i n g p a t h s ( d ) , and ( e ) t h e u l t i m a t e d i p o l e structure. During t h e c o l l a p s e t h e l e a d i n g edge of t h e i n j e c t e d f l u i d r e g i o n h a s t h e appearance of a s h a r p f r o n t , which is caused by t h e 'pushing' i n t e r n a l waves i n t h a t region.
e f f e c t of t h e
The wave v e l o c i t y is h i g h e r t h a n t h e speed of
advance of t h e i n t r u s i o n nose, and t h i s probably r e s u l t s in a r e l a t i v e motion d i r e c t e d away from t h e symmetry a x i s as i n d i c a t e d by t h e arrows i n Fig.
3b. When
t h e c o l l a p s e i s completed, t h e motion h a s become almost h o r i z o n t a l , and t h e s i z e of t h e small e d d i e s g r a d u a l l y increases. An i n t e r e s t i n g f e a t u r e i n t h i s s t a g e is t h e occurrence of two secondary d i p o l e s t r u c t u r e s moving away from t h e symmetry a x i s of t h e dyed c l o u d , as i n d i c a t e d in Fig. 3c. These small c o h e r e n t s t r u c t u r e s
598
f o l l o w a curved p a t h a s s k e t c h e d by t h e dashed a r r o w s , u n t i l t h e y meet a t t h e a x i s , some d i s t a n c e behind t h e l e a d i n g edge of t h e i n t r u d i n g b l o b , see Fig. 3d. During t h i s s t a g e i n t e r n a l waves are s t i l l o b s e r v e d , and dyed f l u i d moves a t a r e l a t i v e l a r g e speed a l o n g t h e a x i s , t h u s m a i n t a i n i n g t h e s h a r p f r o n t . Due t o t h i s s u b s t a n t i a l motion a l o n g t h e a x i s , t h e ' i n n e r ' p a r t s of t h e secondary d i p o l e s a r e advected towards t h e nose, g r a d u a l l y l o o s i n g t h e i r i d e n t i t y ; on t h e o t h e r hand, t h e ' o u t e r ' p a r t s of t h e secondary d i p o l e s a r e l e f t behind, and, being deformed by t h e s h e a r i n g motion i n t h e wake, they f i n a l l y end up as e l o n g a t e d dye p a t c h e s , as c l e a r l y v i s i b l e i n F i g s . 2c and 2d. In t h e u l t i m a t e s t a g e t h e i n t e r n a l waves have decayed, and t h e f l o w h a s o b t a i n e d t h e appearance
of a slowly advancing symmetric d i p o l e ( s e e Fig. 3e). The o c c u r r e n c e of t h e secondary d i p o l e s in t h e c o l l a p s e d s t a g e can be s e e n in Fig. 4 , which i s a photograph t a k e n between t h e photographs shown i n F i g s . 2b and 2c.
Fig. 4. Plan-view photograph showing t h e o c c u r r e n c e of t h e secondary d i p o l e s , The experiment was t h e same as t h a t of Fig. 2 ; t h e photograph was t a k e n 26 s a f t e r t h e i n j e c t i o n was stopped.
In o r d e r t o q u a n t i f y t h e o b s e r v a t i o n s , some c h a r a c t e r i s t i c s of t h e dyed f l u i d cloud were measured from t h e p h o t o g r a p h i c r e c o r d i n g s . The measured q u a n t i t i e s a r e : ( i ) t h e d i s t a n c e L of t h e nose of t h e i n j e c t e d dyed f l u i d r e g i o n r e l a t i v e t o t h e i n j e c t i o n n o z z l e , (ii) t h e maximum width B of t h e dyed f l u i d r e g i o n , and ( i i i ) t h e s i z e A of t h e l a r g e s t i n d i v i d u a l eddy s t r u c t u r e t h a t can be d i s t i n g u i s h e d in t h e dyed cloud. In p a r t i c u l a r i n t h e i n i t i a l phase of t h e
599
experiment the pictures were taken at short time intervals (typically 1 s ) , so that even rapid changes in the flow could be monitored easily. For a characteristic experiment the evolution of the quantities L, B and A I s shown graphically in Fig. 5. In this graph the time is nondimensionalized by the buoyancy frequency N, but the measured quantities L, B and A are expressed in cm. Observations by eye revealed that the transition from 3D to 2D turbulence is almost completed at Nt =- 4 , and that the flow is thus essentially two-
dimensional for Nt
>
4 . In the curves for L, B and A three stages can be
distinguished, which are in Fig. 5 denoted by the Roman symbols I, I1 and 111.
The first stage is characterized by a steady size increase of the vortices, and also the distance L and the width B of the dye blob show a continuous growth. The experimental data for this stage suggest straight lines, which implies a growth with some power of t. Stage I1 begins at approximately Nt = 24 and can be characterized as a stage of 'vortex interaction', in which A increases more rapidly than before. The decrease in B in this stage is caused by the two secondary dipoles
-
as sketched in Fig. 3c
- which now move
backwards to the
axis, thus decreasing the value of the maximum width B of the dye region.
I
I
I
I
I
I
I
I
Fig. 5. Graphical representation of the time evolution of L, B and A (for explanation of these symbols, see the text) as measured from photographic recordings. The straight l i n e s are fitted to the data by eye. Experimental parameters are: Q = 3.66 id,A t = .4 s , N = 1.66 1s.
600
During this stage of vortex interaction, L appears to grow slightly slower than in the initial stage. Once the ultimate dipole is formed, the distance L again
grows proportional to some power of t. In this particular experiment the dipole formation was established at Nt = 90, at which B and A obviously have reached equal values. The horizontal size of the dipole (A,
B) shows a gradual growth
proportional to some power of t. This size increase is most likely caused by frictional effects associated with horizontal shear layers present at the upper and lower sides of the intruding dyed fluid region. The increase can also be inferred from the spiral-shaped dye lines visible in both halves of the ultimate dipole structure, see e.g.
Fig. 2d. Such dye patterns suggest that undyed ambi-
ent fluid flows in at the rear side of the dipole, thus compensating its radial growth. Because of the preliminary character of the laboratory experiments, detailed information about the growth of L, B and A in the various stages of the dipole formation i s not yet available, but will be published in due time.
Fig. 6 . Streakline photograph showing the horizontal motion associated with the dipole structure.
601 Apart from visualization by dye, the horizontal motion in and around the collapsed region was also visualized by bringing small (0.5
m
diameter)
polystyrene tracer particles in the stratified fluid. The density of these particles was 1.038 g/cm3, and in a number of experiments the stratification was chosen such that they settled in a relatively thin layer at approximately halfdepth. The tracer particles were illuminated from the side by a slit light, and against a dark background their motion was recorded from above by a photocamera. A typical example of the streakline photographs thus obtained is shown in
Fig. 6 , from which the dipolar structure in the horizontal flow can be easily observed. By measuring the lengths of the streaks as well as their orientation, horizontal velocity vectors in the flow domain can be determined. Once the flow field is known, it is also possible to calculate numerically the spatial distribution of the vorticity w and the streamfunction Y. This technique is presently being developed, and it i s expected that along this line important information can be obtained about the relationship between Y and w for the dipole structure. In many studies ( s e e e.g.
Batchelor ( 1 9 6 7 ) , Stern ( 1 9 7 5 ) ,
Larichev and Reznik ( 1 9 7 6 ) ) on dipolar vortices
- whether
or not on a p-plane
-
it is assumed that in the interior of the dipole the vorticity w is a linear function of the streamfunction Y. For example, the Lamb dipole as described by 2
2
Batchelor ( 1 9 6 7 ) is found by solving V Y ( r , N = --w = -k Y on a circular region r < a with Y
=
constant on the circle itself; r and 8 are cylindrical
coordinates, and k is a constant. The solution for the streamfunction is given by
with C a constant coefficient and ka
=
3.83 (defining the first zero of J1).
Some streamlines according to this solution are shown in Fig. 7 , and it is clear that this pattern very much resembles the dye line pattern as visible in Fig. 2d. It must be kept in mind, however, that the dye lines are actually streaklines, so that the radial expansion of the dipole causes the dye to flow in spirals rather than in closed curves. The streamlines of the Lamb dipole also
much resemble the flow pattern visualized by the tracer particles, as shown in Fig. 6. It was observed by eye, however, that the dipolar dye pattern occupies a relatively small area, with a horizontal size comparable to the distance between the points of zero velocity in Fig. 6 . The flow outside the dye region is caused by the translation of the dipole structure, in analogy with the motion induced by a cylinder moving through a fluid. Therefore, a correction must be made for the dipole translation in order to be able to make a proper comparison between the velocity field derived from streakline pictures and any theoretical solution
602
for the stationary dipole flow. At the moment appropriate data reduction techniques are being developed, and results will be published in due time. A s mentioned before, it is expected that these measurements will provide
essential information about the relationship between w and Y. Indications of a nonlinear relation between these quantities were found by Legras, Santangelo and Benzi (1988) in a numerical study of forced 2D-turbulence, and the dipole experiments in a horizontal layer of mercury subjected to a vertical magnetic field as performed by Nguyen Duc and Sommeria (1988) yielded similar signs of nonlinearity. It might thus well be that the theoretical dipole models need some
.
refinement
Fig. 7. Streamlines according to Lamb's solution for a dipole.
4. INTERACTION OF DIPOLES Robustness of coherent flow structures is a well-known feature, and in particular the interaction behaviour of dipoles has been studied by a number of investigators. For example, head-on collisions between dipoles were studied numerically by Larichev and Reznik (1976, MeWilliams and Zabusky (1982),
1983),
by llakino et al. (1981),
and by Couder and Basdevant (1986).
by
In some of
these papers attention was also given to overtaking collisions and off-centre head-on collisions. A number of the interaction properties thus found were confirmed by the soap film experiments of Couder and Basdevant (1986) and by the magnetohydrodynamic experiments of Nguyen Duc and Sommeria (1988).
This chapter
concentrates on the interaction behaviour of the dipole structures as arising from turbulence in a stratified fluid, which has been examined experimentally by considering two particular cases, viz. a head-on collision between two dipoles of approximately equal size and strength, and a collision of a single dipole against a solid wall.
603
Fig. 8. Sequence of plan-view p h o t o g r a p h s showing t h e e v o l u t i o n of two c o l l i d i n g d i p o l e s . E x p e r i m e n t a l p a r a m e t e r s a r e : Q = 3.66 ml, A t = 0.5 s , N = 2.25 /s (Q a n d A t were t a k e n e q u a l f o r b o t h j e t s ) . The p i c t u r e s were t a k e n ( a ) 2 s , ( b ) 65 6 , ( c ) 110 s and ( d ) 340 s a f t e r t h e i n j e c t i o n was stopped.
604 4.1
Colliding dipoles A head-on collision of two dipoles was created by injecting fluid
simultaneously from two nozzles placed in juxtaposition at some distance apart. The jets were generated under identical experimental conditions, in order to ensure that the dipoles emerging after the collapse were approximately identical. The jets were dyed with different colours, so that possible exchange of mass during the collision could be easily observed. A sequence of photographs showing the evolution of the collision is presented in Fig. 8. After the collapse of the initially turbulent patches (Fig. 8a), two dipoles are formed (Fig. 8b) as described in the previous sections. The resulting dipoles were not completely symmetric, and at the point of collision their axes were not aligned. Nevertheless, in the next stage (Fig. 8c) the dipoles are seen to exchange their partners, and two new dipoles are formed which move along a straight line away from the collision region (Fig. 8d).
This new axis of motion is not exactly
perpendicular to the original axis, but this effect is attributed to the misalignment of the dipoles before collision. The newly formed dipoles have almost symmetric shapes, in fact even more symmetric than the original dipoles. The exchange of dipole partners was also found in the above-mentioned numerical studies, but the use of different dyes in this laboratory experiment provides some extra information. It appears that hardly any mass is exchanged between the partners. Apart from some 'streamers' wrapped around the other partner (which is most likely due to the non-alignment of the colliding dipoles), each partner seems to conserve its own 'colour', i.e.
its own mass.
Similar experiments were conducted with centred head-on collisions between dipoles of different size, and it was observed that, after the exchange of partners as seen in Fig. 8 , the newly formed dipoles perform a looping excursion and collide again at the original axis. At this second collision the partners are exchanged again, so that the original dipoles are reformed, subsequently separating in their original direction of propagation. This observation confirms the numerical results obtained by Makino et al. (1981) and McWilliams and Zabusky (1982); details of this laboratory experiment will be published elsewhere. 4.2
Dipole collision against a solid wall Another type of interaction is the dipole collision perpendicular to a
solid wall. Maybe one would expect this situation to be equivalent to the case considered in the previous sub-section, viz. that of a centred head-on collision of two identical dipoles. A solid-wall collision has been studied numerically by
605
Larichev (private communication) and indeed, the partners were found to get separated while moving along the wall away from the collision area. In this case the wall in fact acts as a mirror, and the resulting structures can be considered as two new dipoles, consisting of the separated partners and their mirror images on the other side of the wall. However, in the laboratory experiment to be described here an essentially different interaction behaviour was observed. The sequence of observed events is shown by the plan-view photographs in Fig. 9. In (a) it can be seen how the dipole approaches the solid wall, its axis almost perpendicular to it. In the next stage the dipole partners have separated and slowly move along the wall in opposite directions, away from the collision area. Due to frictional effects associated with boundary layers at the wall, oppositely-signed vorticity is created at each partner, leading to the formation of hook-like satellite vortices (see Fig. 9b).
Eventually, these split
structures take on the appearance of strongly a-symmetric dipoles ( s e e Fig. Sc), which move in looping curves back towards the original dipole axis (Fig. 9d).
In
this stage of the experiment (ca. 9 min after the fluid was injected) the motion was extremely slow
- because of
the viscous damping associated with the boundary
layers above and below the moving structures
-
and a collision between the
dipoles was not observed. Assuming that the dipoles contained sufficient momentum, however, the subsequent collision
-
assumed perfectly elastic
- would
result in the formation of a smaller dipole moving along the original dipole axis away from the wall, and a larger dipole moving again towards the wall. Theoretically, this process of consecutive collisions will be repeated ad infinitum, until eventually all the kinetic energy is radiated away from the collision area by the small dipoles that are formed in each collision cycle. It is fascinating to realize that
-
as a consequence of wall friction
-
most of the
kinetic energy and momentum is trapped in a relatively small region at the wall, while a small portion of the energy is leaking away along the original dipole axis. This energy 'leak' can be considered as radiation with a particle-like character: in each collision cycle a small dipole is shot away from the collision area, thus carrying away a small portion of kinetic energy and moment urn. This laboratory experiment nicely illustrates the crucial effect of lateral friction on the collision process: in the absence of boundary layers at the wall (as in Larichev's numerical study) all the kinetic energy would be transported in equal portions along the wall by two identical structures moving i n opposite
directions, while wall friction tends to keep the energy trapped in the collision area.
606
9. Sequence of plan-view photographs showing the evolution of a dipole colliding against a solid wall. Experimental conditions are: Q = 19.5 ml, At = 3.2 s , N = 2.0 /s. The pictures were taken (a) 115 s, (b) 270 6 , (c) 330 6 , and ( d ) 545 s after the injection was stopped.
Fig.
607
5. CONCLUSIONS The experiments on turbulence generated by a pulsed horizonal jet i n a linearly stratified fluid have demonstrated the emergence of coherent dipolar flow structures. It was observed that, after the gravitational collapse of the 3D-turbulent spot, the flow becomes nearly two-dimensional, and smaller eddies were seen to grow into larger ones. Internal waves, generated by the collapse, play an important role in the dipole formation process, in that they push fluid forwards at a higher speed than the intruding mixed fluid advances. This results i n a sharp leading edge and a meridional flow along this front directed away
from the axis. In this stage two secondary dipoles can be observed, which move in looping paths backwards to the axis, finally being 'sucked' into the larger
primary dipole that has gradually been formed. Measurements on the collapsing turbulent region showed that three stages can b e distinguished in the dipole formation process, characterized by different growth rates of the horizontal dimensions of the injected fluid cloud and the size of individual eddies inside it. In order to investigate the interaction behaviour of dipoles, an experiment was performed i n which a head-on collision between two dipoles was established.
The exchange of partners was observed (as found in numerical studies on interacting dipoles), leading to the formation of two new dipoles. The u s e of different dyes in the injected fluids made clear that during the collision hardly any mass was exchanged between the dipole partners, demonstrating the elasticity of the head-on collision. Finally, the experiment on a single dipole colliding against a solid wall revealed that the partners get separated, eventually taking on the appearance of a-symmetric dipoles, an effect attributed to oppositely-signed vorticity production in the boundary layer at the wall. The newly formed dipoles move i n looping curves back to the original axis, and collide, thus forming a small dipole that moves away from the wall, and a larger dipole that moves again towards the wall, upon which the collision process is repeated.
REFERENCES Ahlnss, K., Royer, T.C. and George, T.H., 1987. Multiple dipole eddies i n the Alaska Coastal Current detected with Landsat thematic mapper data. J. Geophys. Res., 92: 13041-13047. 1967. An introduction to fluid dynamics. Cambridge University Batchelor, G.K., Press, 615 pp. Batchelor, G.K., 1969. Computation of the energy spectrum i n homogeneous twodimensional turbulence. Phys. Fluids, 12 (Suppl. 11): 233-240.
608
Couder, Y. and Basdevant, C., 1986. Experimental and numerical study of vortex couples in two-dimensional flows. J. Fluid. Mech., 173: 225-251. Fedorov, K.N. and Ginsburg, A.I., 1986. 'Mushroourlike' currents (vortex dipoles) in the ocean and in a laboratory tank. Ann. Geophys., 4: 507-516. Flierl, G.R., Stern, M.E. and Whitehead, J.A., 1983. The physical significance of modons: laboratory experiments and general integral constraints. Dyn. Atmos. Oceans, 7: 233-263. Griffiths, R.W. and Linden, P.F., 1981. The stability of vortices in a rotating, stratified fluid. J. Fluid Mech., 105: 283-316. 1989a. Vortices in a rotating fluid. Kloosterziel, R.C. and Van Heijst, G.J.F. Submitted to J. Fluid Mech. 1989b. On tripolar vortices. This Kloosterziel, R.C. and Van Heijst, G.J.F., issue. Larichev, V.D. and Reznik, G.M., 1976. Concerning two-dimensional solitary Rossby waves. Dokl. Akad. Nauk. SSSR, 231: 1077-1079. Larichev, V.D. and Reznik, G.M., 1982. On collisions between two-dimensional solitary Rossby waves. Oceanology, 23: 545-552. Legras, B., Santangelo, P. and Benzi, R., 1988. High-resolution numerical experiments for forced two-dimensional turbulence. Europhys. Lett., 5: 3742. Makino, M., Kamimura, T. and Taniuti, T., 1981. Dynamics of two-dimensional solitary vortices in a low-beta plasma with convective motion. J. Phys. SOC. Japan, 50: 980-989. McWilliams, J.C. and Zabusky, N.J., 1982. Interactions of isolated vortices. I: Modons colliding with modons. Geophys. Astrophys. Fluid Dyn., 19: 207-227. Nguyen Due, .I.-M. and Sommeria, J., 1988. Experimental characterization of steady two-dimensional vortex couples. J. Fluid Mech. 192: 175-192. Nycander, J . , 1988. New stationary vortex solutions of the Hasegawa-Mima equation. J. Plasma Phys., 39: 413-430. Salmon, R., 1982. Geostrophic turbulence. In: Topics in ocean physics, P. Malanotte-Rizzoli and A.R. Osborne (eds.), North Holland, Amsterdam: 30-78. Stern, M.E., 1975. Minimal properties of planetary eddies. J. Mar. Res., 40: 57-74. Stockton, P.L. and Lutjeharms, J.R.E., 1988. Observations of vortex dipoles on the Benguela upwelling front. S. Afr. Geogr., in press. Voropayev, S.I. and Filippov, I.A., 1985. Development of a horizontal jet in uniform-density and stratified fluids. Laboratory experiment. Izv. Atm. Oc. Phys., 21: 742-747.
LABORATORY EXPERIMENTS ON DIPOLE STRUCTURES IN A STRATIFIED FLUID G.J.F. VAN HEIJST and J.B. FLOR Institute of Meteorology and Oceanography University of Utrecht Princetonplein 5 3584 CC Utrecht, The Netherlands
ABSTRACT Coherent flow structures (vortices, dipoles) as occurring in geophysical fluid systems have been investigated in the laboratory. This paper concentrates on dipole experiments, in which locally 3D turbulence is generated in a continuously stratified fluid, viz. by horizontal injection of a fixed volume of fluid during a short period of time. The cone-shaped region of 3D turbulent fluid collapses under gravity (due to the stratified ambient fluid), resulting in a nearly 2D flow confined within a thin layer of fluid. A s can be expected from 2Pturbulence theory, one observes a spectral flux of energy towards larger scales, and the collapsed fluid eventually gets organized in a dipole-shaped structure. The formation process leading to these dipoles as well as their dynamics will be discussed. Two experiments were carried out in order to study interaction properties of dipoles, viz. a head-on collision between two dipoles, and a collision of a single dipole against a solid wall. Some preliminary results are presented. 1.
INTRODUCTION Coherent flow structures are typical features of geostrophic or quasi two-
dimensional turbulence, and these structures play an important part in largescale geophysical flows. Essential elements in such flows are rotation and/or stratification, which tend to make the motion two-dimensional. Flow coherency has been observed in the atmosphere as well as in the oceans, and has been studied theoretically (both analytically and numerically) and by laboratory experiments. In previous studies it was found that coherent structures in quasi two-dimensional flows can take the appearance of 'monopoles' (i.e. single vortices) and 'dipoles' (i.e.
vortex pairs), and recently evidence was obtained
for the existence of tripolar flow structures, see e.g. Legras, Santangelo and Benzi (1988) and Kloosterziel and Van Heijst (1989a,b).
The present study
concentrates on a laboratory experiment in which a dipole structure is generated by turbulent horizontal injection of a small fluid volume i n a stratified, nonrotating ambient fluid. The set-up of this experiment is relatively simple, and it provides valuable information about the dipole formation as well as the flow dynamics of the ultimate dipole structure.
592
Observations by satellites have revealed the occurrence of dipolar flow patterns in the ocean. Fedorov and Ginsburg (1986) describe some examples of oceanic dipoles in the Black Sea and the Sea of Okhotsk, and the satellite observations presented by Ahlniis, Royer and George (1987) reveal an abundancy of dipolar structures at the edges of the Alaska Coastal Current where it separates from Kayak Island, These patterns have horizontal dimensions of typically 20 km. Other examples of oceanic dipoles can be found in a recent paper by Stockton and Lutjeharms (1988), which describes observations of dipolar vortex structures on the Benguela upwelling front. Blocking events in the atmosphere are usually characterized by dipolar patterns in the isobars. Because of the large horizonal scale of such dipolar systems, the B-effect plays a non-negligible role in their dynamics. An analytical solution for a coherent dipole structure on a +plane was given by Stern (1975), who named it 'modon'. Motivated by geophysical applications, a number of laboratory experiments on rotating fluids were performed, in which dipolar flow structures were
observed to occur. For example, Griffiths and Linden (1981) found evidence of dipoles in a laboratory study of an unstable density front in a rotating fluid, whereas Kloosterziel and Van Heijst (1989a) observed dipole formation in rotating fluid experiments on unstable anticyclonic barotropic vortices, These experiments demonstrate that dipoles can be generated by a rather non-descript forcing. This was also observed in the experiment described by Flierl, Stern and Whitehead (1983):
they found that the turbulent flow induced by a pulsed
horizontal jet in a homogeneous rotating fluid becomes two-dimensional (due to the rotation of the system) within one or two rotation periods, and a subsequent organization into a single dipole structure was observed. Two-dimensionality of the flow can also be established by other forces than those associated with rotation or stratification. For example, magnetic forces have similar effects on electrically conducting fluids, and the occurrence of coherent structures has indeed been observed in such fluids. Solitary vortices are common features in plasma flows and their dynamics have been studied by numerous investigators (see e.g. Nycander (1988)).
Makino, Kamimura and Taniuti (1981) and
Interesting laboratory experiments on dipole structures in
magnetohydrodynamic flows were recently performed by Nguyen h c and Sommeria (1988).
The basic set-up of their experiment was a thin layer of mercury
subjected to a homogeneous magnetic field perpendicular to the fluid layer; the dipoles were generated by an electric current pulse between two electrodes placed in the fluid at some distance apart. The dynamics of the dipole structures thus created were investigated by visualization of the flow with small tracer particles, which allowed accurate measurement of the flow field and its vorticity. Coherent flow structures were also observed in the soap film
593
experiments by Couder and Basdevant (1986).
As a consequence of the geometry,
turbulence generated in a soap film is essentially two-dimensional, and the formation of single and dipolar vortices in a turbulent wake behind an obstacle moving through the film was indeed observed. The present study concentrates on an experiment with a turbulent jet in a stratified, non-rotating fluid. By horizontally injecting a small volume of fluid a region of three-dimensionally turbulent flow is created, which collapses due to gravity forces associated with the ambient stratification. The resulting flow quickly becomes two-dimensional, and the subsequent formation of coherent structures can be clearly observed, eventually leading to a single dipole structure. In these experiments the flow was visualized by adding dye to the injected fluid or by putting neutrally buoyant tracer particles in the stratified fluid. The set-up of this laboratory experiment is extremely simple, and allows one to study in detail (1) the transition process from threedimensional turbulence to two-dimensional flow, ( 2 ) the subsequent emergence of coherent flow structures, and ( 3 ) the dynamics of the eventually arising dipole structures. It has been found by many investigators that dipolar flow structures are very robust and that they behave in a soliton-like manner in interactions with other dipoles (e.g.
Makino et al. (1981), McWilliams and Zabusky (1982),
Couder and Basdevant (1986)),
and experiments on colliding dipoles to be
discussed in this paper confirm these results. Dipoles were also studied in a similar experiment by Voropayev and Fillipov (1985).
However, their experiment
is essentially different from the one discussed in the present paper, in that in their case fluid was injected continuously rather than in a short pulse, and the injection produced a smooth laminar flow rather than a turbulent jet. The paper is organized as follows: the set-up of the laboratory experiment as well as the observed flow phenomena are described in Section 2. Quantitative observations of the flow before, during and after the gravitational collapse of the turbulent region are presented in Section 3. Interaction properties of dipoles are considered in Section 4 , attention being focussed on two particular cases, viz. a head-on collision between two dipoles, and a collision of a single dipole against a solid wall. Finally, a summary of the results is given in Section 5.
2. LABORATORY EXPERIMENT
In order to study the transition from three-dimensional turbulence to twodimensional turbulence and the subsequent emergence of coherent structures, experiments were carried out with a horizontal turbulent jet in a stratified
594
fluid, The basic set-up of this laboratory experiment is depicted schematically in Fig. 1. A square perspex tank of dimensions 1 m
x
1 m x 0.3 m i s filled with
a linearly stratified salt solution. During a short period of time (At) some fixed volume (Q) of fluid is injected horizontally through a nozzle of diameter d. The density of the injected fluid exactly matches the density of the ambient stratified salt solution at the level of the nozzle, so that a horizontal jet flow is created. In the experiments the injection velocity U is taken large enough to ensure that the jet is turbulent. Typical experimental conditions are: U
FI
1 m/s
,d
a
m, and the kinematic fluid viscosity V =
the Reynolds number Red
=
m2/s, so that
Ud/v based o n the nozzle diameter has a value of
typically lo3. The flow was visualized by adding dye to the injected fluid, and the subsequent deformation of the injected dye cloud was recorded both from above and from the side by remotely controlled photocameras. Because of the short duration of the injection (At of a ' b l o b '
<
1
s),
the induced flow has the appearance
of turbulent flow in an otherwise motionless stratified environment. QAt
Fig.
1. The basic set-up of the laboratory experiment.
A s in a non-stratified environment, this jet
-
or I b l o b '
-
initially has a
conical shape with a circular cross-section, and turbulent entrainment of ambient fluid is observed to occur, both horizontally and vertically. In the next stage, this three-dimensionally turbulent region is observed to collapse under gravity, as a result of the ambient stratification: in this stage the vertical velocity component is suppressed, and the flow shows a transition from 3D to 2D turbulence. A well known property of two-dimensional flow is the so-
called 'inverse energy cascade': in contrast to three-dimensional turbulence, kinematic energy is in two-dimensional turbulence transferred from smaller to larger scales ( s e e e.g. Batchelor (1969) and Salmon (1982)),
leading to an
apparent organization of the flow into larger coherent structures. In this experiment the two-dimensional flow is observed to get organized in a welldefined flat dipole structure, as illustrated by the sequence of photographs presented in Fig. 2. Once this dipole is formed, it moves slowly forward without
595
Fig. 2. A sequence of plan-view photographs showing the evolution of the turbulent volume of h o r i z o n t a l l y i n j e c t e d f l u i d . The p i c t u r e s were taken ( a ) 3 s , (b) 12 s , ( c ) 50 s, and (d) 180 s a f t e r the i n j e c t i o n was stopped. Experimental parameters are: Q = 2.44 ml, A t = .I9 s , N = 2 . 2 7 s.
596
any d r a s t i c changes i n i t s shape. T y p i c a l v a l u e s of i t s speed of advance V and
i t s d i a m e t e r D are: V
m / s and D
10-l m y so t h a t t h e Reynolds number
based on D t a k e s a t y p i c a l v a l u e of l o 2 . The d i p o l e f o r m a t i o n was s t u d i e d i n some 1 5 e x p e r i m e n t s , i n which t h r e e e x p e r i m e n t a l parameters were v a r i e d : t h e i n j e c t e d volume Q ranged from 2.44 m l t o 14.6 m l ,
t h e speed of i n j e c t i o n U through t h e n o z z l e was v a r i e d between 2.1
m / s and 4.4 mfs, and t h e v a l u e of t h e buoyancy f r e q u e n c y N of t h e s t r a t i f i c a t i o n ranged from 1.2 /s t o 2.52 /s. exactly a t half-depth,
I n a l l t h e e x p e r i m e n t s t h e n o z z l e was p l a c e d
and i t s d i a m e t e r measured 1.9 mm.
During and a f t e r t h e c o l l a p s e of t h e t u r b u l e n t r e g i o n , i n t e r n a l waves a r e observed t o propagate away from t h e c o l l a p s i n g f l o w r e g i o n , t h u s i n d u c i n g a p p r e c i a b l e motions in t h e ambient f l u i d . Although t h e s e waves are r e f l e c t e d by t h e s o l i d t a n k walls, t h e y a p p e a r t o decay r a t h e r q u i c k l y , w i t h o u t s e r i o u s l y a f f e c t i n g t h e dipole s t r u c t u r e t h a t a r i s e s i n t h e collapsed region. N e v e r t h e l e s s , t h e i n t e r n a l waves p l a y a n i m p o r t a n t r o l e i n t h e f o r m a t i o n p r o c e s s of t h e d i p o l e , as w i l l be p o i n t e d o u t i n t h e n e x t s e c t i o n ,
3. LABORATORY OBSERVATIONS AND RESULTS An i n t r i g u i n g q u e s t i o n concerns t h e f o r m a t i o n of c o h e r e n t s t r u c t u r e s a f t e r t h e c o l l a p s e of t h e i n j e c t e d t u r b u l e n t f l u i d r e g i o n : i t i s obvious t h a t i n i t i a l l y t h e f l o w is e s s e n t i a l l y three-dimensional,
i t s appearance b e i n g
i r r e g u l a r , w h i l e t h e u l t i m a t e two-dimensional f l o w h a s a r e g u l a r , well-organized s t r u c t u r e , I n o r d e r t o g a i n i n s i g h t i n t o t h e f o r m a t i o n of t h e u l t i m a t e d i p o l e s t r u c t u r e , c a r e f u l o b s e r v a t i o n s were c a r r i e d o u t by r e c o r d i n g t h e dyed c l o u d both from above and from t h e s i d e by photocameras and by a v i d e o camera. The s c e n a r i o of e v e n t s l e a d i n g t o t h e e v e n t u a l d i p o l a r f l o w p a t t e r n is s k e t c h e d i n Fig. 3, showing 5 s t a g e s i n t h e f o r m a t i o n p r o c e s s as observed from above. I n i t i a l l y , t h e t u r b u l e n t j e t v e r y much r e s e m b l e s a h o r i z o n t a l j e t i n a nons t r a t i f i e d ambient f l u i d : b e f o r e t h e c o l l a p s e t h e j e t o c c u p i e s a c o n i c a l volume and t h e t u r b u l e n t motion i n s i d e i t is e s s e n t i a l l y three-dimensional
(Fig. 3a).
I n t h e n e x t s t a g e , t h e t u r b u l e n t c l o u d c o l l a p s e s under g r a v i t y , and i n t e r n a l waves are observed t o p r o p a g a t e away from a n a p p a r e n t c e n t r e l o c a t e d somewhere i n s i d e t h e i n j e c t e d f l u i d volume. T h i s is s k e t c h e d i n Fig.
3b, where t h e
i n t e r n a l wave crests are i n d i c a t e d by t h e arcs. The dominant mode of t h e i n t e r n a l wave motion is t h e m = 2 mode, w i t h a h o r i z o n t a l p l a n e of symmetry a p p r o x i m a t e l y a t t h e i n j e c t i o n l e v e l . The i n t e r n a l waves p r o p a g a t e away from t h e c o l l a p s i n g r e g i o n and t h i s i m p l i e s t h a t a l a r g e p o r t i o n of t h e k i n e t i c energy a s s o c i a t e d w i t h t h e i n i t i a l j e t f l o w is r a d i a t e d away from t h e mixed r e g i o n . Due
597 t o r e f l e c t i o n a t t h e t a n k walls, complicated wave motions can be observed throughout t h e tank, b u t t h e s e appear t o decay r e l a t i v e l y q u i c k l y , a p p a r e n t l y w i t h o u t a f f e c t i n g t h e formation of c o h e r e n t flow s t r u c t u r e s i n t h e c o l l a p s e d region.
before collapse:
apparent centre of radiation
during collapse:
+
-
(b)
internal wave crests
after collapse:
Fig. 3. Schematic plan-view drawing of t h e p r i n c i p a l s t a g e s in t h e d i p o l e formation: ( a ) t h e 3D t u r b u l e n t jet b e f o r e c o l l a p s e , ( b ) t h e o c c u r r e n c e of i n t e r n a l waves d u r i n g (and a f t e r ) t h e c o l l a p s e , ( c ) t h e emergence of two secondary d i p o l e s , which move i n l o o p i n g p a t h s ( d ) , and ( e ) t h e u l t i m a t e d i p o l e structure. During t h e c o l l a p s e t h e l e a d i n g edge of t h e i n j e c t e d f l u i d r e g i o n h a s t h e appearance of a s h a r p f r o n t , which is caused by t h e 'pushing' i n t e r n a l waves i n t h a t region.
e f f e c t of t h e
The wave v e l o c i t y is h i g h e r t h a n t h e speed of
advance of t h e i n t r u s i o n nose, and t h i s probably r e s u l t s in a r e l a t i v e motion d i r e c t e d away from t h e symmetry a x i s as i n d i c a t e d by t h e arrows i n Fig.
3b. When
t h e c o l l a p s e i s completed, t h e motion h a s become almost h o r i z o n t a l , and t h e s i z e of t h e small e d d i e s g r a d u a l l y increases. An i n t e r e s t i n g f e a t u r e i n t h i s s t a g e is t h e occurrence of two secondary d i p o l e s t r u c t u r e s moving away from t h e symmetry a x i s of t h e dyed c l o u d , as i n d i c a t e d in Fig. 3c. These small c o h e r e n t s t r u c t u r e s
598
f o l l o w a curved p a t h a s s k e t c h e d by t h e dashed a r r o w s , u n t i l t h e y meet a t t h e a x i s , some d i s t a n c e behind t h e l e a d i n g edge of t h e i n t r u d i n g b l o b , see Fig. 3d. During t h i s s t a g e i n t e r n a l waves are s t i l l o b s e r v e d , and dyed f l u i d moves a t a r e l a t i v e l a r g e speed a l o n g t h e a x i s , t h u s m a i n t a i n i n g t h e s h a r p f r o n t . Due t o t h i s s u b s t a n t i a l motion a l o n g t h e a x i s , t h e ' i n n e r ' p a r t s of t h e secondary d i p o l e s a r e advected towards t h e nose, g r a d u a l l y l o o s i n g t h e i r i d e n t i t y ; on t h e o t h e r hand, t h e ' o u t e r ' p a r t s of t h e secondary d i p o l e s a r e l e f t behind, and, being deformed by t h e s h e a r i n g motion i n t h e wake, they f i n a l l y end up as e l o n g a t e d dye p a t c h e s , as c l e a r l y v i s i b l e i n F i g s . 2c and 2d. In t h e u l t i m a t e s t a g e t h e i n t e r n a l waves have decayed, and t h e f l o w h a s o b t a i n e d t h e appearance
of a slowly advancing symmetric d i p o l e ( s e e Fig. 3e). The o c c u r r e n c e of t h e secondary d i p o l e s in t h e c o l l a p s e d s t a g e can be s e e n in Fig. 4 , which i s a photograph t a k e n between t h e photographs shown i n F i g s . 2b and 2c.
Fig. 4. Plan-view photograph showing t h e o c c u r r e n c e of t h e secondary d i p o l e s , The experiment was t h e same as t h a t of Fig. 2 ; t h e photograph was t a k e n 26 s a f t e r t h e i n j e c t i o n was stopped.
In o r d e r t o q u a n t i f y t h e o b s e r v a t i o n s , some c h a r a c t e r i s t i c s of t h e dyed f l u i d cloud were measured from t h e p h o t o g r a p h i c r e c o r d i n g s . The measured q u a n t i t i e s a r e : ( i ) t h e d i s t a n c e L of t h e nose of t h e i n j e c t e d dyed f l u i d r e g i o n r e l a t i v e t o t h e i n j e c t i o n n o z z l e , (ii) t h e maximum width B of t h e dyed f l u i d r e g i o n , and ( i i i ) t h e s i z e A of t h e l a r g e s t i n d i v i d u a l eddy s t r u c t u r e t h a t can be d i s t i n g u i s h e d in t h e dyed cloud. In p a r t i c u l a r i n t h e i n i t i a l phase of t h e
599
experiment the pictures were taken at short time intervals (typically 1 s ) , so that even rapid changes in the flow could be monitored easily. For a characteristic experiment the evolution of the quantities L, B and A I s shown graphically in Fig. 5. In this graph the time is nondimensionalized by the buoyancy frequency N, but the measured quantities L, B and A are expressed in cm. Observations by eye revealed that the transition from 3D to 2D turbulence is almost completed at Nt =- 4 , and that the flow is thus essentially two-
dimensional for Nt
>
4 . In the curves for L, B and A three stages can be
distinguished, which are in Fig. 5 denoted by the Roman symbols I, I1 and 111.
The first stage is characterized by a steady size increase of the vortices, and also the distance L and the width B of the dye blob show a continuous growth. The experimental data for this stage suggest straight lines, which implies a growth with some power of t. Stage I1 begins at approximately Nt = 24 and can be characterized as a stage of 'vortex interaction', in which A increases more rapidly than before. The decrease in B in this stage is caused by the two secondary dipoles
-
as sketched in Fig. 3c
- which now move
backwards to the
axis, thus decreasing the value of the maximum width B of the dye region.
I
I
I
I
I
I
I
I
Fig. 5. Graphical representation of the time evolution of L, B and A (for explanation of these symbols, see the text) as measured from photographic recordings. The straight l i n e s are fitted to the data by eye. Experimental parameters are: Q = 3.66 id,A t = .4 s , N = 1.66 1s.
600
During this stage of vortex interaction, L appears to grow slightly slower than in the initial stage. Once the ultimate dipole is formed, the distance L again
grows proportional to some power of t. In this particular experiment the dipole formation was established at Nt = 90, at which B and A obviously have reached equal values. The horizontal size of the dipole (A,
B) shows a gradual growth
proportional to some power of t. This size increase is most likely caused by frictional effects associated with horizontal shear layers present at the upper and lower sides of the intruding dyed fluid region. The increase can also be inferred from the spiral-shaped dye lines visible in both halves of the ultimate dipole structure, see e.g.
Fig. 2d. Such dye patterns suggest that undyed ambi-
ent fluid flows in at the rear side of the dipole, thus compensating its radial growth. Because of the preliminary character of the laboratory experiments, detailed information about the growth of L, B and A in the various stages of the dipole formation i s not yet available, but will be published in due time.
Fig. 6 . Streakline photograph showing the horizontal motion associated with the dipole structure.
601 Apart from visualization by dye, the horizontal motion in and around the collapsed region was also visualized by bringing small (0.5
m
diameter)
polystyrene tracer particles in the stratified fluid. The density of these particles was 1.038 g/cm3, and in a number of experiments the stratification was chosen such that they settled in a relatively thin layer at approximately halfdepth. The tracer particles were illuminated from the side by a slit light, and against a dark background their motion was recorded from above by a photocamera. A typical example of the streakline photographs thus obtained is shown in
Fig. 6 , from which the dipolar structure in the horizontal flow can be easily observed. By measuring the lengths of the streaks as well as their orientation, horizontal velocity vectors in the flow domain can be determined. Once the flow field is known, it is also possible to calculate numerically the spatial distribution of the vorticity w and the streamfunction Y. This technique is presently being developed, and it i s expected that along this line important information can be obtained about the relationship between Y and w for the dipole structure. In many studies ( s e e e.g.
Batchelor ( 1 9 6 7 ) , Stern ( 1 9 7 5 ) ,
Larichev and Reznik ( 1 9 7 6 ) ) on dipolar vortices
- whether
or not on a p-plane
-
it is assumed that in the interior of the dipole the vorticity w is a linear function of the streamfunction Y. For example, the Lamb dipole as described by 2
2
Batchelor ( 1 9 6 7 ) is found by solving V Y ( r , N = --w = -k Y on a circular region r < a with Y
=
constant on the circle itself; r and 8 are cylindrical
coordinates, and k is a constant. The solution for the streamfunction is given by
with C a constant coefficient and ka
=
3.83 (defining the first zero of J1).
Some streamlines according to this solution are shown in Fig. 7 , and it is clear that this pattern very much resembles the dye line pattern as visible in Fig. 2d. It must be kept in mind, however, that the dye lines are actually streaklines, so that the radial expansion of the dipole causes the dye to flow in spirals rather than in closed curves. The streamlines of the Lamb dipole also
much resemble the flow pattern visualized by the tracer particles, as shown in Fig. 6. It was observed by eye, however, that the dipolar dye pattern occupies a relatively small area, with a horizontal size comparable to the distance between the points of zero velocity in Fig. 6 . The flow outside the dye region is caused by the translation of the dipole structure, in analogy with the motion induced by a cylinder moving through a fluid. Therefore, a correction must be made for the dipole translation in order to be able to make a proper comparison between the velocity field derived from streakline pictures and any theoretical solution
602
for the stationary dipole flow. At the moment appropriate data reduction techniques are being developed, and results will be published in due time. A s mentioned before, it is expected that these measurements will provide
essential information about the relationship between w and Y. Indications of a nonlinear relation between these quantities were found by Legras, Santangelo and Benzi (1988) in a numerical study of forced 2D-turbulence, and the dipole experiments in a horizontal layer of mercury subjected to a vertical magnetic field as performed by Nguyen Duc and Sommeria (1988) yielded similar signs of nonlinearity. It might thus well be that the theoretical dipole models need some
.
refinement
Fig. 7. Streamlines according to Lamb's solution for a dipole.
4. INTERACTION OF DIPOLES Robustness of coherent flow structures is a well-known feature, and in particular the interaction behaviour of dipoles has been studied by a number of investigators. For example, head-on collisions between dipoles were studied numerically by Larichev and Reznik (1976, MeWilliams and Zabusky (1982),
1983),
by llakino et al. (1981),
and by Couder and Basdevant (1986).
by
In some of
these papers attention was also given to overtaking collisions and off-centre head-on collisions. A number of the interaction properties thus found were confirmed by the soap film experiments of Couder and Basdevant (1986) and by the magnetohydrodynamic experiments of Nguyen Duc and Sommeria (1988).
This chapter
concentrates on the interaction behaviour of the dipole structures as arising from turbulence in a stratified fluid, which has been examined experimentally by considering two particular cases, viz. a head-on collision between two dipoles of approximately equal size and strength, and a collision of a single dipole against a solid wall.
603
Fig. 8. Sequence of plan-view p h o t o g r a p h s showing t h e e v o l u t i o n of two c o l l i d i n g d i p o l e s . E x p e r i m e n t a l p a r a m e t e r s a r e : Q = 3.66 ml, A t = 0.5 s , N = 2.25 /s (Q a n d A t were t a k e n e q u a l f o r b o t h j e t s ) . The p i c t u r e s were t a k e n ( a ) 2 s , ( b ) 65 6 , ( c ) 110 s and ( d ) 340 s a f t e r t h e i n j e c t i o n was stopped.
604 4.1
Colliding dipoles A head-on collision of two dipoles was created by injecting fluid
simultaneously from two nozzles placed in juxtaposition at some distance apart. The jets were generated under identical experimental conditions, in order to ensure that the dipoles emerging after the collapse were approximately identical. The jets were dyed with different colours, so that possible exchange of mass during the collision could be easily observed. A sequence of photographs showing the evolution of the collision is presented in Fig. 8. After the collapse of the initially turbulent patches (Fig. 8a), two dipoles are formed (Fig. 8b) as described in the previous sections. The resulting dipoles were not completely symmetric, and at the point of collision their axes were not aligned. Nevertheless, in the next stage (Fig. 8c) the dipoles are seen to exchange their partners, and two new dipoles are formed which move along a straight line away from the collision region (Fig. 8d).
This new axis of motion is not exactly
perpendicular to the original axis, but this effect is attributed to the misalignment of the dipoles before collision. The newly formed dipoles have almost symmetric shapes, in fact even more symmetric than the original dipoles. The exchange of dipole partners was also found in the above-mentioned numerical studies, but the use of different dyes in this laboratory experiment provides some extra information. It appears that hardly any mass is exchanged between the partners. Apart from some 'streamers' wrapped around the other partner (which is most likely due to the non-alignment of the colliding dipoles), each partner seems to conserve its own 'colour', i.e.
its own mass.
Similar experiments were conducted with centred head-on collisions between dipoles of different size, and it was observed that, after the exchange of partners as seen in Fig. 8 , the newly formed dipoles perform a looping excursion and collide again at the original axis. At this second collision the partners are exchanged again, so that the original dipoles are reformed, subsequently separating in their original direction of propagation. This observation confirms the numerical results obtained by Makino et al. (1981) and McWilliams and Zabusky (1982); details of this laboratory experiment will be published elsewhere. 4.2
Dipole collision against a solid wall Another type of interaction is the dipole collision perpendicular to a
solid wall. Maybe one would expect this situation to be equivalent to the case considered in the previous sub-section, viz. that of a centred head-on collision of two identical dipoles. A solid-wall collision has been studied numerically by
605
Larichev (private communication) and indeed, the partners were found to get separated while moving along the wall away from the collision area. In this case the wall in fact acts as a mirror, and the resulting structures can be considered as two new dipoles, consisting of the separated partners and their mirror images on the other side of the wall. However, in the laboratory experiment to be described here an essentially different interaction behaviour was observed. The sequence of observed events is shown by the plan-view photographs in Fig. 9. In (a) it can be seen how the dipole approaches the solid wall, its axis almost perpendicular to it. In the next stage the dipole partners have separated and slowly move along the wall in opposite directions, away from the collision area. Due to frictional effects associated with boundary layers at the wall, oppositely-signed vorticity is created at each partner, leading to the formation of hook-like satellite vortices (see Fig. 9b).
Eventually, these split
structures take on the appearance of strongly a-symmetric dipoles ( s e e Fig. Sc), which move in looping curves back towards the original dipole axis (Fig. 9d).
In
this stage of the experiment (ca. 9 min after the fluid was injected) the motion was extremely slow
- because of
the viscous damping associated with the boundary
layers above and below the moving structures
-
and a collision between the
dipoles was not observed. Assuming that the dipoles contained sufficient momentum, however, the subsequent collision
-
assumed perfectly elastic
- would
result in the formation of a smaller dipole moving along the original dipole axis away from the wall, and a larger dipole moving again towards the wall. Theoretically, this process of consecutive collisions will be repeated ad infinitum, until eventually all the kinetic energy is radiated away from the collision area by the small dipoles that are formed in each collision cycle. It is fascinating to realize that
-
as a consequence of wall friction
-
most of the
kinetic energy and momentum is trapped in a relatively small region at the wall, while a small portion of the energy is leaking away along the original dipole axis. This energy 'leak' can be considered as radiation with a particle-like character: in each collision cycle a small dipole is shot away from the collision area, thus carrying away a small portion of kinetic energy and moment urn. This laboratory experiment nicely illustrates the crucial effect of lateral friction on the collision process: in the absence of boundary layers at the wall (as in Larichev's numerical study) all the kinetic energy would be transported in equal portions along the wall by two identical structures moving i n opposite
directions, while wall friction tends to keep the energy trapped in the collision area.
606
9. Sequence of plan-view photographs showing the evolution of a dipole colliding against a solid wall. Experimental conditions are: Q = 19.5 ml, At = 3.2 s , N = 2.0 /s. The pictures were taken (a) 115 s, (b) 270 6 , (c) 330 6 , and ( d ) 545 s after the injection was stopped.
Fig.
607
5. CONCLUSIONS The experiments on turbulence generated by a pulsed horizonal jet i n a linearly stratified fluid have demonstrated the emergence of coherent dipolar flow structures. It was observed that, after the gravitational collapse of the 3D-turbulent spot, the flow becomes nearly two-dimensional, and smaller eddies were seen to grow into larger ones. Internal waves, generated by the collapse, play an important role in the dipole formation process, in that they push fluid forwards at a higher speed than the intruding mixed fluid advances. This results i n a sharp leading edge and a meridional flow along this front directed away
from the axis. In this stage two secondary dipoles can be observed, which move in looping paths backwards to the axis, finally being 'sucked' into the larger
primary dipole that has gradually been formed. Measurements on the collapsing turbulent region showed that three stages can b e distinguished in the dipole formation process, characterized by different growth rates of the horizontal dimensions of the injected fluid cloud and the size of individual eddies inside it. In order to investigate the interaction behaviour of dipoles, an experiment was performed i n which a head-on collision between two dipoles was established.
The exchange of partners was observed (as found in numerical studies on interacting dipoles), leading to the formation of two new dipoles. The u s e of different dyes in the injected fluids made clear that during the collision hardly any mass was exchanged between the dipole partners, demonstrating the elasticity of the head-on collision. Finally, the experiment on a single dipole colliding against a solid wall revealed that the partners get separated, eventually taking on the appearance of a-symmetric dipoles, an effect attributed to oppositely-signed vorticity production in the boundary layer at the wall. The newly formed dipoles move i n looping curves back to the original axis, and collide, thus forming a small dipole that moves away from the wall, and a larger dipole that moves again towards the wall, upon which the collision process is repeated.
REFERENCES Ahlnss, K., Royer, T.C. and George, T.H., 1987. Multiple dipole eddies i n the Alaska Coastal Current detected with Landsat thematic mapper data. J. Geophys. Res., 92: 13041-13047. 1967. An introduction to fluid dynamics. Cambridge University Batchelor, G.K., Press, 615 pp. Batchelor, G.K., 1969. Computation of the energy spectrum i n homogeneous twodimensional turbulence. Phys. Fluids, 12 (Suppl. 11): 233-240.
608
Couder, Y. and Basdevant, C., 1986. Experimental and numerical study of vortex couples in two-dimensional flows. J. Fluid. Mech., 173: 225-251. Fedorov, K.N. and Ginsburg, A.I., 1986. 'Mushroourlike' currents (vortex dipoles) in the ocean and in a laboratory tank. Ann. Geophys., 4: 507-516. Flierl, G.R., Stern, M.E. and Whitehead, J.A., 1983. The physical significance of modons: laboratory experiments and general integral constraints. Dyn. Atmos. Oceans, 7: 233-263. Griffiths, R.W. and Linden, P.F., 1981. The stability of vortices in a rotating, stratified fluid. J. Fluid Mech., 105: 283-316. 1989a. Vortices in a rotating fluid. Kloosterziel, R.C. and Van Heijst, G.J.F. Submitted to J. Fluid Mech. 1989b. On tripolar vortices. This Kloosterziel, R.C. and Van Heijst, G.J.F., issue. Larichev, V.D. and Reznik, G.M., 1976. Concerning two-dimensional solitary Rossby waves. Dokl. Akad. Nauk. SSSR, 231: 1077-1079. Larichev, V.D. and Reznik, G.M., 1982. On collisions between two-dimensional solitary Rossby waves. Oceanology, 23: 545-552. Legras, B., Santangelo, P. and Benzi, R., 1988. High-resolution numerical experiments for forced two-dimensional turbulence. Europhys. Lett., 5: 3742. Makino, M., Kamimura, T. and Taniuti, T., 1981. Dynamics of two-dimensional solitary vortices in a low-beta plasma with convective motion. J. Phys. SOC. Japan, 50: 980-989. McWilliams, J.C. and Zabusky, N.J., 1982. Interactions of isolated vortices. I: Modons colliding with modons. Geophys. Astrophys. Fluid Dyn., 19: 207-227. Nguyen Due, .I.-M. and Sommeria, J., 1988. Experimental characterization of steady two-dimensional vortex couples. J. Fluid Mech. 192: 175-192. Nycander, J . , 1988. New stationary vortex solutions of the Hasegawa-Mima equation. J. Plasma Phys., 39: 413-430. Salmon, R., 1982. Geostrophic turbulence. In: Topics in ocean physics, P. Malanotte-Rizzoli and A.R. Osborne (eds.), North Holland, Amsterdam: 30-78. Stern, M.E., 1975. Minimal properties of planetary eddies. J. Mar. Res., 40: 57-74. Stockton, P.L. and Lutjeharms, J.R.E., 1988. Observations of vortex dipoles on the Benguela upwelling front. S. Afr. Geogr., in press. Voropayev, S.I. and Filippov, I.A., 1985. Development of a horizontal jet in uniform-density and stratified fluids. Laboratory experiment. Izv. Atm. Oc. Phys., 21: 742-747.