The evolution of an isolated turbulent region in a two‐layer fluid

A turbulent region is generated by horizontal pulsed injection at the interface of a two‐layer fluid. Flow visualization studies reveal the existence of three stages in the evolution of the vertical size of this region: growth, maximum height, and collapse. A scaling analysis for the height of the turbulent region is presented, which appears to be in good agreement with the measurements. Comparable results were obtained by Fernando, van Heijst, and Fonseka (submitted to J. Fluid Mech.) for similar experiments in a linearly stratified fluid. Thorpe‐scale measurements of the turbulent region reveal that the ratio of the rms displacement Lt and the maximum displacement Ltmax remain constant with time. The eventual formation process of the dipolar vortices after the collapse and the influence of interfacial wave motions on these dipolar vortices are discussed.


I. INTRODUCTION
In many geophysical flows stable stratification plays a In the laboratory, the evolution of an isolated turbu lent region in a linearly density-stratifi ed fluid has been investigated by Fernando, van Heijst, and Fonseka, 5 here after referred to as FvHF. In that study, an isolated tur

II. EXPERIMENTAL SETUP
The experiments reported in this paper were conducted in two different Perspex tanks: one narrow tank of dimen sions 183X30X40 (em) and one square tank of dimen sions 90X90X40 (em), respectively. For the study of the growth and collapse of the turbulent patch, the narrow tank was used, while the square tank was used to study the dipole formation. In order to assure that side wall effects in the small tank did not bias the collapse significantly, the collapse stage was studied in the large tank. A schematic of the experimental arrangement is shown in Fig. 1 interface was siphoned off using small perforated horizon tal tubes. By this method the interface thickness oh was reduced to a thickness of approximately 0.6-2.5 em, which was always smaller than the size of the turbulent patch. The total depth of the water was varied from 20 to 31 em, and the position of the interface, measured from the bot tom of the tank, was varied between 12 and 14 em. The buoyancy jump across the interface, represented by with p 1 and p 2 the densities of the upper and lower layer, respectively, and g the gravitational acceleration, was in the range from 0.37 to 0.77 m/s 2 • In order to measure the density profile a conductivity probe was traversed verti cally through the fluid, from which the interfacial position was determined. Then the nozzle was placed at this level and a small drop of the dyed injection fluid-with density ( p 1 + p 2) /2-was released at the level of the nozzle. Due to the strong buoyancy force at the interface the drop col lapses instantly to a thin spot of less than a millimeter thickness and marked the midlevel of the interface quite accurately. The fluid was injected horizontally at this level through a nozzle of diameter 2 mm by a computer controlled injection mechanism. The density of the injected fluid was adjusted to the intermediate density, (p1 + p 2 )/2. The injection velocity in these experiments was varied from 1.8 to 11.7 m/s, and the injection volume V0 ranged from 1 to 13 ml. The time of injection 8t was varied between 0.01-0.2 s, which was sufficiently short to create an iso lated turbulent patch of fluid. The Reynolds number Re= U0d!v at the injection, where U0 is the injection ve locity, d is the nozzle diameter, and v is the kinematic viscosity, ranged from 3500 to 22 000, thus ensuring that the jet was turbulent. The resulting turbulent patch was visualized by coloring the injected fluid with fluorescein and by illuminating it using a vertical sheet of light. The extent of dye diffusion was considered as the extent to which the turbulence diffuses; that is, the turbulent Prandtl number is considered to be of the order unity (Refs. 8 and

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Phys. Fluids, Vol. 6, No. 1, January 1994 9). A horizontal light sheet was used for visualizing the dipolar vortex formation in the square tank. The evolution of the flow was recorded by a video camera and by a frame by-frame examination of these recordings the time evolu tion of the vertical patch size was determined. The moment at which the fi rst spot of colored fl uid is ejected by the nozzle, was taken as time t=O. The vertical density distribution was measured by us ing a fast traversing microscale conductivity probe that was installed above the square tank. This traversing mech anism could be translated horizontally to track and capture the moving patch during the course of an experiment. The plunging speed of the probe had a maximum value of 30 cm/s, which is fast compared with the observed maximum translation speed of the patch (about 5 cm/s). These in stantaneous vertical density profi les were used to calculate Thorpe displacements, i.e., the size of the vertical displace ments of fluid parcels relative to a stable profile, which was obtained by rearranging the observed density profile. The method employed is similar to that used by Dillon. 10 The time and position at which the probes were shot, were recorded by video.

Ill. QUALITATIVE OBSERVATIONS
The evolution of the turbulent patch after the injection is presented in Fig. 2. During and shortly after the injec tion, an approximately cone-shaped jet forms and evolves into an isolated three-dimensional turbulent patch and moves in the direction of injection. During the initial stage, the vertical inertia forces of the eddies are large compared to the buoyancy forces, and entrainment of homogeneous fluid from above and below the interface causes the turbu lent region to grow continuously [see Fig. 2(b)]. Mixing of jet fluid and ambient fluid implies that the kinetic energy is converted into potential energy. When the vertical inertia forces become of the same order as the buoyancy forces, the turbulent patch reaches its maximum vertical size and then starts to collapse under gravity, thus reducing its ver tical size [for example, see Fig. 2(c)]. During the collapse, the mixed fluid intrudes horizontally into the ambient fluid. In some cases one could observe the formation of two separated intrusion noses at the front of the intrusion: one above and one below the interface [see, e.g., Figs. 2(c) and 2(d)]. The occurrence of the double intrusion can be at tributed to incomplete mixing within the patch, due to the finite thickness of the initial interface, 11 as well as interfa cial friction between the two intrusions. The collapse con tinues until the patch attains a thickness comparable to that of the interface. Interfacial waves, generated during the collapse, could not be distinguished from the waves that were forced by the initial impulse of the jet. In most of the experiments it is observed that these waves move away from the turbulent region along the upper and lower , boundaries of the intrusion. These traveling waves are dis sipated rapidly, in particular, after having been reflected at the tank walls. From the video recordings, however, it ap pears that in some cases a soliton-like disturbance moves at the front of the patch, giving rise to a vertical bulge near the nose of the intrusion. This bulge vanishes slowly com-  pared to the collapse rate of the patch. In a few experi ments with high injection velocities, a soliton-like feature was observed to detach from the collapsing region and travel up and down the tank a few times (see Fig. 3). Similar wave motions have been observed by Gilreath and Brandt 12 in the wake of an obstacle towed through a den sity interface. At the earlier stages of the collapse, the in trusion front is contorted and is observed to be distorted by flattened large eddies, which persist for longer times than the small-scale turbulence. At later stages, the front changes into one single, sharp intrusion. Mter the collapse the motion is mainly horizontal. A dipolar vortex structure emerges, which moves slowly in a straight line along its axis of symmetry. This dipolar structure takes on the same Phys. Fluids, Vol. 6, No. 1, January 1994 . When the dipolar vortex structure emerges, only the horizontal motions remain at the interface and due to molecular diffusion between the sharp intrusions, an intrusion with a single nose appears.
In some experiments it was observed that remnant stand ing interfacial wave motions interact with the dipole vor tices. In contrast to the traveling internal waves, these waves decay with a much longer time scale. The standing internal waves induce pulsations in vortex size, which cause stretching and compression of the vortices. This mechanism leads to changes in the vertical vorticity com ponent of the dipolar structure. As a consequence, the translation velocity and the horizontal size of the dipolar structure show oscillations around their mean values. Eventually, these standing waves are damped out and the dipole translates at a low velocity.

IV. COLLAPSE OF THE TURBULENT REGION
For a typical experiment, the maximum vertical patch size and the maximum horizontal spreading of the region were measured from video recordings. The vertical patch size as a function of time is shown in Fig. 5 (a), whereas the combination of the maximum horizontal and vertical patch sizes is presented in Fig. 5(b). Initially, the turbulent re gion grows equally in horizontal and vertical directions due to entrainment of ambient fluid. Once the infl uence of the buoyancy forces sets in, the patch growth is retarded these, however, are considered to be of minor importance to the evolution of the vertical patch size, and are thus neglected in the scaling analysis, which will be discussed below.
A. The growth of the turbulent region During the initial growth, the turbulent motions are dominated by the inertia forces. Thus, a scaling analysis similar to that advanced by FvHF for the growth of a turbulent region in a linearly stratified fluid can be used.
The growth of the turbulent region lasts for a very short period of time, so that the kinetic energy during that period can be assumed to be a constant, viz., where U0 is the injection velocity, V0 is the injected volume at time t=O, and u'2(t) and V(t) are the mean square horizontal turbulent velocity and volume of the turbulent region at time t, respectively. During the evolution, the turbulent region can be approximated by a self-similar con ical shape of characteristic dimension h(t), and thus its volume can be approximated by V(t) �h(t)3• For the tur bulence within the turbulent region it is assumed that u'2 w'2, with w'2 1 12 being the rms vertical velocity. Using the scaling for u'2 and V(t), one may write

B. The maximum patch size
The growth of the turbulent blob continues until the vertical inertia forces (w'21h per unit mass) of the eddies have decreased to such an extent that the buoyancy forces ( f>b per unit mass) become of comparable magnitude. At that stage, the patch reaches its maximum vertical size, and the inertia forces are balanced by the buoyancy forces, viz., (6) where he is the maximum vertical size of the patch. In other words, this relationship implies that the patch col lapse occurs when the bulk Richardson number he 8b/w'2 increases to a critical value B 4 . By using (3) and ( 6), it is possible to obtain where B5 = ( B1 B4) 114• The experimental results for the maximum patch height are presented in Fig. 7, and it is obvious that they show good agreement with the proposed scaling. The best fit required for the data yields B5=0.5. By using (5) and (7) the time tc, at which the patch reaches its maximum size he, can be written as  Table I.

C. The collapsing stage
After the patch has reached its maximum height, the eddies of the integral scale are too feeble to entrain the ambient fluid. An isolated patch of mixed fluid with a den sity different from the surrounding fluid remains, while the turbulence within the patch is weakened and fi nally is transformed into small-scale wave-like motions. Simple hy drostatic calculations show that there will be a horizontal pressure gradient between the inner and outer fluid, which causes the mixed patch to physically collapse and the mixed fluid to intrude in the ambient fluid at rest. The speed of the intrusion can be evaluated by a balance be tween the pressure gradient force and the inertia force of the intrusion: 16 where h(t) and r(t) are the total vertical patch size and the horizontal dimension, respectively, and C 1 is a constant. Assuming a double conical shape for the mixed patch, the horizontal spreading rate dr!dt can be estimated by h�=2?h(t).  Table I.
By using (9), ( 10), and the expression for the maximum vertical patch size (7), it is possible to obtain 1 1 where C3=2 {iB5 3 12C 1• A comparison of data with (11) is presented in Fig. 8. The values of C3 for individual ex periments are listed in Table I. The constant c3 = 0.30 ± 0.09 is obtained by taking the mean value of c3 over all measurements. Although the data for each indi vidual experiment are in good agreement with the pro posed scaling, there is some spread of the values of C3 over all the experiments. This is probably due to the interaction of the turbulent patch with interfacial wave motions, and the neglect of the development of the density gradient within the patch during initial mixing. The value C 1 can be evaluated by using c3 and yields cl =0.04.

D. The Thorpe-scale measurements
In a turbulent stratified fl uid heavy fluid elements are displaced above the lighter ones by turbulent stirring mo tions, causing density inversions. One method to estimate the scale of such overturning motions is to determine the vertical displacements of fluid elements relative to a stable, monotonic density profile. This density profi le can be ob tained by rearranging the measured instantaneous profile into a stable one by displacing fluid particles to their neu tral buoyancy depth by a so-called "bubble sort" routine. 10 The required displacements are called Thorpe displace ments, which provide information about the overturning motions of the eddies. The rms value of these displace ments is known as the Thorpe length scale Lt.
In order to measure the Thorpe scale of a decaying turbulent region, a number of experiments were performed under identical conditions, i.e., the same injection velocity and volume, and stratifi cation. For each experiment, two traversing conductivity probes-with a distance 2.9 em   of the injection parameters and the evaluated constants of the laboratory experiments. U0, V0, and lib are the injection velocity, injection volume, and buoyancy jump across the interface, respectively. The experimental constants for the initial growth and the collapse B3 and C3, respectively (see the text) are given for each experiment, together with their correlation coefficients R2 of the approximations with the measurements.
For experiments 20-22 a soliton-like wave motion occurred, and no regular collapse was observed. apart-were positioned at a given distance from the injec tion nozzle. These probes were shot vertically downward through the fluid at the instant that the turbulent region was at the same position as the probes. Information about the spatiotemporal evolution of the turbulent region was obtained by changing the position of the probes for each experiment. By using the measured density profi les, the maximum Thorpe displacement L1 max, the Thorpe scale L1, and the vertical patch size L P were determined: L P was defi ned as the region that has displacements exceeding 5% of L1 m ax. This threshold of 5% was necessary to suppress the noise peaks generated at the start and end position of the probe traverse. In addition, the patch size h was mea sured using video records, and the results are shown in Fig.  9. The latter showed consistently higher values. This is not  ing the collapse of the patch the difference between these two patch size measurements becomes larger. Under these conditions, the color visualization technique indicates the physical size of the collapsing region, whereas the Thorpe displacement technique indicates the return of buoyant fl uid parcels to their respective stable position. The latter occurs faster, as was indicated by the measurements of FvHF.
The Thorpe scale L1 and the nondimensional Thorpe scale L/ Lrmax are plotted against time in Fig. 10. The graph shows that L1 decays approximately linearly with time. The normalized Thorpe scale L/ L1 max appears to · remain constant ( =0.4) during the evolution of the patch. This is in fair agreement with the previous observations of Itsweire 17 and DeSilva and Fernando, 18 which showed Lrmaxl L1 = 2.75, for decaying stratifi ed turbulence and forced stratifi ed turbulence, respectively. Shown in Fig. 11 is a plot of L1 vs L P ; a linear increase can be seen with L1 where z1 and oz are the level and thickness of the interface, respectively. When the wave amplitude is much smaller than the layer thickness, it is possible to obtain the disper sion relation as k,= (2n+ l) a l!oz o}-8b!8z _ 4(n 2+ n) a l /8 � ]112 where n is the wave mode and k11 is the wave number (see, e.g., Ref. 21). Waves interacting with the dipolar vortex should have at least a wavelength of twice the diameter of the dipole in order to stretch and squeeze the structure as a whole; from Fig. 13, it follows that the dipoles have a mean diameter of approximately D=35 em, and thus waves with wave numbers k11<.2TT/2D=.9 m -I are the pos sible candidates. The frequency of these oscillations mea sured from Fig. 13 is approximately w=0.25 rad/s. Figure  14 shows that-according to relation ( 13 ) same frequency as the observed oscillation frequency (0.25 rad/s) of the dipolar vortex. Apparently, it is the wave mode 2 that causes the dipolar oscillation.

VI. CONCLUSIONS
A laboratory study on the evolution of a turbulent region injected at the interface of a two-layer fluid was performed. An isolated blob of turbulent fl uid in a two layer fl uid reaches its maximum height when the turbulent eddies in the blob come to a balance dictated by the vertical inertia forces of the eddies and the buoyancy forces asso ciated with them. Before this occurs, the effect of stratifi cation appears to be overshadowed by the inertia forces. After reaching this maximum height, the patch starts to collapse. This collapse leads to the formation of pancake like eddies, which may be modulated by interfacial waves. The results can be summarized as follows.
(i) The vertical thickness of the region grows indepen dent of the stratification for a time period tc= 1. 7 ( V0U 61ob 5 ) 11 8 • (ii) At tc, the stratification becomes important and arrests the vertical growth; the vertical size of the patch under this critical condition is determined by a balance between vertical inertia and buoyancy forces of the eddies, and is given by hc=.0.5 ( V0U 6/8b) 114• (iii) Measurements indicate an initial increase of the Thorpe scale to a maximum value (coinciding with he), and a decrease thereafter. The decrease is slow, but is still faster than the decrease of the patch size, owing to the physical collapse. The ratio of the Thorpe scale to the max imum Thorpe displacement was found to remain constant over the time period investigated. The relationship between the Thorpe scale and the patch size was measured to be L1::::; 0.18L P " The maximum Thorpe displacement was found to be given by L1max1 L P = 0.45. This' is somewhat low compared to that observed in previous studies, and the development of stratification within the mixed layer during its evolution may be the reason for this observation.
(iv) The dipolar structure observed in the present two layer experiments has the same appearance as in a linearly stratified fluid. However, in a two-layer fluid internal wave motions appear to be more energetic than in a linearly stratified fluid because of their trapping at the interface; thus the interaction between the waves and the vortices is strong. These internal waves are mainly of mode 2, and subsequently stretch and squeeze the vortices, inducing os cillations of the translation speed and of the horizontal size of the dipole.
( v) The growth of a turbulent patch in two-layer and linearly stratifi ed fluids (studied by FvHF) appear to be governed by similar dynamics, but some differences can be observed in detail. For example, in linearly stratified fluids the entrained fluid into the patch is subjected to mixing over the entire turbulent fl uid column, whereas in the two layer case a sharp interface may develop at the plane of symmetry, thus inhibiting the exchange of fluid between upper and lower halves of the patch. Thus the overturning distances can be smaller in the latter case, and the con stants that appear in the different balances (scaling argu ments) can be different for the two cases.