Coupled convection and internal gravity waves excited in water around its density maximum at $4°\mathrm{C}$

Coupled mixed convective and stratified systems are common in natural flows. To study experimentally the associated dynamics, we use a singular property of water: its nonlinear equation of state is characterized by a maximum density close to $4°\mathrm{C}$. By heating the top of a tank at $35°\mathrm{C}$ and cooling the bottom at $0°\mathrm{C}$, a two-layer configuration spontaneously appears. The convective motion in the bottom layer consists mostly of a large-scale circulation and rising cold plumes. This turbulent flow generates propagating internal gravity waves in the upper stably stratified layer. Particle image velocimetry measurements are performed and spectral characteristics of the convection and internal gravity waves are presented. An horizontal large-scale reversing flow in the stratified layer is observed which is viscously driven by a third, intermediate layer. This buffer layer is located between the convective and stratified layers and is thermally coupled with the convective one, hence sustaining a strong horizontal shear. Three-dimensional direct numerical simulations with geometry and physical parameters close to the experimental ones corroborate our experimental results.

Figure 1

2D sketch of the tank. A cylinder (light grey shaded area) is placed in a larger cubic tank. The system is cooled down at the bottom at $0°\mathrm{C}$ and heated up at the top at $35°\mathrm{C}$. The bottom half is convective with an almost constant density, apart from the bottom boundary layer. The upper half is stably stratified and waves are generated due to the fluid motions in the convective layer. The graph on the right shows the theoretical profile for the buoyancy frequency $N^{*}$. It is computed considering a linear temperature profile and the equation of state of water (1). The dashed line is the global buoyancy frequency $N$ calculated on the stratified layer. The various length scales are the cylinder diameter $D$, the vertical extent of the tank $H$ and the vertical extent of the convective layer $h$. $\delta$ is the minimal width between the outer square tank and the inner cylinder. The $x, y$ and $z$-velocity components are noted $u, v$ and $w$ respectively.

Figure 2

Left: Instantaneous velocity field. An ascending plume is visible at $x=150$ mm, and transported by the large-scale circulation. Right: Large-scale circulation in the convective layer obtained by time-averaging velocities over a 50 minute signal. The large-scale circulation is a counterclockwise cell. No inversion of the circulation has been seen in our experiment. The velocities under $z=10$ mm are noisy and thus not shown here. Maximum instantaneous velocities are 2.5 times bigger than the maximum averaged velocities. The left edge of the cylinder is located at $x=19$ mm and the center of the cylinder is at $x=160$ mm. Approximately 40 mm of the right side of the cylinder is not captured with our PIV measurments.

Figure 3

Time evolution of the vertical velocity $w$ within (left) upward plumes at $x=200$ mm, $z=45$ mm and (right) downward structures at $x=100$ mm, $z=95$ mm.

Figure 4

PSD for the vertical velocity fluctuations. Left: PSD computed at a single point $x=100$ mm, $z=95$ mm (signal shown in Fig. 3 right). The plume forcing frequency can be seen around $f={10}^{-2}$ Hz (red dashed line). Right: PSD spatially averaged over the whole convective cell in the measured ($x,z$) plane (all points above $z=10$ mm and below $z=110$ mm).

Figure 5

Probability density function of the vertical velocities in the convective layer. All PIV points under $z=110$ mm have been used to compute the PDF.

Figure 6

Time evolution of the horizontal average of the horizontal velocity, noted $u_X$. Red (resp. blue) regions correspond to mean flow going towards the right (resp. left).

Figure 7

Sketch of the thermal coupling between the convective and buffer layers. On the left, a cold plume moves upwards towards the interface between the two layers. On the right, isotherms are deflected due to the impact.

Figure 8

Velocity fields showing IGWs propagating. Top: Velocities in $(x,z)$ plane. The signal is frequency filtered to enhance the visualization of oscillatory motions: only frequencies between 0.02 and 0.03 Hz are shown, propagating at an angle of roughly $75°$ with the vertical. The angle of propagation is the angle between the constant phase line and the vertical. Bottom: Velocities in $(x,y)$ plane located at $z\approx 125$ mm. In the $(x,y)$ plane, IGWs take the form of oscillating rings. Note that this figure is from a previous experiment without any internal cylinder and is therefore only displayed here as an illustration of the IGWs seen from above.

Figure 9

Left: Power spectral density of the absolute velocity $\sqrt{u^2+w^2}$ above the convective layer. The grey curve shows the theoretical buoyancy frequency profile, assuming a linear profile for the temperature, from $4°\mathrm{C}$ to $35°\mathrm{C}$. Right: Two selected profiles (taken at frequencies shown by dashed lines on the left graph) of the rescaled PSDs by the PSD at the top of the convective layer, i.e., $z=120$ mm (denoted ${\mathrm{PSD}}_{\mathrm{o}}$). The PIV measurements are performed for 50 minutes and results (using the pwelch matlab function) are horizontally averaged at each height to obtain the averaged power spectral density.

Figure 10

Horizontal velocity vector fields in the stratified layer at different times. The laser sheet is located at $z=150$ mm. The large-scale flow reverses from (a) to (c). Time between (a) and (c) is approximately half an hour. Maximum velocities are 0.1 mm/s.

Figure 11

Criterion used to extract a significant value for the large-scale flow and its direction: the azimuthal velocity is averaged over the ring shown in red.

Figure 12

Reversals of the large-scale flow. $z=115\text{–}120$ mm is the convective/buffer layer interface. Ascending plumes often perturb the buffer layer flow. The velocity was measured at 11 heights, marked by each tick in the vertical axis of the figure, and interpolated in between. The slope of the black dot lines represents the viscous coupling phase velocity.

Figure 13

Velocity vector fields in the horizontal plane. Different columns represent different sweeping cycles $t^{*}$ (one sweeping cycle corresponds to the 11 steps needed to go from the lowest position $z=110$ mm to the highest position $z=160$ mm). Different rows represent different heights within the same sweeping cycle: the first row is the top of the convection $z=110$ mm, the second row is in the buffer layer $z=120$ mm, and the third row is in the stratified layer $z=145$ mm. Convective plumes are easily noticeable on the first row fields.

Figure 14

Same as Fig. 13 but for different sweeping cycles. Note that in this set of velocity fields the buffer and stratified layers are less correlated than they are in Fig. 13.

Figure 15

Velocity correlation between the three layers. Dashed black lines are the $-0.5$ and 0.5 values. Correlation coefficient are window averaged over 10 min to smooth the curve.

Figure 16

Left: Instantaneous velocity field. An ascending plume is visible at $x=230$ mm. Right: Large-scale circulation in the convective layer obtained by time averaging velocities over a 50 minute recording. The large-scale circulation is a counterclockwise cell. Maximum instantaneous velocities are 3 times bigger than the maximum averaged velocities.

Figure 17

Left: Horizontal average of the horizontal velocity $u$ over a vertical cross section in the middle of the tank. The buffer layer can be seen above $z=120$ mm. A stationary large-scale circulation is present in the convective layer, even if it appears quite perturbed at the end of the signal. Right: Temperature profiles along the $z$ axis.

Figure 18

Velocity field and temperature isotherms at the end of an upward plume impact on the interface.

Figure 19

Left: Spatio-temporal diagram of the horizontal velocity $u$ at $z=128$ mm. Right: Spatio-temporal diagram of the vertical velocity $w$ at $z=108$ mm. The event at $t\approx 1.052\times {10}^5$ s is shown in Fig. 18.

Figure 20

Left: Power spectral density of the absolute velocity $\sqrt{u^2+w^2}$ in the buffer and stratified layers. The grey curve shows the buoyancy frequency profile computed from the spatial and temporal average of the temperature field. Right: Two selected profiles (taken at frequencies shown by dashed lines on the left graph) of the rescaled PSDs by the PSD at the top of the convective layer, i.e., $z=118$ mm.

Figure 21

Spatio-temporal diagrams of the azimuthal averaging of the azimuthal velocity inside a virtual cylinder of radius $r=140\mathrm{mm}$. The bottom figure is a zoom on the stratified zone, delimited in the top figure by the black square. The slope of the black lines shows the theoretical viscous diffusive time.

Figure 22

Horizontal average of the horizontal velocity $u$ over a vertical cross section in the middle of the numerical domain for $\text{Pr}=0.1$.