Phys. Fluids 32, 095105 (2020); https://doi.org/10.1063/5.0016083 32, 095105

© 2020 Author(s).

Detection of standing internal gravitywaves in experiments with convection overa wavy heated wallCite as: Phys. Fluids 32, 095105 (2020); https://doi.org/10.1063/5.0016083Submitted: 11 June 2020 . Accepted: 13 August 2020 . Published Online: 02 September 2020

L. Barel, A. Eidelman, T. Elperin, G. Fleurov , N. Kleeorin , A. Levy , I. Rogachevskii , and O.Shildkrot

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Detection of standing internal gravity wavesin experiments with convection over a wavyheated wall

Cite as: Phys. Fluids 32, 095105 (2020); doi: 10.1063/5.0016083Submitted: 11 June 2020 • Accepted: 13 August 2020 •Published Online: 2 September 2020

L. Barel, A. Eidelman, T. Elperin, G. Fleurov, N. Kleeorin, A. Levy, I. Rogachevskii,a) and O. Shildkrot

AFFILIATIONSThe Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-GurionUniversity of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

a)Author to whom correspondence should be addressed: gary@bgu.ac.il

ABSTRACT

Convection over a wavy heated bottom wall in the air flow has been studied in experiments with the Rayleigh number of ∼108. It is shownthat the mean temperature gradient in the flow core inside a large-scale circulation is directed upward, which corresponds to the stablystratified flow. In the experiments with a wavy heated bottom wall, we detect large-scale standing internal gravity waves (IGWs) excited inthe regions with the stably stratified flow. The wavelength and the period of these waves are much larger than the turbulent spatial and timescales, respectively. In particular, the frequencies of the observed large-scale waves vary from 0.006 Hz to 0.07 Hz, while the turbulent time inthe integral scale is about 0.5 s. The measured spectra of these waves contain several localized maxima that imply an existence of waveguideresonators for large-scale standing IGWs. For comparisons, experiments with convection over a smooth plane bottom wall at the same meantemperature difference between the bottom and upper walls have also been conducted. In these experiments, various locations with a stablystratified flow are also found, and large-scale standing IGWs are observed in these regions.

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I. INTRODUCTION

Temperature stratified turbulence in convective and stablystratified flows has been investigated theoretically, experimentally,and in numerical simulations due to numerous applications in geo-physical, astrophysical, and industrial flows.1–10 One of the keyingredients of stably stratified flows is internal gravity waves (IGWs).In atmospheric and oceanic turbulence, they have been a subject ofintense research.11–31 In the atmosphere, internal gravity waves existat scales ranging from meters to kilometers and are measured bydirect probing or remote sensing using radars and lidars.17,25 Thesources of internal gravity waves can be flows over complex terrain,strong wind shears, convective and other local-scale motions under-lying the stably stratified layer, and wave–wave interactions.17,18

The internal gravity wave propagation is complicated by variablewind and density profiles causing refraction, reflection, focusing,and ducting.

Internal gravity waves can strongly affect the small-scale tur-bulence. In particular, these waves create additional productions of

turbulent energy and additional vertical turbulent fluxes of momen-tum and heat. In particular, the waves emitted at a certain level prop-agate upward, and the losses of wave energy cause the productionof turbulence energy. These effects have been studied theoretically,where the energy- and flux-budget (EFB) turbulence closure the-ory, which accounts for large-scale internal gravity waves (IGWs)for stably stratified atmospheric flows, has been developed.32,33 Forthe stationary (in statistical sense) and homogeneous turbulence, theEFB theory without large-scale IGW yields universal dependenciesof the main turbulence parameters on the flux Richardson num-ber [defined as the ratio of the consumption of turbulent kineticenergy (TKE) needed for overtaking buoyancy forces to the TKEproduction by the velocity shear].34–37 Due to the large-scale IGW,these dependencies lose their universality.32,33 The maximum valueof the flux Richardson number (universal constant of ≈0.25 in theabsence of the large-scale IGW) becomes strongly variable. In thevertically homogeneous stratification, the flux Richardson numberincreases with increasing wave energy and can even exceed 1. Forheterogeneous stratification, when internal gravity waves propagate

Phys. Fluids 32, 095105 (2020); doi: 10.1063/5.0016083 32, 095105-1

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toward stronger stratification, the maximal flux Richardson numberdecreases with increasing wave energy and can even reach very smallvalues.32,33

Internal gravity waves also reduce the anisotropy of turbulence:in contrast to the mean wind shear, which generates only horizon-tal TKE, internal gravity waves generate both horizontal and verticalTKE. A well-known effect of internal gravity waves is their directcontribution to the vertical transport of momentum. Depending onthe direction of the wave propagation (downward or upward), inter-nal gravity waves either strengthen or weaken the total vertical fluxof momentum.32,33

Even in a convective turbulence, stably stratified regions can beformed, where the mean temperature gradient in the flow core insidethe large-scale circulation is directed upward.38,39 In these regions,internal gravity waves are generated, which affect the turbulence.Despite many studies of stratified turbulence and internal gravitywaves, a mechanism of formation of the stably stratified regions inconvective turbulence and generation of internal gravity waves inthese regions is not comprehensively studied and understood.

In various flows, a complex terrain (i.e., canopy and varioustopographies) strongly affects the stratified turbulence.40 It changeslocal temperature gradients and heat and mass fluxes and affects alocal structure of fluid flows. Therefore, one of the important ques-tions is what is the effect of complex terrain on convective and stablystratified turbulence.

To model the effect of complex terrain, the Rayleigh–Bénardconvection (RBC) with modulated boundaries has been investi-gated.38,41–47 In particular, the influence of a modulated bound-ary on the RBC by a lithographically fabricated periodic textureon the bottom plate has been studied experimentally.44 The dif-ferent convection patterns have been obtained by varying theRayleigh number and the wave number of the modulated bound-ary. For small Rayleigh numbers, convection takes the form ofstraight parallel rolls. With increasing Rayleigh number, a secondaryinstability is excited and the convection has more complex pat-terns.44 This secondary instability has been studied theoretically andnumerically.45

The roughness effect on the heat transport in the RBC has beeninvestigated in two-dimensional numerical simulations by varyingthe height and wavelength of the roughness elements where the sinu-soidal roughness profile has been chosen.38,46 The ultimate regime ofthermal convection (when the boundary layers undergo a transitionleading to the generation of smaller scales near the boundaries thatincrease the system’s efficiency in transporting the heat) has recentlybeen found.38 This regime in which the heat flux becomes indepen-dent of the molecular properties of the fluid has been predicted.49,50

The first experiment designed to use roughness to reach the ulti-mate regime at accessible Rayleigh numbers has been conducted.47

One of the key roles of the roughness elements is the productionand release of the plumes from the roughness elements, resulting inthe formation of larger plumes. This can cause an increase in theefficiency of the heat transfer.38,48 The existence of two universalregimes in the RBC, namely, the ultimate regime and the classicalboundary-layer-controlled regime with increased Rayleigh number,has been demonstrated.46 The transition from the first to the sec-ond regime is determined by the competition between bulk andboundary-layer flow. The bulk-dominated regime corresponds tothe ultimate regime.

In the present study, we investigate other aspects related to theroughness effect on turbulent convection. In particular, we studythe formation of the stably stratified regions and excitation of large-scale standing internal gravity waves in laboratory experiments withturbulent convection over a wavy heated bottom wall in air as theworking fluid. We also compare results of these experiments to thoseobtained in experiments with a smooth plane bottom wall at thesame temperature difference between the bottom and upper walls.

This paper is organized as follows. In Sec. II, we present theo-retical analysis, which allows us to determine the frequencies of thestanding internal gravity waves in stably stratified flow. In Sec. III, wedescribe the experimental setup and instrumentation for the labora-tory study of the internal gravity waves. The results of the laboratoryexperiments are discussed in Sec. IV. Finally, conclusions are drawnin Sec. V.

II. INTERNAL GRAVITY WAVESLet us first consider the internal gravity waves in a stably strat-

ified fluid flow in the absence of turbulence and neglecting dissipa-tions. These waves are described by the linearized momentum andentropy equations written in the Boussinesq approximation,1,13–15

∂VW

∂t= −∇PW

ρeq+ g SWe, (1)

∂SW

∂t= −g−1N2 VW ⋅ e, (2)

where

SW = cv[(1 − γ)PW

Peq+ γ

TW

Teq]. (3)

Here, the wave fields VW, PW, SW, and TW are the perturbationsof the velocity, pressure, entropy, and temperature, respectively, eis the vertical unit vector, g is the acceleration due to gravity, andγ = cp/cv is the ratio of the specific heats, where cp and cv are thespecific heats at constant pressure and volume, respectively. TheBrunt–Väisälä frequency is N(z) = (g∇zSeq)1/2, where

∇zSeq = cv[(1 − γ)∇z ln Peq + γ∇z ln Teq], (4)

and the fields Seq, Peq, Teq, and ρeq are the entropy, pressure, tem-perature, and density at an equilibrium given by the following equa-tions: Veq = 0 and ∇Peq = ρeqg, where Veq is the velocity at equi-librium. For conditions pertinent to the laboratory experimentsdiscussed in Secs. III and IV, ∇Peq ≈ 0 so that ∇zSeq ≈ cp∇z ln Teq.The classical Boussinesq approximation with div VW = 0 is appliedhere.

Equations (1) and (2) yield the frequency ω of the internalgravity waves,

ω = N(z)kh

k, (5)

where k = kh + ekz is the wave vector and kh = (kx, ky) is the wavevector in the horizontal direction. Propagation of the internal gravitywaves in the stably stratified flow in the approximation of geomet-rical optics is determined by the following Hamiltonian equations:

∂r∂t= ∂ω

∂k, (6)

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∂k∂t= −∂ω

∂r, (7)

where r is the radius-vector of the center of the wave packet.51 Sincethe Brunt–Väisälä frequency N = N(z), the only non-zero spatialderivative, ∇zω ≠ 0, is in the vertical direction. Therefore, Eq. (7)yields kh = const. The vertical component of the wave vector kz= kz(z) is determined from Eq. (5),

kz(z) = kh(N2(z)ω2 − 1)

1/2

. (8)

We perform standard calculations by taking twice curl toexclude the pressure term in Eq. (1), calculating the time derivativeof the obtained equation and using Eq. (2). This procedure yields thefollowing equation:

∂2

∂t2 ΔVW = [∇(e ⋅∇) − eΔ]VWz N2(z), (9)

which is equivalent to the system of equations for the vertical andhorizontal velocity components,

∂2

∂t2 ΔVWz = −N2(z)Δ⊥VW

z , (10)

∂2

∂t2 ΔVW⊥ = ∇⊥∇z[VW

z N2(z)]. (11)

Here, VW = VW⊥ + eVW

z with VW⊥ = (VW

x , VWy ) being the hor-

izontal velocity and Δ = Δ⊥ + ∇2z . The solution of Eq. (10) for the

vertical velocity VWz (t, r)we seek for in the form of standing internal

gravity waves existing in the range zmin ≤ z ≤ zmax. It is as follows:

VWz (t, r) = V∗ cos(ωt) cos(kh ⋅ r) sin(∫

z

zmin

kz(z′)dz′ + φ), (12)

where V∗ is the amplitude of the vertical velocity for the wave field.Substituting Eq. (12) into Eq. (11), we determine perturbations ofthe horizontal velocity VW

⊥ (t, r) for the wave field as follows:

VW⊥ (t, r) = −kh

kz(z)k2

hN2 V∗ cos(ωt) sin(kh ⋅ r)

× cos(∫z

zmin

kz(z′)dz′ + φ). (13)

Using Eqs. (2) and (12), we obtain the solution for perturba-tions of the entropy SW(t, r),

SW(t, r) = N2(z)g ω

V∗ sin(ωt) cos(kh ⋅ r) sin(∫z

zmin

kz(z′)dz′ + φ).

(14)

Solution (12) should satisfy the following boundary conditions:VW

z (z = zmin) = 0 and VWz (z ≈ zmax) = 0. The latter boundary

condition at the vicinity of z ≈ zmax implies that

∫zmax

zmin

kz(z′)dz′ = π(m +14), (15)

where zmax is the reflection (or “turning”) point in which kz(z = zmax)= 0. To get condition (15), we use an analogy with the behavior

of the wave function near the turning points in the semi-classicallimit applied in quantum mechanics.52,53 The wave functions aredescribed in terms of the Airy functions, and Eq. (15) is analogousto the Bohr–Sommerfeld quantization condition. In particular, thephase φ in Eqs. (12)–(14) is determined using an asymptotic solutionof Eq. (10) in the vicinity of the turning points,

VWz =

12ξ−1/4 exp(−2

3ξ3/2) for ξ ≫ 1,

VWz =

12∣ξ∣−1/4 sin(2

3∣ξ∣3/2 +

π4) for ∣ξ∣≪ 1,

(16)

where ξ ∝ zmax − z. Equation (15) allows us to determine the fre-quencies of the standing internal gravity waves for the quadraticprofile of the Brunt–Väisälä frequency N2(z) = N2

0(1 − z2/L2N). In

particular, calculating the integral ∫zmaxzmin kz(z′)dz′, we obtain

ωm =N0

khLN

⎧⎪⎪⎨⎪⎪⎩[(m +

14)

2+ (khLN)2]

1/2

− (m +14)⎫⎪⎪⎬⎪⎪⎭

, (17)

where LN is the characteristic scale of the Brunt–Väisälä frequencyvariations and m = 0, 1, 2, . . .. Equation (17) implies the existenceof a discrete spectrum of the standing internal gravity waves. In thelong wavelength limit, khLN ≪ 1, and Eq. (17) yields

ωm =2N0

4 m + 1khLN . (18)

We will apply Eq. (17) in the experimental study of convec-tion over a wavy heated bottom surface, where the large-scale inter-nal gravity waves are excited in the stably stratified regions formedinside the flow core of the large-scale circulation.

III. EXPERIMENTAL SETUPIn this section, we describe the experimental setup. The experi-

ments have been conducted in air as the working fluid in rectangularchamber with dimensions Lx × Ly × Lz , where Lx = Lz = 26 cm, Ly= 56 cm, and the axis z is in the vertical direction. The side walls ofthe chamber are made of transparent Perspex with the thickness of1 cm.

A vertical mean temperature gradient in the turbulent air flowis formed by attaching two aluminum heat exchangers to the bot-tom and top walls of the test section (a heated bottom and a cooledtop wall of the chamber). A thickness of the massive aluminumheat exchangers is 2 cm. The top plate is a bottom wall of the tankwith cooling water. Cold water is pumped into the cooling systemthrough two inlets and flows out through two outlets located at theside wall of the cooling system. The bottom plate is attached tothe electrical heater with wire tightly laid in the grooves milled inthe aluminum plate and provided uniform heating. Energy suppliedto the heater is varied in order to obtain the necessary tempera-ture difference between the heater and cooler. The characteristictime of the heater is approximately 90 min that stabilizes the appliedtemperature during measurements.

We study the effects of complex terrain on the structure ofthe velocity and temperature fields in temperature stratified turbu-lence. In the laboratory experiments, complex terrain is modeledby a wavy bottom surface of the chamber, which is manufacturedfrom aluminum. It is produced from a 4 cm thick plate. The wavy

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FIG. 1. A sketch of the experimental setup: the optics (1) and the laser light sheet(2) of the PIV system; the chamber (4); the cooler (3) at the top surface and theheater (5) at the bottom surface; the data recorder (6) for temperature measure-ment system; and the CCD camera (7) and the generator of incense smoke (8) forthe PIV measurements.

bottom surface has a sinusoidal modulation containing 7 periodswith a wavelength of 8 cm and amplitude of 1 cm. A sketch of theexperimental setup is shown in Figs. 1 and 2. The results of theseexperiments are compared to those obtained in experiments witha smooth plane bottom surface at the same temperature differencebetween the top and bottom plates.

The temperature field is measured with a temperature probeequipped with 12 E-thermocouples (with the diameter of 0.13 mmand the sensitivity of ≈65 μV/K) attached to a rod with a diameter4 mm. The spacing between thermocouples along the rod is 22 mm.Each thermocouple is inserted into a case with 1 mm diameter and45 mm length. A tip of a thermocouple protruded at the length of15 mm out of the case. Thermocouples of type E are used for thetemperature measurements in the core flow, while thermocouples

FIG. 2. A sketch of the chamber: the cooler (1) at the top surface; the heater (4) atthe bottom surface; the temperature probe (2) equipped with 12 E-thermocouples;and a wavy bottom surface (3) with a sinusoidal modulation.

of type K are used for temperature measurements at the heater andthe cooler. All thermocouples are built by a manufacturer.54 Cali-brations of all E-thermocouples in the temperature probe have beenperformed for three experiments with boiled water (T = 373 K), coldwater with ice (T = 273 K), and water with intermediate tempera-ture (T = 296 K). Comparisons have been made using a precisiontemperature measurement manufactured device.

The temperature is measured for 11 rod positions with 20 mmintervals in the horizontal direction. This probe is used in the exper-iments with a smooth plane bottom surface, while a temperatureprobe equipped with 13 E-thermocouples is used in the experimentswith the wavy bottom surface. A sequence of 500 temperature read-ings for every thermocouple at every rod position is recorded andprocessed. We have measured the temperature field in many loca-tions. Performing direct continuous measurements of the tempera-tures at the cooled top surface and at the heated bottom surface andusing a standard device for supporting constant temperature differ-ence ΔT between the top and bottom surfaces (Contact voltage reg-ulator TDGC-2K), we control the constant temperature differenceΔT during the experiments.

The velocity field is measured by stereoscopic particle imagevelocimetry.55–57 In the experiments, we use the LaVision FlowMaster III system. A double-pulsed light sheet is provided by anNd:YAG laser (Continuum Surelite 2 × 170 mJ). The light sheetoptics includes spherical and cylindrical Galilei telescopes with tun-able divergence and adjustable focus length. We use the progressive-scan 12 bit digital CCD camera (with a pixel size 6.7 × 6.7 μm2

and 1280 × 1024 pixels) with a dual-frame-technique for cross-correlation processing of captured images. A programmable timingunit (PC interface card) generated sequences of pulses to control thelaser, camera, and data acquisition rate.

An incense smoke with sub-micron particles is used as a tracerfor the PIV measurements. Smoke is produced by high temperaturesublimation of solid incense grains. Analysis of smoke particles usinga microscope (Nikon, Epiphot with an amplification of 560) and aPM-300 portable laser particulate analyzer shows that these particleshave an approximately spherical shape and that their mean diameteris of the order of 0.7 μm. The maximum tracer particle displacementin the experiment is of the order of 1/4 of the interrogation window.The average displacement of tracer particles is of the order of 2.5pixels. The average accuracy of the velocity measurements is of theorder of 4% for the accuracy of the correlation peak detection in theinterrogation window of the order of 0.1 pixels.55–57

We determine the mean and the rms velocities, two-point cor-relation functions, and an integral scale of turbulence from the mea-sured velocity fields. Series of 520 pairs of images acquired with afrequency of 2 Hz are stored for calculating velocity maps and forensemble and spatial averaging of turbulence characteristics. Thecenter of the measurement region coincides with the center of thechamber. We measure velocity in a flow domain 256× 503 mm2 witha spatial resolution of 393 μm/pixels. The velocity field in the probedregion is analyzed with interrogation windows of 32 × 32 pixels. Inevery interrogation window, a velocity vector is determined fromwhich velocity maps comprising 27 × 53 vectors are constructed.The mean and rms velocities for every point of a velocity map arecalculated by averaging over 520 independent velocity maps, and theobtained averaged velocity map is averaged also over the central flowregion.

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The two-point correlation functions of the velocity field aredetermined for every point of the central part of the velocity map(with 16 × 16 vectors) by averaging over 520 independent velocitymaps, which yields 16 correlation functions in horizontal and ver-tical directions. The two-point correlation function is obtained byaveraging over the ensemble of these correlation functions. An inte-gral scale of turbulence, ℓ, is determined from the two-point corre-lation functions of the velocity field. The turbulence time scale at theintegral scale is τ = ℓ/

√⟨u2⟩, where u are the velocity fluctuations

and√⟨u2⟩ is the rms of the velocity fluctuations. In the experiments,

we evaluated the variability between the first and the last 20 velocitymaps of the series of the measured velocity field. Since very smallvariability is found, these tests show that 520 velocity maps containenough data to obtain reliable statistical estimates. The size of theprobed region does not affect our results.

In the experimental study, we employ a triple decomposition,whereby the instantaneous temperature Ttot = T + θ, where θ arethe temperature fluctuations and T is the temperature determinedby sliding averaging of the instantaneous temperature field over thetime (5 s), that is, by one order of magnitude larger than the char-acteristic turbulence time (0.5 s). The temperature T is given by asum, T = T + TW, where TW are the long-term variations of thetemperature T due to the large-scale standing internal gravity wavesaround the mean value T. The mean temperature T is obtainedby the additional averaging of the temperature T over the time500 s.

The time interval during which the temperature field is mea-sured at every point is 500 s. In the temperature measurements, theacquisition frequency of the temperature is 1.25 Hz, and the corre-sponding acquisition time is 0.8 s. It is larger than the characteristicturbulence time (see below) and is much smaller than the periodof the long-term oscillations of the mean temperature caused byinternal gravity waves. Therefore, the acquisition frequency of tem-perature is high enough to provide sufficiently long time series forstatistical estimation of the mean temperature T and the long-termvariations TW of the temperature due to the large-scale internal grav-ity waves. Similar experimental setups, measurement techniques fortemperature and velocity fields, and data processing procedures havebeen used previously in our experimental study of different aspectsof turbulent convection58,59 and stably stratified turbulence.60,61

IV. EXPERIMENTAL RESULTSIn this section, we describe experimental results related to

the formation of stably stratified regions in turbulent convectionand excitation of internal gravity waves. We perform two sets ofexperiments with a wavy bottom surface and a smooth plane bot-tom surface for the same imposed mean temperature difference ΔTbetween the bottom and upper surfaces of the chamber. The velocitymeasurements show that, in the both sets of experiments, a singlelarge-scale circulation is observed for the temperature differencesΔT = 40 K and ΔT = 50 K.

In Fig. 3, we show the mean velocity patterns obtained inthe experiments with the wavy bottom surface (upper panel) andsmooth plane bottom surface (bottom panel). A difference in themean velocity patterns is seen at the vicinity of the wavy bottomsurface where the flows with the sinusoidal modulation of the mean

FIG. 3. The mean velocity patterns in the yz plane for the experiments with thewavy bottom surface (upper panel) and the smooth plane bottom surface (bottompanel) obtained at ΔT = 50 K. Here, z and y are measured in mm and velocity ismeasured in m/s.

velocity field are observed (see the upper panel in Fig. 3). For lowerand higher temperature differences between the bottom and topsurfaces of the chamber, two large-scale circulations are observedand the mean temperature field is more complicated. This case willbe investigated in a separate study. Experiments with smaller Lzincrease the aspect ratio of the chamber, where more than two large-scale circulations can be observed and the mean temperature fieldwill be much more complicated. This case will be also investigatedin a separate study.

Let us discuss parameters in the experiments. The characteristicturbulence time is τ = 0.28 s–0.62 s, while the characteristic periodfor the large-scale circulatory flow is about 10 s, which is by the orderof magnitude larger than the turbulence time τ. These two character-istic times are much smaller than the time during which the veloc-ity fields are measured (∼260 s). The maximum Rayleigh number,Ra = α g L3

z ΔT/(ν κ), in the turbulent convection is about 108, whereα is the thermal expansion coefficient, ν is the kinematic viscosity, κis the thermal diffusivity, and Lz is the height measured from thelower point of a wavy surface with the sinusoidal modulation to theupper surface of the chamber.

The temperature measurements in the experiments with a wavybottom surface show that the mean temperature gradient in the flowcore inside the large-scale circulation is directed upward (∇zT > 0),which corresponds to a stably stratified flow. For instance, in Figs. 4and 6, we plot isolines of the vertical mean temperature gradient∇zTinside the large-scale circulation in the yz plane where we show theregions with the positive vertical mean temperature gradient only.In Figs. 5 and 7, we also plot mean profiles of the vertical mean

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FIG. 4. Isolines in the yz plane of the vertical mean temperature gradient ∇zTinside the large-scale circulation in the experiments with the wavy bottom surfacefor ΔT = 50 K. Only the regions with positive vertical mean temperature gradientare shown. Here, y and z are measured in mm, and the mean temperature gradientis measured in K/cm.

temperature gradient ∇zT obtained in these experiments by aver-aging over 15 vertical profiles measured in different cross sectionsfor different y inside the large-scale circulation. We discuss here theresults of the experiments performed for two values of the tempera-ture difference ΔT between the bottom and top surfaces: ΔT = 50 K(Figs. 4 and 5) and ΔT = 40 K (Figs. 6 and 7). All measurementshave been conducted in the plane x = 0.5Lx. The accuracy of themean temperature measurements is about 0.07% (i.e., the accuracyis about 0.2 K for typical mean temperatures ≈300 K). The accu-racy of the mean temperature gradient measurements is about 5%for∇zT ≈ 0.1 K/cm, and it is about 10% for∇zT ≈ 0.01 K/cm.

As can be seen in Fig. 4, there are two regions with positivemean temperature gradient separated by the region with negativemean temperature gradient in the center of the core flow. In theregions with negative vertical mean temperature gradient, the valueof N2 ≤ 0 and the internal gravity waves cannot exist in these regions.This means that, in the experiments with ΔT = 50 K, there are twowaveguides for the internal gravity waves. On the other hand, as canbe seen in Fig. 6 (ΔT = 40 K), there is one large region with positivemean temperature gradient which corresponds to a one waveguide.Similar trends can be seen in Figs. 5 and 7. In these experiments, the

FIG. 5. Mean profile of the vertical mean temperature gradient∇zT in the experi-ments with the wavy bottom surface for ΔT = 50 K obtained by averaging over 15vertical profiles measured in different cross sections for different y inside the large-scale circulation. Here, the height z is measured in mm, and the mean temperaturegradient is measured in K/cm.

FIG. 6. Isolines in the yz plane of the vertical mean temperature gradient ∇zTinside the large-scale circulation in the experiments with the wavy bottom surfacefor ΔT = 40 K. Only the regions with positive vertical mean temperature gradientare shown. Here, y and z are measured in mm, and the mean temperature gradientis measured in K/cm.

maximum gradient Richardson number, Ri = N2/Sh2, based on themean velocity shear, Sh, of the large-scale circulation is about 1.

For comparison of results in two types of experiments withwavy and smooth plane bottom surfaces, in Figs. 8 and 10, we showisolines of the vertical mean temperature gradient ∇zT in the yzplane inside the large-scale circulation obtained in the experimentswith the smooth plane bottom surface. Similarly, in Figs. 9 and 11,we also show mean profiles of the vertical mean temperature gradi-ent∇zT obtained in these experiments by averaging over 11 verticalprofiles measured in different cross sections for different y insidethe large-scale circulation. Here, Figs. 8 and 9 correspond to theexperiments with ΔT = 50 K, while Figs. 10 and 11 are for theexperiments with ΔT = 40 K. Since the large-scale circulation is notsymmetric due to the formation of small additional vortices near theright wall of the chamber (see Fig. 3 at y > 400 mm), we show, inFigs. 4, 6, 8, and 10, the isolines of the vertical mean temperaturegradient ∇zT inside the large-scale circulation only in the range ofy ≤ 400 mm.

Figures 4–11 demonstrate that the spatial distributions of thevertical mean temperature gradient ∇zT in the experiments with

FIG. 7. Mean profile of the vertical mean temperature gradient∇zT in the experi-ments with the wavy bottom surface for ΔT = 40 K obtained by averaging over 15vertical profiles measured in different cross sections for different y inside the large-scale circulation. Here, the height z is measured in mm, and the mean temperaturegradient is measured in K/cm.

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FIG. 8. Isolines in the yz plane of the vertical mean temperature gradient ∇zTinside the large-scale circulation in the experiments with the smooth plane bottomsurface for ΔT = 50 K. Only the regions with positive vertical mean temperaturegradient are shown. Here, y and z are measured in mm, and the mean temperaturegradient is measured in K/cm.

FIG. 9. Mean profile of the vertical mean temperature gradient∇zT in the experi-ments with the smooth plane bottom surface for ΔT = 50 K obtained by averagingover 11 vertical profiles measured in different cross sections for different y insidethe large-scale circulation. Here, the height z is measured in mm, and the meantemperature gradient is measured in K/cm.

FIG. 10. Isolines in the yz plane of the vertical mean temperature gradient ∇zTinside the large-scale circulation in the experiments with the smooth plane bottomsurface for ΔT = 40 K. Only the regions with positive vertical mean temperaturegradient are shown. Here, y and z are measured in mm, and the mean temperaturegradient is measured in K/cm.

FIG. 11. Mean profile of the vertical mean temperature gradient∇zT in the exper-iments with the smooth plane bottom surface for ΔT = 40 K obtained by averagingover 11 vertical profiles measured in different cross sections for different y insidethe large-scale circulation. Here, the height z is measured in mm, and the meantemperature gradient is measured in K/cm.

the smooth plane bottom surface are slightly different from thoseobtained in the experiments with the wavy bottom surface. In partic-ular, in the experiments with the smooth plane bottom surface, thereare fewer locations with positive vertical mean temperature gradient∇zT than those in the experiments with the wavy bottom surface.

To study internal gravity waves, we determine the spectrum ofthe long-term variations of the temperature field characterizing thelarge-scale waves. From this spectral analysis, we obtain main fre-quencies of internal gravity waves. The spectrum function ET( f )= TW( f ) TW∗( f ) for the temperature field (containing 500 fre-quency data points) has been determined at 80 locations of the sta-bly stratified region, where TW( f ) is the Fourier component of thetemperature TW(t). For every frequency f, the obtained spectrumfunctions ET(f ) have been averaged over 80 locations.

In Figs. 12 and 13, we show the volume averaged spectrumfunction ET( f ) = ⟨ET( f )⟩vol of the temperature field in the experi-ments with the wavy bottom surface for the temperature differencesΔT = 50 K (Fig. 12) and ΔT = 40 K (Fig. 13), where the brack-ets ⟨⋯⟩vol denote the averaging over 80 locations. For comparison,in Figs. 14 and 15, we show the volume averaged spectrum func-tion ET( f ) = ⟨ET( f )⟩vol of the temperature field in the experimentswith the smooth plane bottom surface for ΔT = 50 K (Fig. 14) and

FIG. 12. The averaged spectrum function ET( f ) of the temperature field obtainedin the experiments with the wavy bottom surface for the temperature differencesΔT = 50 K. The main frequencies of the large-scale internal gravity waves mea-sured in Hz are indicated above the maxima. The function T( f ) is measuredin K2/Hz.

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FIG. 13. The averaged spectrum function ET( f ) of the temperature field obtainedin the experiments with the wavy bottom surface for the temperature differencesΔT = 40 K. The main frequencies of the large-scale internal gravity waves mea-sured in Hz are indicated above the maxima. The function T( f ) is measuredin K2/Hz.

FIG. 14. The averaged spectrum function ET( f ) of the temperature field obtainedin the experiments with the smooth bottom surface for the temperature differencesΔT = 50 K. The main frequencies of the large-scale internal gravity waves mea-sured in Hz are indicated above the maxima. The function T( f ) is measuredin K2/Hz.

FIG. 15. The averaged spectrum function ET( f ) of the temperature field obtainedin the experiments with the smooth bottom surface for the temperature differencesΔT = 40 K. The main frequencies of the large-scale internal gravity waves mea-sured in Hz are indicated above the maxima. The function T( f ) is measuredin K2/Hz.

FIG. 16. The normalized frequencies f /f 0 of the internal gravity waves vs the nor-malized horizontal wavelength λh/LN : theoretical curves for different modes m = 0(solid), m = 1 (dashed), and m = 3 (dashed–dotted) and measured frequencies inthe experiments with the wavy bottom surface for the temperature differences ΔT= 50 K (triangles) and ΔT = 40 K (snowflakes) and in the experiments with thesmooth bottom surface for ΔT = 50 K (diamonds) and ΔT = 40 K (circles).

ΔT = 40 K (Fig. 15). It is seen in Figs. 12–15 that the measured spec-trum contains several localized maxima in all experiments with thewavy and smooth plane bottom surfaces, which can be interpretedas large-scale standing internal gravity waves excited in the regionswith the stably stratified flow.

Figure 16 shows the theoretical dependencies for the normal-ized frequencies f /f 0 of the standing internal gravity waves vs thenormalized horizontal wavelength λh/LN given by Eq. (17), whereLN ≈ 10 cm is the characteristic vertical scale of the Brunt–Väisälafrequency variations, λh = 2π/kh is the horizontal wavelength, and f 0= N0/2π. The theoretical curves are plotted for the main modes of thestanding internal gravity waves. When m > 2, the theoretical curvesare located very close to each other, so we plot, in Fig. 16, the theo-retical curves for m = 0, 1, 3. Comparing these theoretical curves andthe measured frequencies of the large-scale standing internal grav-ity waves obtained in the temperature measurements, we determinethe horizontal wave-numbers for the observed large-scale standinginternal gravity waves.

In Fig. 16, we show the measured frequencies of the large-scalestanding internal gravity waves in the experiments with the wavyand smooth plane bottom surfaces for the temperature differencesΔT = 50 K and ΔT = 40 K. In the experiments with the wavy bottomsurface, the frequency f 0 based on the maximum Brunt–Väisäla isf 0 = 8.2 × 10−2 Hz for ΔT = 50 K and f 0 = 7.1 × 10−2 Hz for ΔT= 40 K. In the experiments with the smooth plane bottom surface,the frequency f 0 = 1.08 × 10−1 Hz for ΔT = 50 K and f 0 = 6.5 × 10−2

Hz for ΔT = 40 K. As follows from Figs. 12–15, in all experiments,there are eight main frequencies of the large-scale standing internalgravity waves. Since we show, in Fig. 16, only the limited range ofthe horizontal wavelength, 0 < λh/LN < 15 of the standing internalgravity waves, for higher frequencies, 0.6 < f /f 0 < 1, we observe threemodes for m = 0, 1, 3, for the intermediate range, 0.1 < f /f 0 < 0.5,we observe two modes for m = 1, 3, and for lower frequencies, f /f 0< 0.1, we observe only one mode for m = 3.

The measured wavelength and the period of these waves aremuch larger than the turbulent spatial and time scales, respectively.In particular, the frequencies of the observed large-scale waves varyfrom 0.006 Hz to 0.07 Hz in the experiments with the wavy bottom

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surface and from 0.03 Hz to 0.065 Hz in the experiments with thesmooth plane bottom surface, while the turbulent time in the inte-gral scale is about 0.5 s. Note that, inside the chamber, we observestanding but not propagating internal gravity waves because thechamber is a small closed volume. Actually, interactions of propa-gating and reflecting waves cause the standing internal gravity waves,which are observed in our experiments.

We also compare the results of measurements of velocity fluctu-ations in the two sets of experiments with the wavy and smooth planebottom surfaces, conducted at the same mean temperature differ-ence between the bottom and upper surfaces. In Fig. 17, we show thevertical profiles of the rms velocity components urms

y =√⟨u2

y⟩ and

urmsz =

√⟨u2

z⟩, where the spatial average is over the reduced domain(excluding the end boundaries). In the experiments with the wavybottom surface, there is a minimum in the vertical profile of the rmsvelocities in the center part of the chamber. On the other hand, inthe experiments with the smooth plane bottom surface, the verticaldistribution of the rms velocities is essentially less inhomogeneous.Also, the values of turbulent velocities in the experiments with thewavy bottom surface are smaller compared to velocity fluctuationsin the experiments with the smooth plane bottom surface.

The reason for these differences may be as follows. In the exper-iments with the wavy bottom surface, the intensity of the velocity

FIG. 17. The vertical profiles of the rms velocities urmsy (snowflakes) and urms

z(circles) in the experiments with the wavy bottom surface (upper panel) and thesmooth plane bottom surface (bottom panel) obtained at ΔT = 50 K. Here, z ismeasured in mm, and the rms velocities urms

y and urmsz are measured in cm/s.

fluctuations are partially depleted by the production of the tempera-ture fluctuations in stably stratified regions, where the part of turbu-lent kinetic energy is transformed to the turbulent potential energycaused by the temperature fluctuations.34,37 In another words, instably stratified flows, the production of turbulent kinetic energy isdepleted by destruction caused by the buoyancy. On the other hand,in the experiments with the smooth plane bottom surface, there areless locations of the stably stratified regions, and the level of veloc-ity fluctuations is larger in comparison with that observed in theexperiments with the wavy bottom surface.

V. CONCLUSIONSIn the present study, we perform laboratory experiments in

convection over a wavy heated bottom surface. An interaction of thelarge-scale circulation with the wavy heated bottom surface stronglyaffects the spatial structure of the mean temperature field and gen-eration of the large-scale standing internal gravity waves in the flowcore. In particular, we have found that there are many locations withstably stratified regions in the flow core of the large-scale circula-tion, and the large-scale standing internal gravity waves are observedin these regions. The wavelength and the period of these waves aremuch larger than the turbulent spatial and time scales. The spectrumof these waves contains several localized maxima, which is an indi-cation of the existence of the waveguide resonators for the internalgravity waves.

In the experiments with a smooth plane bottom surface at thesame temperature difference between bottom and upper surfaces,there are fewer locations with a stably stratified turbulence. How-ever, the large-scale standing internal gravity waves are detected inboth experiments with the wavy and smooth plane bottom surfaces.

The turbulence in the region with stably stratified flows in theexperiments with the wavy bottom surface is inhomogeneous, e.g.,there is a minimum in the vertical profiles of the rms velocities in thecenter part of the chamber, where the intensity of the velocity fluc-tuations are partially depleted by the production of the temperaturefluctuations. On the other hand, in the experiments with the smoothplane bottom surface, the vertical distribution of the rms velocitiesis nearly homogeneous.

ACKNOWLEDGMENTSThis paper is dedicated to Professor T. Elperin (1949-2018),

who initiated this work. The authors thank A. Krein for his assis-tance in construction of the experimental setup and E. Elmakiesfor his assistance in data analysis. The detailed comments on ourmanuscript by the anonymous referees are very much appreciated.This research was supported in part by the Israel Ministry of Sci-ence and Technology (Grant No. 3-16516) and the PAZY Founda-tion of the Israel Atomic Energy Commission (IAEC) (Grant No.122-2020).

NOMENCLATURE

ΔT temperature difference between the bottomand upper surfaces of the chamber

α thermal expansion coefficientγ = cp/cv ratio of the specific heats

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θ temperature fluctuationsκ thermal diffusivityλh horizontal wavelengthν kinematic viscosityρeq density at an equilibriumτ turbulence time scale at the integral scale of

turbulenceφ phase of the velocity perturbationsω frequency of the internal gravity wavescp specific heat at constant pressurecv specific heat at constant volumee vertical unit vectorET( f ) spectrum function for the temperature fieldET( f ) volume averaged spectrum function for the

temperature fieldf frequencyg acceleration due to gravityk wave vectorkh = (kx, ky) horizontal component of the wave vectorkz vertical component of the wave vectorℓ integral scale of turbulenceLN characteristic vertical scale of the Brunt–

Väisälä frequency variationsLx and Ly horizontal sizes of the chamber along the x

and y axesLz height measured from the lower point of a

wavy surface with the sinusoidal modulationto the upper surface of the chamber

N(z) = (g∇zSeq)1/2 Brunt–Väisälä frequencyPeq pressure at equilibriumPW pressure wave field (the long-term variations

of the pressure due to the large-scale internalgravity waves)

r radius-vector of the center of the wave packetRa Rayleigh numberRi gradient Richardson numberSeq entropy at equilibriumSW entropy wave field (the long-term variations

of the entropy due to the large-scale internalgravity waves)

Sh mean velocity shear of the large-scale circula-tion

T = T + TW temperature determined by sliding averagingof the instantaneous total temperature fieldover the time (5 s), which is by one orderof magnitude larger than the characteristicturbulence time (0.5 s)

T mean temperature obtained by the averagingof the temperature T over the time 500 s

Teq temperature at equilibriumTtot = T + TW + θ total instantaneous temperatureTW = T − T temperature wave field (the long-term vari-

ations of the temperature due to the large-scale internal gravity waves)

TW( f ) Fourier component of the temperature TW(t)u velocity fluctuationsurms

y rms of the turbulent horizontal velocity

urmsz rms of the turbulent vertical velocity

V∗ amplitude of the vertical velocity for the wavefield

Veq velocity at equilibriumVW velocity wave field (the long-term variations

of the velocity due to the large-scale internalgravity waves)

VW⊥ = (VW

x , VWy ) horizontal component of the velocity wave

fieldzmax reflection (or “turning”) point

DATA AVAILABILITY

The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.

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