12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

SMALL-SCALE DENSITY FLUCTUATIONS IN GRID-GENERATEDTURBULENCE IN A SALT-STRATIFIED FLUID

Shinya Okino
Department of Mechanical Engineering and Science
Kyoto University
Kyoto daigaku-katsura 4, Nishikyo-ku, Kyoto 615-8540, Japanokino.shinya.8n@kyoto-u.ac.jp

Soichiro Minakami
Department of Mechanical Engineering and Science
Kyoto University
Kyoto daigaku-katsura 4, Nishikyo-ku, Kyoto 615-8540, Japan
minakami.soichiro.46x@st.kyoto-u.ac.jp

Yuki Imanishi
Department of Mechanical Engineering and Science
Kyoto University
Kyoto daigaku-katsura 4, Nishikyo-ku, Kyoto 615-8540, Japan
imanishi.yuki.23m@st.kyoto-u.ac.jp

Hideshi Hanazaki
Department of Mechanical Engineering and Science
Kyoto University
Kyoto daigaku-katsura 4, Nishikyo-ku, Kyoto 615-8540, Japanhanazaki.hideshi.5w@kyoto-u.ac.jp

fusivity, so that the Schmidt number Sc(= ν /κ ) is as large as
700 (ν is the kinematic viscosity of ﬂuid and κ is the diffusion coefﬁcient). According to a theory established by Batchelor (1959), a high-Sc(≫ 1) scalar in turbulence diffuses at the
Batchelor wavenumber kB = Sc1/2 kK (kK is the Kolmogorov
wavenumber) if the scalar does not change the density of theﬂuid and can be assumed to be passive.

ABSTRACTThe density and velocity ﬁelds of grid-generated turbulence in a salt-stratiﬁed ﬂuid are measured by the laser-induced
ﬂuorescence (LIF) and the particle image velocimetry (PIV)
for several different mesh Froude numbers. The potentialenergy spectrum shows a local reduction
p near the primitive
wavenumber of stratiﬁed ﬂuid kP = N/ν (N is the BruntVäisälä frequency, and ν is the kinematic viscosity of ﬂuid),
which strongly supports the recent results from direct numerical simulations by Okino & Hanazaki (2020). The examination of the Froude number dependence suggests that for verystrong density stratiﬁcation, density perturbations below theprimitive scale are hardly generated, even for high Schmidt
numbers.

Most of the experiments on density stratiﬁed turbulence
have been used saltwater, with which the density stratiﬁcation
can be easily generated (e.g. Stillinger et al. 1983; Itsweire etal. 1986; Fincham et al. 1996; Praud et al. 2005). Density
ﬂuctuations in grid-generated stratiﬁed turbulence were measured by a conductivity probe in a series of water-channel experiments by Stillinger et al. (1983) and Itsweire et al. (1986),
but the spatial resolution of the probe was not high enough toresolve scales smaller than the Kolmogorov scale.

INTRODUCTION

Recently, direct numerical simulations of decaying stratiﬁed turbulence of high Schmidt numbers have been performed
to reveal the behaviour of small-scale density ﬂuctuations below the Kolmogorov scale (Okino & Hanazaki 2019, 2020).
In particular, Okino & Hanazaki (2020) studied the case of
Sc = 700 (corresponding to saltwater) and showed that thepower spectrum of potential energy (or density variance) decreases locally at the primitive wavenumber of stratiﬁed tur-

The atmosphere and the ocean, when averaged in time,
are the density stratiﬁed ﬂuids with larger density at lower altitude, and their dynamics are greatly inﬂuenced by the density stratiﬁcation. For example, the meridional circulation of
the deep ocean is driven by the density difference of seawater,
whose density is determined by temperature and salinity.
The salinity, in particular, has a very small molecular dif-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1. Schematic of the experimental facilities. A uniform density stratiﬁcation is initially generated in the test tank.
pbulence kP = N/ν (N is the Brunt-Väisälä frequency) after
the initial turbulence decays so much that the buoyancy effect reaches the Kolmogorov scale. The local reduction of thepotential energy spectrum was found to be caused by the persistently counter-gradient vertical density ﬂux at small scales(cf. Hanazaki & Hunt 1996; Komori & Nagata 1996).
In the present study, we have measured the density (orsalinity) ﬁeld of grid-generated turbulence in a salt-stratiﬁed
ﬂuid by the laser induced ﬂuorescence (LIF) with a spatial
resolution high enough to resolve sub-Kolmogorov scales,with the aim of validating the numerical prediction (Okino &Hanazaki 2020) and investigating the Froude number dependence of the small-scale density ﬂuctuations.

mission wavelength of 534.5 ± 21.5 nm. A lens with a focal
length of 50 mm (Nikkor 50mm f/1.2, Nikon) is used to observe the area of 18 × 18 cm2 at the centre of the cross-sectionof the test tank.
Since the dye concentration is made to be proportional to
the salinity (and the density) in the initial quiescent ﬂuid, thedensity remains generally proportional to the dye concentration even after turbulence is excited by the grid. Using theprinciple that the intensity of the ﬂuorescence increases withthe concentration of the ﬂuorescent dye (the Beer-Lambertlaw), the local density of the ﬂuid can be determined by thelight intensity photographed by the CCD camera. We shouldnote that the proportionality between the density and the dyeconcentration is incomplete below their diffusive scale due tothe difference in the Schmidt numbers of the ﬂuorescent dye(Sc ∼ 2100–3500) and saltwater (Sc ∼ 700). However, we
will restrict our focus to scales several times smaller than the
Kolmogorov scale, and will not discuss scales as small as the
Batchelor scale.
The velocity ﬁeld is measured by the particle image velocimetry (PIV) using a double-pulsed Nd:YAG laser (DualPower 65-15, Dantec Dynamics; Nano S PIV, Litron Lasers;
532 nm, 65 mJ, 15 Hz) and an acrylic emulsion as the seeding
particles.
The values of the mesh Reynolds number ReM = UM/νand the mesh Froude number FrM = U/(NM) used for thepresent experiments are listed in table 1.
The horizontal Reynolds number and the horizontal
Froude number are deﬁned by Reh = v4rms /(νε ) and Frh =ε /(Nv2rms ), respectively, where vrms is the root-mean-square(r.m.s) horizontal velocity and ε is the kinetic energy dissipation rate. The buoyancy Reynolds number deﬁned by
Reb = ε /(ν N 2 ) is a measure of stratiﬁcation effects at small
scales and is related to the horizontal Reynolds and Froudenumbers by Reb = Reh Frh2 . These non-dimensional numbers
at the time when the kinetic energy dissipation rate becomesmaximum are also given in table 1. The initial Froude numbersin the present experiments are smaller than those of Itsweire etal. (1986) (Frh > 2) though the initial Reynolds numbers are
similar (Reh ∼ 200). The initial buoyancy Reynolds number
is sufﬁciently larger than unity, so that small-scale ﬂuctuations
would not initially be affected by buoyancy, unlike the experiments by Praud et al. (2005).
The following two non-dimensional times are used to describe the results. One is the advection time tL = Ut/M and theother is the buoyancy time tN = t/TBV = Nt/(2π ), where TBV
is the Brunt-Väisälä period. The ratio of the two is determined

METHODSIn the present experiments, we excited turbulence bymoving a square grid horizontally in a salt-stratiﬁed ﬂuid generated in a test tank, and measured the density and velocity
ﬁelds in their decaying process. The test tank with a squarebottom of 60×60 cm2 was ﬁlled with a linearly stratiﬁed water
up to a height of 28 cm by a two-tank method (Fortuin 1960)(ﬁgure 1). The refractive index in the ﬂuid was made constantusing ethanol (index matching; cf. Daviero et al. (2001)) to
improve the accuracy of the LIF measurement. The unperturbed density distribution ρ (z) (z is the vertical coordinate) is
obtained by sampling a small amount of ﬂuid from cocks attached to one of the sidewalls of the test tank and measuring
its density by a density meter (DMA4500M, Anton Paar) withan accuracy of 5 × 10−5pg cm−3 . The Brunt-Väisälä frequency
is determined by N = (−g/ρ0 )(dρ (z)/dz), where ρ0 is thedensity at the bottom, and g is the gravitational acceleration.
We used two grids with different mesh widths M androd diameters d, i.e.
(M, d) = (4.0 cm, 0.4 cm) and(8.0 cm, 0.8 cm), but with the same solidity σ = (d/M)(2 −d/M) = 0.19. The square grid is moved horizontally along
the rail at the top of the tank by a DC motor (BXS230A-10S,
Oriental Motor) at a constant speed U.
The density distribution in a vertical plane is measured
by LIF using a continuous-wave laser (488 nm, 4 W). The
laser beam is vertically expanded into a laser sheet using acylindrical lens. As the ﬂuorescent dye, we used Rhodamine
110, whose peak absorption wavelength is 496 nm and whose
peak ﬂuorescence wavelength is 520 nm. The ﬂuorescence ofRhodamine 110 was photographed by a CCD camera (Hisense
4MC, Dantec Dynamics; C9300-024, Hamamatsu Photonics;
12 bits, 11 Hz) equipped with a band-pass ﬁlter with a trans-

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. List of the parameter used for the experiments. The values at the time when the kinetic energy dissipation rate becomesmaximum are denoted by the subscript 0.

FrM

ReM

M (cm)U (cm s−1 )N (s−1 )ε0 (cm2 s−3 )vrms,0 (cm s−1 )

Reh,0

Frh,0

Reb,0

16

9200

4.0

23.0

0.37

1.07

1.33

290

1.6

780

7.0

9200

4.0

23.0

0.83

0.978

1.28

270

0.72

140

1.8

9200

8.0

11.5

0.83

0.633

1.12

250

0.61

92

number of FrM = 1.8 (ﬁgure 3b), the buoyancy time becomesso large (tN ∼ 3.0) that the potential energy distribution exhibits horizontal cloud structures, resembling those observedfor FrM = 16 at the same buoyancy time (ﬁgure 2g). This suggests that the pattern of the potential energy distribution is better determined by the buoyancy time rather than the advection
time.
Figure 4 shows the temporal variation of the vertical spectrum of potential energy,

by the mesh Froude number, i.e.
tL
= 2π FrM .
tN(1)

RESULTSWe ﬁrst show in ﬁgure 2 the temporal variation of the distribution of potential energy, (g2 /2ρ02 N 2 )ρ ′ 2 , and kinetic energy, (1/2)(v2 + w2 ), in a vertical plane for FrM = 16, where
ρ ′ is the density perturbation, v the horizontal velocity and w
the vertical velocity. At an early time of tL = 29.3, thin sheetlike structures of high potential energy are distributed isotropically without a speciﬁc direction (ﬁgure 2a), since the buoyancy time is smaller than unity (tN = tL /(2π FrM ) ∼ 0.3) andthe buoyancy effect has not been signiﬁcant at small scales.
Then, the distribution of kinetic energy is also isotropic (ﬁgure2b). The smallest scale of potential energy is much smaller
than that of kinetic energy because the Schmidt number ismuch larger than unity (Sc ∼ 700 for saltwater). Figures 2(c,d)
show that similar structures can be seen even at a later timeof tL = 78.8 when the buoyancy time is still less than unity(tN ∼ 0.8). We note that the thin sheet-like structures of highSchmidt number scalar is observed also in isotropic turbulencewith a constant concentration-gradient of a passive (i.e. nonbuoyant) scalar (Brethouwer et al. 2003).
After the buoyancy time exceeds unity (ﬁgure 2e whentN ∼ 1.8), the distribution of the potential energy becomesanisotropic in that the ﬁne structures of high potential energy
cluster together to form a horizontally long structure (whose
length is ∼ M). The kinetic energy distribution also has suchhorizontal structures (ﬁgure 2f) though the structures as ﬁne as
those seen in the potential energy distribution does not exist.
As time passes further (tN ∼ 3.0), small-scale structures ofpotential energy have largely decayed, but large-scale clouds
composed of small-scale streaks can be identiﬁed (ﬁgure 2g).
The horizontal and vertical scales of the cloud are comparable to those of high kinetic energy (ﬁgure 2h). The temporalvariation of the structure of the potential energy distribution
from thin sheets to clouds is consistent with the recent resultof direct numerical simulation at Sc = 700 (Okino & Hanazaki2020).
We next compare the potential energy distributions forthree mesh Froude numbers (FrM = 16, 7.0 and 1.8) at a ﬁxed
advection time tL (∼ 30). It should be noted here that the buoyancy times tN (= tL /(2π FrM )) are different for different mesh
Froude numbers. As mentioned earlier, the potential energy
distribution for FrM = 16 (ﬁgure 2a) is isotropic and does not
signiﬁcantly affected by buoyancy since the buoyancy timeis tN ∼ 0.3(≪ 1). Similarly, for FrM = 7.0 (ﬁgure 3a), thebuoyancy time is still less than unity (tN ∼ 0.8), so that the
sheet-like structures of the potential energy are distributed almost isotropically. However, for the smallest mesh Froude

EP (kz ) =g2
2ρ0 2 N 2 ∑
ky



2

ρb′ (ky , kz ) + ρb′ (ky , −kz )

2



1
,
kmin

(2)
where ρb′ (ky , kz ) is the two-dimensional Fourier componentof the density perturbation ρ ′ (y, z) with a two-dimensional
wave-vector (ky , kz ). In decaying stratiﬁed turbulence, theKolmogorov wavenumber kK decreases with time, while theOzmidov wavenumber kO increases, since the kinetic energy
dissipation rate decreases. In the present experiments, kK andkO coincide at a buoyancy time of tN ∼ 1.
When FrM = 16 (ﬁgure 4a), the potential energy spectrum
exhibits Batchelor’s k−1 law near the Kolmogorov wavenumber at tN = 0.6. As time passes, however, EP (kz ) decreases locally andp bends at the primitive wavenumber of stratiﬁed ﬂuid
kP = N/ν , in agreement with the result of DNS (Okino &Hanazaki, 2020). The local reduction of the potential energy
at the primitive wavenumber is observed also for the stronger
stratiﬁcation of FrM = 7.0 and 1.8 (ﬁgure 4b,c) though thebending at kz = kP is less conspicuous.
The temporal variation of the amount of the potential energy at each wavenumber can be seen more readily from thepre-multiplied spectrum (ﬁgure 5), where the area below the
spectral curve corresponds to the amount of energy. Initially(tN ≃ 0.6), the pre-multiplied spectrum for FrM = 16 (ﬁgure5a) has a plateau (kz EP (kz ) ≃ const.) near the Kolmogorov
wavenumber, which corresponds to the k−1 law observed inﬁgure 4(a). When kK and kO approaches (tN ≃ 1.0), the potential energy decrease locally at kz = kP , generating two peaks inthe pre-multiplied spectrum (kz M ∼ 10 and 102 ).
Similarly for stronger stratiﬁcation (FrM = 7.0 and 1.8),
we can identify the local reduction of the potential energy
at the primitive wavenumber and the resulting two potentialenergy peaks (ﬁgure 5b,c). It should be noted, however, thatthe energy below kP increases relative to the energy above
kP as the stratiﬁcation increases. In other words, for weakstratiﬁcation of FrM = 16 (ﬁgure 5a), the potential energy
is contained more at small scales (kz > kP ) than large scales(kz < kP ), whereas for strong stratiﬁcation of FrM = 1.8 (ﬁgure 5c), the energy is contained more at large scales (kz < kP ).
The relative increase in the potential energy at large scales
would be because the potential energy is persistently converted
into the kinetic energy at the primitive wavenumber (Okino &3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
×10−3

(a)

(b)
×10−3
4.0

1.0

3.2

0.5

2.4
0.0
1.6

−0.5

0.8

−1.0

0.0
−1.0 −0.5

0.0
𝑦/𝑀

0.5

1.5
1.2

0.5

𝑧/𝑀

𝑧/𝑀

×10−3
1.0

0.9
0.0
0.6

−0.5
−1.0
−1.0

1.0

0.3
0.0
−0.5

0.0

0.5

𝑦/𝑀

(c)

1.0

×10−4

(d)
×10−3
2.5

1.0

1.0

2.0

0.5

1.5
0.0
1.0

−0.5

0.0
𝑦/𝑀

0.5

3.0
0.0
2.0

−1.0
−1.0

0.0
−1.0 −0.5

4.0

−0.5

0.5

−1.0

5.0

0.5

𝑧/𝑀

𝑧/𝑀

×10−4

1.0

1.0
0.0
−0.5

0.0

0.5

𝑦/𝑀

(e)

1.0

×10−4

(f)
×10−4
1.0

1.0

4.0

0.5

3.0
0.0
2.0

−0.5

0.0
𝑦/𝑀

0.5

0.9
0.0
0.6

−1.0
−1.0

0.0
−1.0 −0.5

1.2

−0.5

1.0

−1.0

1.5

0.5

𝑧/𝑀

𝑧/𝑀

×10−4

5.0

1.0

0.3
0.0
−0.5

0.0

0.5

𝑦/𝑀

(g)

1.0

×10−5

(h)
×10−4
5.0

1.0

4.0

0.5

3.0
0.0
2.0

−0.5

1.0

−1.0

0.0
−1.0 −0.5

0.0
𝑦/𝑀

0.5

5.0
4.0

0.5

𝑧/𝑀

𝑧/𝑀

×10−5
1.0

3.0
0.0
2.0

−0.5
−1.0
−1.0

1.0

1.0
0.0
−0.5

0.0

0.5

1.0

𝑦/𝑀

Figure 2. Temporal evolution of the distribution of (a,c,e,g) potential energy (g2 /2ρ02 N 2 )ρ ′ 2 /U 2 and (b,d,f,h) kinetic energy(1/2)(v2 + w2 )/U 2 in the vertical plane for FrM = 16. (a,b) tN = 0.3,tL = 29.3, (c,d) tN = 0.8,tL = 78.8, (e,f) tN = 1.8,tL = 173.1, and(g,h) tN = 3.0,tL = 288.1.

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)

(a)
1.0

𝑡 𝑁 = 0.6 (𝑡 𝐿 = 58.1)
𝑡 𝑁 = 0.8 (𝑡 𝐿 = 78.8)

10−5

𝑡 𝑁 = 1.0 (𝑡 𝐿 = 100.6)

𝐸 𝑃 (𝑘 𝑧 )/(𝑀𝑈 2)

3.2

0.5
𝑧/𝑀

10−4

×10−3
4.0

2.4
0.0
1.6

−0.5

0.8

−1.0

𝐸 𝑃 ∝ 𝑘 −1𝑧

10−6𝑡 𝑁 = 1.5 (𝑡 𝐿 = 144.3)
𝑡 𝑁 = 2.1 (𝑡 𝐿 = 201.8)
𝑡 𝑁 = 2.7 (𝑡 𝐿 = 259.3)

10−7
10−8

0.0
−1.0 −0.5

0.0
𝑦/𝑀

0.5

10−9 0
10

1.0

101

102

𝑘𝑧𝑀

103

104

(b)

(b)
1.0

𝑡 𝑁 = 0.7 (𝑡 𝐿 = 29.3)

𝐸 𝑃 (𝑘 𝑧 )/(𝑀𝑈 2)

2.4
0.0
1.6

−0.5
−1.0
−1.0 −0.5

0.0
𝑦/𝑀

0.5

𝑡 𝑁 = 1.0 (𝑡 𝐿 = 43.1)

10−5

3.2

0.5
𝑧/𝑀

10−4

×10−3
4.0

10−6

𝑡 𝑁 = 1.5 (𝑡 𝐿 = 65.0)
𝑡 𝑁 = 1.8 (𝑡 𝐿 = 78.8)
𝑡 𝑁 = 2.6 (𝑡 𝐿 = 115.6)

10−7

0.8

10−8

0.0

10−9 010

1.0

𝑡 𝑁 = 1.3 (𝑡 𝐿 = 58.1)

𝐸 𝑃 ∝ 𝑘 −1
𝑧

101

102

𝑘𝑧𝑀

103

104

(c)
10−4

Figure 3. Froude number dependence of the potential energy
distribution (g2 /2ρ02 N 2 )ρ ′ 2 /U 2 in the vertical plane at tL ∼30. (a) FrM = 7.0,tN = 0.8,tL = 36.2, and (b) FrM = 1.8,tN =3.0,tL = 33.2.

𝑡 𝑁 = 0.8 (𝑡 𝐿 = 8.8)

𝐸 𝑃 (𝑘 𝑧 )/(𝑀𝑈 2)

10−5

Hanazaki 2020) earlier than it is transferred to high wavenumbers.
As the stratiﬁcation becomes further stronger, the potential energy is expected to be absent at wavenumbers larger thanthe primitive wavenumber, even at high Schmidt numbers. To
conﬁrm this conjecture, we have performed a direct numericalsimulation at the initial Froude number of Fr0 = U0 /(NL0 ) =0.1 and Sc = 700, where L0 is the initial integral length and U0
the initial r.m.s. velocity. This initial Froude number is several
times smaller than the horizontal Froude number in the present
experiments (Frh > 0.6; cf. table 1). The numerical methods
are same as Okino & Hanazaki (2020).
Figure 6 shows the temporal evolution of the premultiplied potential-energy spectrum at Fr0 = 0.1. Indeed, thepotential energy is not transferred to wavenumbers above theprimitive wavenumber (indicated by the vertical dotted line) atall, and the most of the energy remains at the initial integral
scale (kL0 ∼ 3).

𝐸 𝑃 ∝ 𝑘 −1𝑧𝑡 𝑁 = 1.0 (𝑡 𝐿 = 11.1)
𝑡 𝑁 = 1.5 (𝑡 𝐿 = 16.5)
𝑡 𝑁 = 2.0 (𝑡 𝐿 = 22.0)
𝑡 𝑁 = 2.5 (𝑡 𝐿 = 27.5)

10−6

𝑡 𝑁 = 3.1 (𝑡 𝐿 = 33.2)

10−7
10−8
10−9 0
10

101

102

𝑘𝑧𝑀

103

104

Figure 4. The temporal evolution of the vertical spectrumof potential energy EP (kz ) for (a) FrM = 16, (b) FrM = 7.0,
and (c) FrM = 1.8. Symbols on each spectral curve show theKolmogorov wavenumber kK (circle), the Ozmidov wavenumber kO (square) and the primitive wavenumber of stratiﬁed
ﬂuid kP (triangle), which are deﬁned by kK = (ε /ν 3 )1/4 ,kO = (N 3 /ε )1/2 and kP = (N/ν )1/2 , respectively.

Kolmogorov wavenumber. As time elapses and the buoyancy effect extends beyond the Kolmogorov wavenumber, thepotential-energy spectrum shows a local reduction near theprimitive wavenumber of stratiﬁed ﬂuid, which is in good
agreement with the recent result from a direct numerical simulation by Okino & Hanazaki (2020).
When the density stratiﬁcation is very strong, density perturbations below the primitive scale are hardly generated, evenfor high Schmidt numbers such as saltwater. This would be
because the potential energy is persistently converted into thekinetic energy at the primitive wavenumber earlier than it is

CONCLUSIONS
We have performed tank experiments using a saltstratiﬁed water to investigate the time evolution and the Froudenumber dependence of the density ﬁeld of grid-generated turbulence. The distributions of the density and velocity in a vertical plane are measured by LIF and PIV, respectively.
Initially when the buoyancy effect has not reached
wavenumbers higher than the Kolmogorov wavenumber, thespectrum of potential energy exhibits the k−1 law near the5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)

0.16

×10−4

0.5

𝑡 𝑁 = 1.0
𝑡 𝑁 = 1.4
𝑡 𝑁 = 2.1
𝑡 𝑁 = 3.0
𝑡 𝑁 = 4.0
𝑡 𝑁 = 4.9
𝑡 𝑁 = 6.0

0.14
𝑡 𝑁 = 0.6 (𝑡 𝐿 = 58.1)
𝑡 𝑁 = 0.8 (𝑡 𝐿 = 78.8)

0.12

𝑡 𝑁 = 1.0 (𝑡 𝐿 = 100.6)

𝑘 × 𝐸 𝑃 (𝑘)/𝑈02

𝑘 𝑧 × 𝐸 𝑃 (𝑘 𝑧 )/𝑈 2

0.4

𝑡 𝑁 = 1.5 (𝑡 𝐿 = 144.3)
𝑡 𝑁 = 2.1 (𝑡 𝐿 = 201.8)

0.3

𝑡 𝑁 = 2.7 (𝑡 𝐿 = 259.3)

0.2

0.1
0.08
0.06
0.04

0.1
0.02

0.0 0
10

101

102

𝑘𝑧𝑀

103

0 −1
10

104

100

101
𝑘 𝐿0

102

103

(b)
×10−4

Figure 6. Numerical result of the radial spectrum of potential energy in the pre-multiplied form, i.e. k × EP (k), for verystrong stratiﬁcation. The initial Froude and Reynolds numbers
are Fr0 = U0 /(NL0 ) = 0.1 and Re0 = U0 L0 /ν = 50, respectively. Symbols on each spectral curve show the Kolmogorov
wavenumber kK (open circle), the Ozmidov wavenumber kO(closed circle) and the Batchelor wavenumber kB (triangle).
The vertical dotted line indicates the primitive wavenumber ofstratiﬁed ﬂuid kP .

1.0

𝑡 𝑁 = 0.7 (𝑡 𝐿 = 29.3)
𝑡 𝑁 = 1.0 (𝑡 𝐿 = 43.1)

𝑘 𝑧 × 𝐸 𝑃 (𝑘 𝑧 )/𝑈 2

0.8

𝑡 𝑁 = 1.3 (𝑡 𝐿 = 58.1)
𝑡 𝑁 = 1.5 (𝑡 𝐿 = 65.0)
𝑡 𝑁 = 1.8 (𝑡 𝐿 = 78.8)

0.6

𝑡 𝑁 = 2.6 (𝑡 𝐿 = 115.6)

0.4
0.2
0.0 0
10

101

102

𝑘𝑧𝑀

103

104

5, 113–133.
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2003 Micro-structure and Lagrangian statistics of the scalarﬁeld with a mean gradient in isotropic turbulence. J. Fluid
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Energy dissipation and vortex structure in freely decaying
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821–855.
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122.

(c)
×10−4

5.0

𝑡 𝑁 = 0.8 (𝑡 𝐿 = 8.8)𝑡 𝑁 = 1.0 (𝑡 𝐿 = 11.1)

𝑘 𝑧 × 𝐸 𝑃 (𝑘 𝑧 )/𝑈 2

4.0

𝑡 𝑁 = 1.5 (𝑡 𝐿 = 16.5)
𝑡 𝑁 = 2.0 (𝑡 𝐿 = 22.0)
𝑡 𝑁 = 2.5 (𝑡 𝐿 = 27.5)

3.0

𝑡 𝑁 = 3.0 (𝑡 𝐿 = 33.2)

2.0
1.0
0.0 0
10

101

102

𝑘𝑧𝑀

103

104

Figure 5. The temporal evolution of the vertical spectrum ofpotential energy in the pre-multiplied form, i.e. kz × EP (kz ),for (a) FrM = 16, (b) FrM = 7.0, and (c) FrM = 1.8. Symbols
on each spectral curve show the Kolmogorov wavenumber kK(circle), the Ozmidov wavenumber kO (square) and the primitive wavenumber of stratiﬁed ﬂuid kP (triangle).

transferred to high wavenumbers.
This study was supported by JSPS KAKENHI grant nos
21H01247, 21K03875, and used computational resources provided by Tohoku University through the HPCI System Research Project (hp210042) and by the Earth Simulator Centerof JAMSTEC.

REFERENCES
Batchelor, G. K. 1959 Small-scale variation of convected
quantities like temperature in turbulent ﬂuid. Part 1. General
discussion and the case of small conductivity. J. Fluid Mech.

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