Turbulent skies have often inspired artists, particularly in the iconic swirls of Vincent van Gogh's The Starry Night. For an extended period, debate has raged over whether the flow pattern in this masterpiece adheres to Kolmogorov's theory of turbulence. In contrast to previous studies that examined only part of this painting, all and only the whirls/eddies in the painting are taken into account in this work, following the Richardson–Kolmogorov's cascade picture of turbulence. Consequently, the luminance's Fourier power spectrum spontaneously exhibits a characteristic Kolmogorov-like power-law. This result suggests that van Gogh had a very careful observation of real flows, so that not only the sizes of whirls/eddies in The Starry Night but also their relative distances and intensity follow the physical law that governs turbulent flows. Moreover, a “–1”-like power-law persists in the spectrum below the scales of the smallest whirls, hinting at Batchelor-type scalar turbulence with a high Schmidt number. Our study, thus, unveils the hidden turbulence captured within The Starry Night.
REFERENCES
1.
U.
Frisch
, Turbulence: The Legacy of an Kolmogorov
(
Cambridge University Press
, 1995
).2.
H. H.
Wensink
,
J.
Dunkel
,
S.
Heidenreich
,
K.
Drescher
,
R. E.
Goldstein
,
H.
Löwen
, and
J. M.
Yeomans
, “
Meso-scale turbulence in living fluids
,” Proc. Natl. Acad. Sci. U. S. A.
109
, 14308
–14313
(2012
).3.
X.
Qiu
,
L.
Ding
,
Y.
Huang
,
M.
Chen
,
Z.
Lu
,
Y.
Liu
, and
Q.
Zhou
, “
Intermittency measurement in two-dimensional bacterial turbulence
,” Phys. Rev. E
93
, 062226
(2016
).4.
Y.
Zhou
, “
Turbulence theories and statistical closure approaches
,” Phys. Rep.
935
, 1
–117
(2021
).5.
Z.
Warhaft
, “
The art of turbulence
,” Am. Sci.
110
, 360
–367
(2022
).6.
G.
Chen
,
S.
Yang
, and
N.
Jiang
, “
Leonardo da Vinci and fluid mechanics
,” Mech. Eng.
41
, 634
–639
(2019
) (in Chinese).7.
M.
Raissi
,
A.
Yazdani
, and
G. E.
Karniadakis
, “
Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
,” Science
367
, 1026
–1030
(2020
).8.
I.
Marusic
and
S.
Broomhall
, “
Leonardo da Vinci and fluid mechanics
,” Annu. Rev. Fluid Mech.
53
, 1
–25
(2021
).9.
A.
Colagrossi
,
S.
Marrone
,
P.
Colagrossi
, and
D.
Le Touzé
, “
Da Vinci's observation of turbulence: A French–Italian study aiming at numerically reproducing the physics behind one of his drawings, 500 years later
,” Phys. Fluids
33
, 115122
(2021
).10.
J. H.
Cartwright
and
H.
Nakamura
, “
What kind of a wave is Hokusai's great wave off Kanagawa?
,” Notes Rec. R. Soc.
63
, 119
–135
(2009
).11.
J. M.
Dudley
,
V.
Sarano
, and
F.
Dias
, “
On Hokusai's great wave off Kanagawa: Localization, linearity and a rogue wave in sub-Antarctic waters
,” Notes Rec. R. Soc.
67
, 159
–164
(2013
).12.
S.
Ornes
, “
Science and culture: Dissecting the great wave
,” Proc. Natl. Acad. Sci. U. S. A.
111
, 13245
–13245
(2014
).13.
J. L.
Aragón
,
G. G.
Naumis
,
M.
Bai
,
M.
Torres
, and
P. K.
Maini
, “
Turbulent luminance in impassioned van Gogh paintings
,” J. Math. Imaging Vision
30
, 275
–283
(2008
).14.
D. W.
Olson
, Celestial Sleuth: Using Astronomy to Solve Mysteries in Art, History and Literature
(
Springer
, 2014
).15.
16.
W. H.
Finlay
, “
The midrange wavenumber spectrum of van Gogh's Starry Night does not obey a turbulent inertial range scaling law
,” J. Turbul.
21
, 34
–38
(2020
).17.
M. L.
Spreafico
and
E.
Tramuns
, “
The Starry Night among art, maths, and origami
,” J. Math. Arts
15
, 1
–18
(2021
).18.
A.
Sherman
and
D.
Anderson
, “
How art contributes to scientific knowledge
,” Philos. Psychol.
36
, 1
–21
(2023
).19.
K.
Wright
, “
Arts & culture: Turbulence in the starry night
,” 2019
, see https://physics.aps.org/articles/v12/45.20.
L.
Richardson
, Weather Prediction by Numerical Process
(
Cambridge University Press
, 1922
).21.
A.
Alexakis
and
L.
Biferale
, “
Cascades and transitions in turbulent flows
,” Phys. Rep.
767–769
, 1
–101
(2018
).22.
A. N.
Kolmogorov
, “
Local structure of turbulence in an incompressible fluid at very high Reynolds numbers
,” Dokl. Akad. Nauk SSSR
30
, 301
(1941
).23.
24.
A.
Tsinober
, An Informal Conceptual Introduction to Turbulence
(
Springer Verlag
, 2009
).25.
H.
Tennekes
and
J. L.
Lumley
, A First Course in Turbulence
(
MIT Press
, 1972
).26.
A.
Groisman
and
V.
Steinberg
, “
Elastic turbulence in a polymer solution flow
,” Nature
405
, 53
–55
(2000
).27.
X.
Jian
,
W.
Zhang
,
Q.
Deng
, and
Y.
Huang
, “
Turbulent lithosphere deformation in the Tibetan Plateau
,” Phys. Rev. E
99
, 062122
(2019
).28.
K. R.
Sreenivasan
, “
Turbulent mixing: A perspective
,” Proc. Natl. Acad. Sci. U. S. A.
116
, 18175
–18183
(2019
).29.
A. M.
Obukhov
, “
Structure of the temperature field in a turbulent flow
,” Izv. Acad. Nauk SSSR Ser. Geog. Geofiz
13
, 58
–69
(1949
).30.
S.
Corrsin
, “
On the spectrum of isotropic temperature fluctuations in an isotropic turbulence
,” J. Appl. Phys.
22
, 469
–473
(1951
).31.
Z.
Warhaft
, “
Passive scalars in turbulent flows
,” Annu. Rev. Fluid Mech.
32
, 203
–240
(2000
).32.
G. K.
Batchelor
, “
Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity
,” J. Fluid Mech.
5
, 113
–133
(1959
).33.
D. A.
Donzis
,
K.
Sreenivasan
, and
P.
Yeung
, “
The Batchelor spectrum for mixing of passive scalars in isotropic turbulence
,” Flow. Turbul. Combust.
85
, 549
–566
(2010
).34.
C.
Gibson
and
W.
Schwarz
, “
The universal equilibrium spectra of turbulent velocity and scalar fields
,” J. Fluid Mech.
16
, 365
–384
(1963
).35.
X.
Wu
,
B.
Martin
,
H.
Kellay
, and
W.
Goldburg
, “
Hydrodynamic convection in a two-dimensional Couette cell
,” Phys. Rev. Lett.
75
, 236
–239
(1995
).36.
R.
Antonia
and
P.
Orlandi
, “
Effect of Schmidt number on small-scale passive scalar turbulence
,” Appl. Mech. Rev.
56
, 615
–632
(2003
).37.
P.
Yeung
,
S.
Xu
,
D.
Donzis
, and
K.
Sreenivasan
, “
Simulations of three-dimensional turbulent mixing for Schmidt numbers of the order 1000
,” Flow, Turbul. Combust.
72
, 333
–347
(2004
).38.
Y.
Amarouchene
and
H.
Kellay
, “
Batchelor scaling in fast-flowing soap films
,” Phys. Rev. Lett.
93
, 214504
(2004
).39.
P.
Götzfried
,
M. S.
Emran
,
E.
Villermaux
, and
J.
Schumacher
, “
Comparison of Lagrangian and Eulerian frames of passive scalar turbulent mixing
,” Phys. Rev. Fluids
4
, 044607
(2019
).40.
M.
Mohaghar
,
L. P.
Dasi
, and
D. R.
Webster
, “
Scalar power spectra and turbulent scalar length scales of high-Schmidt-number passive scalar fields in turbulent boundary layers
,” Phys. Rev. Fluids
5
, 084606
(2020
).41.
K.
Iwano
,
J.
Hosoi
,
Y.
Sakai
, and
Y.
Ito
, “
Power spectrum of high Schmidt number scalar in a turbulent jet at a moderate Reynolds number
,” Exp. Fluids
62
, 129
(2021
).42.
J.
Bedrossian
,
A.
Blumenthal
, and
S.
Punshon-Smith
, “
The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics at fixed Reynolds number
,” Commun. Pure Appl. Math.
75
, 1237
–1291
(2022
).43.
I.
Saito
,
T.
Watanabe
, and
T.
Gotoh
, “
Spectrum of passive scalar carried by particles in isotropic turbulence
,” Phys. Rev. Fluids
9
, 054601
(2024
).44.
We conducted the same analysis for each individual channel (not shown here). Apart from the blue channel, the Fourier power spectra of the red and green channels exhibited the same
and −1 scalings as found from the gray-scale field, indicating that the flow-like structures are well maintained.
45.
Y.
Gao
,
F. G.
Schmitt
,
J. Y.
Hu
, and
Y. X.
Huang
, “
Scaling analysis of the China France Oceanography SATellite along-track wind and wave data
,” J. Geophys. Res. Oceans
126
, e2020JC017119
, https://doi.org/10.1029/2020JC017119 (2021
).46.
F. G.
Schmitt
and
Y.
Huang
, Stochastic Analysis of Scaling Time Series: From Turbulence Theory to Applications
(
Cambridge University Press
, 2016
).47.
Y.
Huang
,
F.
Schmitt
,
Z.
Lu
,
P.
Fougairolles
,
Y.
Gagne
, and
Y.
Liu
, “
Second-order structure function in fully developed turbulence
,” Phys. Rev. E
82
, 026319
(2010
).48.
Y.
Huang
,
L.
Biferale
,
E.
Calzavarini
,
C.
Sun
, and
F.
Toschi
, “
Lagrangian single particle turbulent statistics through the Hilbert–Huang Transforms
,” Phys. Rev. E
87
, 041003(R)
(2013
).49.
Y.
Huang
,
F.
Schmitt
,
J.-P.
Hermand
,
Y.
Gagne
,
Z.
Lu
, and
Y.
Liu
, “
Arbitrary-order Hilbert spectral analysis for time series possessing scaling statistics: Comparison study with detrended fluctuation analysis and wavelet leaders
,” Phys. Rev. E
84
, 016208
(2011
).50.
M. P.
Clay
, “
Strained turbulence and low-diffusivity turbulent mixing using high performance computing
,” Ph.D. thesis,
Georgia Institute of Technology
, 2017
.51.
A.
Einstein
, “
On the movement of particles suspended in resting liquids required by the molecular kinetic theory of heat
,” Ann. Phys.
322
, 549
–560
(1905
).52.
R. H.
Kraichnan
, “
Small-scale structure of a scalar field convected by turbulence
,” Phys. Fluids
11
, 945
–953
(1968
).53.
G.
He
,
G.
Jin
, and
Y.
Yang
, “
Space-time correlations and dynamic coupling in turbulent flows
,” Annu. Rev. Fluid Mech.
49
, 51
–70
(2017
).54.
A.
Einstein
, “
Eine neue bestimmung der moleküldimensionen
,” Ann. Phys.
324
, 289
–306
(1906
).55.
We estimate here the order of Schmidt number, therefore the value of this ratio does not change our conclusion.
56.
J. H.
Rogers
, “
The accelerating circulation of Jupiter's Great Red Spot
,” J. Br. Astron. Assoc.
118
, 14
–20
(2008
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
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