Kraichnan’s seminal ideas on inverse cascades yielded new tools to study common phenomena in geophysical turbulent flows. In the atmosphere and the oceans, rotation and stratification result in a flow that can be approximated as two-dimensional at very large scales but which requires considering three-dimensional effects to fully describe turbulent transport processes and non-linear phenomena. Motions can thus be classified into two classes: fast modes consisting of inertia-gravity waves and slow quasi-geostrophic modes for which the Coriolis force and horizontal pressure gradients are close to balance. In this paper, we review previous results on the strength of the inverse cascade in rotating and stratified flows and then present new results on the effect of varying the strength of rotation and stratification (measured by the inverse Prandtl ratio N/f, of the Coriolis frequency to the Brunt-Väisäla frequency) on the amplitude of the waves and on the flow quasi-geostrophic behavior. We show that the inverse cascade is more efficient in the range of N/f for which resonant triads do not exist, 1/2N/f2. We then use the spatio-temporal spectrum to show that in this range slow modes dominate the dynamics, while the strength of the waves (and their relevance in the flow dynamics) is weaker.

1.
R. H.
Kraichnan
, “
Inertial ranges in two-dimensional turbulence
,”
Phys. Fluids
10
,
1417
(
1967
).
2.
R. H.
Kraichnan
and
D.
Montgomery
, “
Two-dimensional turbulence
,”
Rep. Prog. Phys.
43
,
547
(
1980
).
3.
H. J. H.
Clercx
and
G. J. F.
van Heijst
, “
Energy spectra for decaying 2D turbulence in a bounded domain
,”
Phys. Rev. Lett.
85
,
306
(
2000
).
4.
A.
Bracco
,
J. C.
McWilliams
,
G.
Murante
,
A.
Provenzale
, and
J. B.
Weiss
, “
Revisiting freely decaying two-dimensional turbulence at millennial resolution
,”
Phys. Fluids
12
,
2931
(
2000
).
5.
H.
Kellay
and
W. I.
Goldburg
, “
Two-dimensional turbulence: A review of some recent experiments
,”
Rep. Prog. Phys.
65
,
845
(
2002
).
6.
L.
Biferale
,
S.
Musacchio
, and
F.
Toschi
, “
Inverse energy cascade in three-dimensional isotropic turbulence
,”
Phys. Rev. Lett.
108
,
164501
(
2012
).
7.
G.
Boffetta
and
R. E.
Ecke
, “
Two-dimensional turbulence
,”
Annu. Rev. Fluid Mech.
44
,
427
(
2011
).
8.
P. D.
Mininni
and
A.
Pouquet
, “
Inverse cascade behavior in freely decaying two-dimensional fluid turbulence
,”
Phys. Rev. E
87
,
033002
(
2013
).
9.
A.
Pouquet
, “
On two-dimensional magnetohydrodynamic turbulence
,”
J. Fluid Mech.
88
,
1
(
1978
).
10.
A. C.
Ting
,
W. H.
Matthaeus
, and
D.
Montgomery
, “
Turbulent relaxation processes in magnetohydrodynamics
,”
Phys. Fluids
29
,
3261
(
1986
).
11.
M.
Christensson
,
M.
Hindmarsh
, and
A.
Brandenburg
, “
Inverse cascade in decaying three-dimensional magnetohydrodynamic turbulence
,”
Phys. Rev. E
64
,
056405
(
2001
).
12.
P. D.
Mininni
,
D. C.
Montgomery
, and
A. G.
Pouquet
, “
A numerical study of the alpha model for two-dimensional magnetohydrodynamic turbulent flows
,”
Phys. Fluids
17
,
035112
(
2005
).
13.
A.
Alexakis
,
P. D.
Mininni
, and
A.
Pouquet
, “
On the inverse cascade of magnetic helicity
,”
Astrophys. J.
640
,
335
(
2006
).
14.
P. D.
Mininni
, “
Inverse cascades and α effect at a low magnetic Prandtl number
,”
Phys. Rev. E
76
,
026316
(
2007
).
15.
P.
Démoulin
and
E.
Pariat
, “
Modelling and observations of photospheric magnetic helicity
,”
Adv. Space Res.
43
,
1013
(
2009
).
16.
E. N.
Lorenz
, “
The predictability of a flow which possesses many scales of motion
,”
Tellus
21
,
289
(
1969
).
17.
C. E.
Leith
, “
Atmospheric predictability and two-dimensional turbulence
,”
J. Atmos. Sci.
28
,
145
(
1971
).
18.
C. E.
Leith
and
R. H.
Kraichnan
, “
Predictability of turbulent flows
,”
J. Atmos. Sci.
29
,
1041
(
1972
).
19.
G.
Boffetta
and
S.
Musacchio
, “
Predictability of the inverse energy cascade in 2D turbulence
,”
Phys. Fluids
13
,
1060
(
2001
).
20.
J. G.
Charney
, “
Geostrophic turbulence
,”
J. Atmos. Sci.
28
,
1087
(
1971
).
21.
J. R.
Herring
, “
The inverse cascade range of quasi-geostrophic turbulence
,”
Meteorol. Atmos. Phys.
38
,
106
(
1988
).
22.
G.
Boffetta
,
F.
De Lillo
, and
S.
Musacchio
, “
Inverse cascade in Charney-Hasegawa-Mima turbulence
,”
Europhys. Lett.
59
,
687
(
2002
).
23.
S.
Fox
and
P. A.
Davidson
, “
The competition between quadratic and integral invariants in inviscid truncated two-dimensional and quasigeostrophic shallow-water turbulence
,”
Phys. Fluids
21
,
125102
(
2009
).
24.
A.
Vallgren
and
E.
Lindborg
, “
Charney isotropy and equipartition in quasi-geostrophic turbulence
,”
J. Fluid Mech.
656
,
448
(
2010
).
25.
W.-C.
Müller
and
M.
Thiele
, “
Scaling and energy transfer in rotating turbulence
,”
Europhys. Lett.
77
,
34003
(
2007
).
26.
P. A.
Davidson
,
Turbulence in Rotating Stratified and Electrically Conducting Fluids
(
Cambridge University Press
,
Cambridge
,
2013
).
27.
G. D.
Nastrom
,
K. S.
Gage
, and
W. H.
Jasperson
, “
Kinetic energy spectrum of large-and mesoscale atmospheric processes
,”
Nature
310
,
36
(
1984
).
28.
G. D.
Nastrom
and
K. S.
Gage
, “
A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft
,”
J. Atmos. Sci.
42
,
950
(
1985
).
29.
S.
Sukoriansky
,
N.
Dikovskaya
, and
B.
Galperin
, “
On the arrest of inverse energy cascade and the Rhines scale
,”
J. Atmos. Sci.
64
,
3312
(
2007
).
30.
R. B.
Scott
and
F.
Wang
, “
Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry
,”
J. Phys. Oceanogr.
35
,
1650
(
2005
).
31.
F.
Schlösser
and
C.
Eden
, “
Diagnosing the energy cascade in a model of the North Atlantic
,”
Geophys. Res. Lett.
34
,
L02604
, (
2007
).
32.
M. K.
Verma
, “
Variable enstrophy flux and energy spectrum in two-dimensional turbulence with Ekman friction
,”
Europhys. Lett.
98
,
14003
(
2011
).
33.
D. K.
Lilly
, “
Stratified turbulence and the mesoscale variability of the atmosphere
,”
J. Atmos. Sci.
40
,
749
(
1983
).
34.
R.
Salmon
,
Lectures on Geophysical Fluid Dynamics
(
Oxford University Press
,
New York
,
1998
).
35.
E.
Lindborg
, “
The effect of rotation on the mesoscale energy cascade in the free atmosphere
,”
Geophys. Res. Lett.
32
,
L01809
, (
2005
).
36.
J. J.
Riley
and
E.
Lindborg
, “
Stratified turbulence: A possible interpretation of some geophysical turbulence measurements
,”
J. Atmos. Sci.
65
,
2416
(
2008
).
37.
L. M.
Smith
and
F.
Waleffe
, “
Generation of slow large scales in forced rotating stratified turbulence
,”
J. Fluid Mech.
451
,
145
(
2002
).
38.
J.-P.
Laval
,
J. C.
McWilliams
, and
B.
Dubrulle
, “
Forced stratified turbulence: Successive transitions with Reynolds number
,”
Phys. Rev. E
68
,
036308
(
2003
).
39.
M. L.
Waite
and
P.
Bartello
, “
Stratified turbulence dominated by vortical motion
,”
J. Fluid Mech.
517
,
281
(
2004
).
40.
M. L.
Waite
and
P.
Bartello
, “
The transition from geostrophic to stratified turbulence
,”
J. Fluid Mech.
568
,
89
(
2006
).
41.
A.
Sen
,
P. D.
Mininni
,
D.
Rosenberg
, and
A.
Pouquet
, “
Anisotropy and nonuniversality in scaling laws of the large-scale energy spectrum in rotating turbulence
,”
Phys. Rev. E
86
,
036319
(
2012
).
42.
C.
Rorai
,
D.
Rosenberg
,
A.
Pouquet
, and
P. D.
Mininni
, “
Helicity dynamics in stratified turbulence in the absence of forcing
,”
Phys. Rev. E
87
,
063007
(
2013
).
43.
C.
Rorai
,
P. D.
Mininni
, and
A.
Pouquet
, “
Turbulence comes in bursts in stably stratified flows
,”
Phys. Rev. E
89
,
043002
(
2014
).
44.
K. L.
Polzin
and
Y. V.
Lvov
, “
Toward regional characterizations of the oceanic internal wavefield
,”
Rev. Geophys.
49
,
RG4003
, (
2011
).
45.
G. N.
Ivey
,
K. B.
Winters
, and
J. R.
Koseff
, “
Density stratification, turbulence, but how much mixing?
,”
Annu. Rev. Fluid Mech.
40
,
169
(
2008
).
46.
P. C.
di Leoni
and
P. D.
Mininni
, “
Absorption of waves by large-scale winds in stratified turbulence
,”
Phys. Rev. E
91
,
033015
(
2015
).
47.
G.
Brethouwer
,
P.
Billant
,
E.
Lindborg
, and
J.-M.
Chomaz
, “
Scaling analysis and simulation of strongly stratified turbulent flows
,”
J. Fluid Mech.
585
,
343
(
2007
).
48.
E.
Lindborg
and
G.
Brethouwer
, “
Stratified turbulence forced in rotational and divergent modes
,”
J. Fluid Mech.
586
,
83
(
2007
).
49.
H.
Aluie
and
S.
Kurien
, “
Joint downscale fluxes of energy and potential enstrophy in rotating stratified Boussinesq flows
,”
Europhys. Lett.
96
,
44006
(
2011
).
50.
M. L.
Waite
, “
Stratified turbulence at the buoyancy scale
,”
Phys. Fluids
23
,
066602
(
2011
).
51.
S.
Almalkie
and
S. M.
de Bruyn Kops
, “
Kinetic energy dynamics in forced, homogeneous, and axisymmetric stably stratified turbulence
,”
J. Turbul.
13
,
N29
(
2012
).
52.
Y.
Kimura
and
J. R.
Herring
, “
Energy spectra of stably stratified turbulence
,”
J. Fluid Mech.
698
,
19
(
2012
).
53.
A. J.
Barker
and
Y.
Lithwick
, “
Non-linear evolution of the tidal elliptical instability in gaseous planets and stars
,”
Mon. Not. R. Astron. Soc.
435
,
3614
(
2013
).
54.
T. L.
Reun
,
B.
Favier
,
A.
Barker
, and
M. L.
Bars
, “
Inertial wave turbulence driven by elliptical instability
,”
Phys. Rev. Lett.
119
,
034502
(
2017
).
55.
P.
Bartello
, “
Geostrophic adjustment and inverse cascades in rotating stratified turbulence
,”
J. Atmos. Sci.
52
,
4410
(
1995
).
56.
O.
Métais
,
P.
Bartello
,
E.
Garnier
,
J. J.
Riley
, and
M.
Lesieur
, “
Inverse cascade in stably stratified rotating turbulence
,”
Dyn. Atmos. Oceans
23
,
193
(
1996
).
57.
S.
Kurien
,
B.
Wingate
, and
M. A.
Taylor
, “
Anisotropic constraints on energy distribution in rotating and stratified turbulence
,”
Europhys. Lett.
84
,
24003
(
2008
).
58.
L. M.
Smith
,
J. R.
Chasnov
, and
F.
Waleffe
, “
Crossover from two- to three-dimensional turbulence
,”
Phys. Rev. Lett.
77
,
2467
(
1996
).
59.
P. D.
Mininni
and
A.
Pouquet
, “
Rotating helical turbulence. I. Global evolution and spectral behavior
,”
Phys. Fluids
22
,
035105
(
2010
).
60.
R.
Marino
,
P. D.
Mininni
,
D. L.
Rosenberg
, and
A.
Pouquet
, “
Large-scale anisotropy in stably stratified rotating flows
,”
Phys. Rev. E
90
,
023018
(
2014
).
61.
G. K.
Vallis
,
Atmospheric and Oceanic Fluid Dynamics
(
Cambridge University Press
,
Cambridge
,
2008
).
62.
D. G.
Dritschel
and
W. J.
McKiver
, “
Effect of Prandtl’s ratio on balance in geophysical turbulence
,”
J. Fluid Mech.
777
,
569
(
2015
).
63.
H.
Hanazaki
, “
Linear processes in stably and unstably stratified rotating turbulence
,”
J. Fluid Mech.
465
,
157
(
2002
).
64.
R.
Marino
,
P. D.
Mininni
,
D.
Rosenberg
, and
A.
Pouquet
, “
Inverse cascades in rotating stratified turbulence: Fast growth of large scales
,”
Europhys. Lett.
102
,
44006
(
2013
).
65.
C.
Cambon
and
L.
Jacquin
, “
Spectral approach to non-isotropic turbulence subjected to rotation
,”
J. Fluid Mech.
202
,
295
(
1989
).
66.
F.
Waleffe
, “
The nature of triad interactions in homogeneous turbulence
,”
Phys. Fluids A
4
,
350
(
1992
).
67.
F.
Waleffe
, “
Inertial transfers in the helical decomposition
,”
Phys. Fluids A
5
,
677
(
1993
).
68.
C.
Cambon
,
N. N.
Mansour
, and
F. S.
Godeferd
, “
Energy transfer in rotating turbulence
,”
J. Fluid Mech.
337
,
303
(
1997
).
69.
C.
Cambon
, “
Turbulence and vortex structures in rotating and stratified flows
,”
Eur. J. Mech. - B/Fluids
20
,
489
(
2001
).
70.
M.
Nikurashin
,
G. K.
Vallis
, and
A.
Adcroft
, “
Routes to energy dissipation for geostrophic flows in the Southern Ocean
,”
Nat. Geosci.
6
,
48
(
2012
).
71.
L. H.
Shih
,
J. R.
Koseff
,
G. N.
Ivey
, and
J. H.
Ferziger
, “
Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations
,”
J. Fluid Mech.
525
,
193
(
2005
).
72.
P. D.
Mininni
,
D.
Rosenberg
, and
A.
Pouquet
, “
Isotropization at small scales of rotating helically driven turbulence
,”
J. Fluid Mech.
699
,
263
(
2012
).
73.
A.
Delache
,
C.
Cambon
, and
F.
Godeferd
, “
Scale by scale anisotropy in freely decaying rotating turbulence
,”
Phys. Fluids
26
,
025104
(
2014
).
74.
C.
Rorai
,
P. D.
Mininni
, and
A.
Pouquet
, “
Stably stratified turbulence in the presence of large-scale forcing
,”
Phys. Rev. E
92
,
013003
(
2015
).
75.
J. Y. N.
Cho
,
Y.
Zhu
,
R. E.
Newell
,
B. E.
Anderson
,
J. D.
Barrick
,
G. L.
Gregory
,
G. W.
Sachse
,
M. A.
Carroll
, and
G. M.
Albercook
, “
Horizontal wavenumber spectra of winds, temperature, and trace gases during the Pacific Exploratory Missions: 1. Climatology
,”
J. Geophys. Res.
104
,
5697
, (
1999
).
76.
D. G.
Vincent
and
T. W.
Schlatter
, “
Evidence of deep convection as a source of synoptic-scale kinetic energy
,”
Tellus
31
,
493
(
1979
).
77.
L.
Liechtenstein
,
F. S.
Godeferd
, and
C.
Cambon
, “
Nonlinear formation of structures in rotating stratified turbulence
,”
J. Turbul.
6
,
N24
(
2005
).
78.
P.
Billant
and
J.-M.
Chomaz
, “
Self-similarity of strongly stratified inviscid flows
,”
Phys. Fluids
13
,
1645
(
2001
).
79.
A.
Babin
,
A.
Mahalov
,
B.
Nicolaenko
, and
Y.
Zhou
, “
On the asymptotic regimes and the strongly stratified limit of rotating Boussinesq equations
,”
Theor. Comput. Fluid Dyn.
9
,
223
(
1997
).
80.
K.
Julien
,
E.
Knobloch
, and
J.
Werne
, “
A new class of equations for rotationally constrained flows
,”
Theor. Comput. Fluid Dyn.
11
,
251
(
1998
).
81.
L. M.
Smith
and
F.
Waleffe
, “
Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence
,”
Phys. Fluids
11
,
1608
(
1999
).
82.
F.
Bellet
,
F. S.
Godeferd
, and
J. F.
Scott
, “
Wave turbulence in rapidly rotating flows
,”
J. Fluid Mech.
562
,
83
(
2006
).
83.
H. A.
Kafiabad
and
P.
Bartello
, “
Balance dynamics in rotating stratified turbulence
,”
J. Fluid Mech.
795
,
914
(
2016
).
84.
A.
Alexakis
, “
Rotating Taylor–Green flow
,”
J. Fluid Mech.
769
,
46
(
2015
).
85.
P. C.
di Leoni
and
P. D.
Mininni
, “
Quantifying resonant and near-resonant interactions in rotating turbulence
,”
J. Fluid Mech.
809
,
821
(
2016
).
86.
L. M.
Smith
and
Y.
Lee
, “
On near resonances and symmetry breaking in forced rotating flows at moderate Rossby number
,”
J. Fluid Mech.
535
,
111
(
2005
).
87.
S.
Nazarenko
,
Wave Turbulence
(
Springer
,
New York
,
2011
).
88.
R. H.
Kraichnan
, “
Inertial-range spectrum of hydromagnetic turbulence
,”
Phys. Fluids
8
,
1385
(
1965
).
89.
P. C.
di Leoni
,
P. J.
Cobelli
,
P. D.
Mininni
,
P.
Dmitruk
, and
W. H.
Matthaeus
, “
Quantification of the strength of inertial waves in a rotating turbulent flow
,”
Phys. Fluids
26
,
035106
(
2014
).
90.
S.
Chen
and
R. H.
Kraichnan
, “
Sweeping decorrelation in isotropic turbulence
,”
Phys. Fluids A
1
,
2019
(
1989
).
91.
B.
Dubrulle
and
L.
Valdettaro
, “
Consequences of rotation in energetics of accretion disks
,”
Astron. Astrophys.
263
,
387
(
1992
).
92.
Y.
Zhou
, “
A phenomenological treatment of rotating turbulence
,”
Phys. Fluids
7
,
2092
(
1995
).
93.
A.
Pouquet
and
P. D.
Mininni
, “
The interplay between helicity and rotation in turbulence: Implications for scaling laws and small-scale dynamics
,”
Philos. Trans. R. Soc., A
368
,
1635
(
2010
).
94.
A.
Campagne
,
B.
Gallet
,
F.
Moisy
, and
P.-P.
Cortet
, “
Direct and inverse energy cascades in a forced rotating turbulence experiment
,”
Phys. Fluids
26
,
125112
(
2014
).
95.
C.
Cambon
,
R.
Rubinstein
, and
F. S.
Godeferd
, “
Advances in wave turbulence: Rapidly rotating flows
,”
New J. Phys.
6
,
73
(
2004
).
96.
J.
Paret
and
P.
Tabeling
, “
Experimental observation of the two-dimensional inverse energy cascade
,”
Phys. Rev. Lett.
79
,
4162
(
1997
).
97.
J.
Sommeria
, “
Experimental study of the two-dimensional inverse energy cascade in a square box
,”
J. Fluid Mech.
170
,
139
(
1986
).
98.
P. D.
Mininni
,
A.
Alexakis
, and
A.
Pouquet
, “
Nonlocal interactions in hydrodynamic turbulence at high Reynolds numbers: The slow emergence of scaling laws
,”
Phys. Rev. E
77
,
036306
(
2008
).
99.
R.
Marino
,
A.
Pouquet
, and
D.
Rosenberg
, “
Resolving the paradox of oceanic large-scale balance and small-scale mixing
,”
Phys. Rev. Lett.
114
,
114504
(
2015
).
100.
J. N.
Reinhaud
,
D. G.
Dritschel
, and
C. R.
Koudella
, “
The shape of vortices in quasi-geostrophic turbulence
,”
J. Fluid Mech.
474
,
175
(
2003
).
101.
C.
Herbert
,
A.
Pouquet
, and
R.
Marino
, “
Restricted equilibrium and the energy cascade in rotating and stratified flows
,”
J. Fluid Mech.
758
,
374
(
2014
).
102.
R.
Marino
,
D.
Rosenberg
,
C.
Herbert
, and
A.
Pouquet
, “
Interplay of waves and eddies in rotating stratified turbulence and the link with kinetic-potential energy partition
,”
Europhys. Lett.
112
,
49001
(
2015
).
103.
P. C.
di Leoni
,
P. J.
Cobelli
, and
P. D.
Mininni
, “
The spatio-temporal spectrum of turbulent flows
,”
Eur. Phys. J. E
38
,
1
(
2015
).
104.
E.
Yarom
and
E.
Sharon
, “
Experimental observation of steady inertial wave turbulence in deep rotating flows
,”
Nat. Phys.
10
,
510
(
2014
).
105.
A.
Campagne
,
B.
Gallet
,
F.
Moisy
, and
P.-P.
Cortet
, “
Disentangling inertial waves from eddy turbulence in a forced rotating-turbulence experiment
,”
Phys. Rev. E
91
,
043016
(
2015
).
106.
P.
Sagaut
and
C.
Cambon
,
Homogeneous Turbulence Dynamics
(
Cambridge University Press
,
Cambridge
,
2008
).
107.
B.
Miquel
and
N.
Mordant
, “
Nonlinear dynamics of flexural wave turbulence
,”
Phys. Rev. E
84
,
066607
(
2011
).
108.
H.
Greenspan
,
The Theory of Rotating Fluids
(
Cambridge University Press
,
Cambridge
,
1968
).
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