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DSS of Jets to Vortices
Direct Statistical Simulation of Jets and Vortices in 2D FlowsS.
M. Tobias1, a) and J. B. Marston2, b)
1)Department of Applied Mathematics, University of Leeds. Leeds,
LS2 9JT,
UK2)Department of Physics, Box 1843, Brown University,
Providence, RI 02912-1843 USA
(Dated: 19 August 2017)
In this paper we perform Direct Statistical Simulations of a
model of two-dimensional flow that exhibitsa transition from jets
to vortices. The model employs two-scale Kolmogorov forcing, with
energy injecteddirectly into the zonal mean of the flow. We compare
these results with those from Direct Numerical Simula-tions. For
square domains the solution takes the form of jets, but as the
aspect ratio is increased a transitionto isolated coherent vortices
is found. We find that a truncation at second order in the
equal-time but nonlocalcumulants that employs zonal averaging
(zonal CE2) is capable of capturing the form of the jets for a
rangeof Reynolds numbers as well as the transition to the vortex
state, but, unsurprisingly, is unable to reproducethe correlations
found for the fully nonlinear (non-zonally symmetric) vortex state.
This result continues theprogram of promising advances in
statistical theories of turbulence championed by Kraichnan.
PACS numbers: 47.27.De, 47.32.cd, 47.27.eb, 94.05.JqKeywords:
two-dimensional turbulence, coherent structures, statistical
theories
I. DIRECT STATISTICAL SIMULATION: THE LEGACY
OF KRAICHNAN
Robert Kraichnan’s vision of a statistical mechanics
ofturbulence notably emphasized essential differences be-tween
flows in two and three spatial dimensions1. Twodimensional flows
are striking for the frequent emergenceof coherent structures. The
structures are of two basictypes: Vortices and jets2,3. The Juno
mission to Jupiterhas recently returned beautiful images of both
types ofstructures, with jets dominating at low latitudes, and
aproliferation of vortices near the poles4. In this paperwe
investigate a simple model of two dimensional fluidflow that
exhibits a transition between jets and vortices.We employ both
Direct Numerical Simulation (DNS) andDirect Statistical Simulation
(DSS). DSS is a rapidly de-veloping set of tools that attempt to
describe, directly,the statistics of turbulent flows, bypassing the
traditionalway of accumulation of those statistics (for example
meanflows and two-point correlation functions) by DNS.
Thesestatistical methods lead to a deeper understanding offluid
flows that should guide researchers to regimes notaccessible
through DNS.The pioneering work of Kraichnan largely focused on
flows with isotropic and homogeneous statistics5. Thestatistical
description of forward and/or inverse cascadesof energy between
different scales, the topic explored inhis seminal 1967 paper1, is
particularly clear in this con-text. Equally important,
translational and rotationalsymmetries reduce the technical
complexity of statisti-cal theories. Most fluid flows in nature,
however, areboth anisotropic and heterogeneous. In DSS this is
seenas a feature, rather than a defect, as anisotropy and
a)Electronic mail: [email protected])Electronic mail:
[email protected]
inhomogeneity can lessen nonlinearity of the flows andmake the
statistics accessible to perturbative computa-tion. We show that a
particularly simple version of DSS,one in which the equations of
motion for the spatially-averaged statistics are closed at the
level of second-ordermoments or cumulants6,7, is already able to
reproducemany features of the model two-dimensional flow.
TheKolmogorov forcing we employ is purely deterministicand no
parameters are tuned at the level of the second-order cumulant
expansion (CE2), permitting a fair andunbiased comparison between
DNS and DSS.We introduce the model in Section II. Results from
DNS and DSS are presented in Section III. Compari-son between
the two approaches is made in Section IVand some conclusions and
possible directions for furtherexploration are discussed in Section
V.
II. SET-UP OF THE MODEL: FORMULATION,
EQUATIONS, AND FORCING.
The models we study are of incompressible fluid mov-ing on a
two-torus — i.e a two dimensional Cartesiandomain (0 ≤ x < Lx, 0
≤ y < Ly = 2π) with peri-odic boundary conditions in both
directions. The fluidmotion is damped by viscosity ν and driven by
a time-independent forcing (described below). Owing to
thetwo-dimensionality of the system the dynamics is com-pletely
described by the time-evolution of the vorticity
ζ ≡ ẑ · (~∇× ~v), which (in dimensional units) is given by
ζ̇ + J(ψ, ζ) = ν∇2ζ + g(y), (1)
where g(y) is the forcing term and J(A,B) is the Jaco-bian
operator given by J(A,B) = AxBy − AyBx. Wenote that, in contrast to
some earlier models, here we donot consider the effects of rotation
via a β-effect.The forcing g(y) is a generalisation of the
Kolmogorov
forcing (see e.g. 8) to two meridional wavenumbers; that
-
DSS of Jets to Vortices 2
is we set
g(y) = A1 cos(y) + 4.0 ∗A4 cos(4y). (2)
We set A1 = −1 and A4 = −2, which leads to non-trivialdynamics
in the fluid system. We note here that thischoice of deterministic
forcing injects energy directly intothe zonal flow, and should be
contrasted with previousstudies that impose stochastic forcing in
the small zonalscales. Once the forcing and the length scale in y
(say)is fixed then the dynamics (and indeed statistics) of theflow
is determined by the choice of the viscosity ν (whichcontrols the
Reynolds number of the flow, which may becalculated a posteriori)
and the aspect ratio (determinedby Lx).The aim of this paper is to
determine how successful
Direct Statistical Simulation truncated at second
order(sometimes termed CE2) is at describing the transitionsthat
occur as the parameters are varied. In particular weshall
investigate how well DSS reproduces the strengthof the mean shear
flows and the transition from solutionsdominated by jets to those
dominated by coherent vortexpairs.
III. RESULTS
In this section we describe results from DNS (in sub-section
IIIA) and those obtained from DSS using CE2(in subsection III B)
before comparing them in SectionIV.
A. Direct Numerical Simulation
Direct Numerical simulation is performed using apseudo-spectral
code optimised for use on parallel archi-tectures with typical
resolutions of 5122. In all cases theresolution is increased to
this level to obtain until con-vergent results. Initially we
integrate the equations forthree different values of the viscosity
in a square domainwith Lx = Ly = 2π. The time series for the
result-
ing spatially-averaged enstrophy ζ2 and kinetic energyψ2y +
ψ
2x where
A ≡1
LxLy
∫∫
Adxdy, (3)
are shown in Figure 1 for three values of the viscosity.This
figure clearly shows that, as expected, as the viscos-ity is
decreased (with the forcing fixed) both the enstro-phy (top panel)
and kinetic energy (bottom panel) of thesolutions increases.Figure
2 (multimedia view) shows snapshots from
movies of the evolution of the vorticity for the cases withLx =
2π (which are included in the supplementary ma-terial). The flow is
reasonably laminar although the so-lution has already undergone a
bifurcation from a steadystate. After some initial transients the
solution becomes
0 100 200 300 400 500 600 700 800 900
t
0
200
400
600
800
1000
1200
1400
ζ2
ν = 0.03
ν = 0.022
ν = 0.02
(a)
0 100 200 300 400 500 600 700 800 900
t
0
200
400
600
800
1000
1200
u2+v2
ν = 0.03
ν = 0.022
ν = 0.02
(b)
FIG. 1. Enstrophy and Kinetic Energy timeseries for the
so-lutions at three different viscosities for Lx = Ly = 2π. Bothof
these increase with decreasing viscosity.
time periodic, with a strong band/jet of positive vorticityin
the domain and weaker negative vorticity (in the formof a vortex)
at the edges. This corresponds to a right-ward jet in the upper
half of the domain and a reversejet in the lower half (see later).
Both the jet and the vor-tex remain fixed in space though pulse in
time. Thoughtime-dependence is present, these vortex regions
possessa well defined zonal mean, which can be calculated by
av-eraging over a suitably long time. The average Reynoldsnumber
for this flow is given by Re ≈ 730.
As the viscosity is decreased from this solution the dy-namics
becomes more irregular and time-dependent, asshown in the two
movies. Decrease of the viscosity leadsto stronger patches of
vorticity and faster flows. Forν = 0.022, Re ≈ 1370, whilst for ν =
0.02, Re ≈ 1650.For both of these solutions the non-zonally
symmetricpart i.e. the part of the solution with kx 6= 0 and
thevortex travel in space, rather than remaining fixed as forthe
earlier case. The strength of the vortex patches and
-
DSS of Jets to Vortices 3
jets increases with decreasing ν (as does the correspond-ing
mean flows and vorticities — see later) as the inertialterms play
an increasingly important role.When the aspect ratio is increased
so that Lx = 4π, the
nature of the solution changes. The driving which is
in-dependent of x no longer puts substantial power into thekx = 0
modes, but instead drives a fully nonlinear quasi-steady vortex
pair solution as shown in Figure 4 (multi-media view) and the
corresponding movie. This state isreminiscent of the localised
states analysed extensively inRef. 8. For this state the average
enstrophy and vorticityare significantly lower than for the jet
states (as shownin the time series in Figure 3). Moreover, as we
shallsee, this state has little energy in the
zonally-averagedvorticity and flow and so can not be characterised
as ajet state. This remarkable transition appears to be theopposite
of a zonostrophic instability (see Ref. 9); therea small-scale
forcing with zero zonal mean drives flowthat interacts with
rotation to put significant amount ofenergy into a zonally averaged
jet. Here the forcing isdesigned to drive strong zonal flows, but
nonlinear in-teractions prefer to put energy into vortex states
withweak zonal flows, and may therefore be termed a “vor-tostrophic
instability.” We note that the aspect ratiocontrols a similar
transition between jets and vortices inother two-dimensional
models10–13. The non-trivial non-linear dynamics provides an
interesting testing groundfor the types of statistical theories
favoured by Kraichnanand so, in the next section, we compare the
results ob-tained here via Direct Numerical Simulation, with
thoseobtained by Direct Statistical Simulation truncated atsecond
order (CE2).
B. Direct Statistical Simulation: The Cumulant
Equations
In this section we perform DSS for the system for thesame range
of parameters as above. The approach wetake is based upon
truncating the hierarchy of equationsof motion for the equal-time
cumulants at low order. It isrelated to stochastic structural
stability theory (S3T)14,15
and other approaches16,17 that do not assume spatial
ho-mogeneity or isotropy in the statistics. Here we definethe
cumulants in terms of zonal averages over the x-direction (see
Refs. 6, 7, 18–20) as opposed to ensembleaverages21–23. Thus
cζ(y) = 〈ζ〉, (4)
where 〈〉 indicates a zonal average, is the first cumulantand
cζζ(y, y′, ξ) = 〈ζ ′(x, y) ζ ′(x+ ξ, y′)〉, (5)
is the second cumulant (or two-point correlation func-tion). We
note that owing to the translational symmetryof the system
(including the forcing) the first cumulantis a function only of y
and the second cumulant is a func-tion of three rather than four
dimensions18,19. There are
similar definitions for the first and second cumulants
in-volving the streamfunction (i.e. cψ and cψζ), but thesecan be
related straightforwardly to the cumulants for thevorticity.The
cumulant hierarchy can be derived in a number of
ways19. Truncated at second order (CE2) this takes theform of
evolution equations for cζζ(y, y
′) and cζ(y) (seeRef. 18):
∂
∂tcζ(y) =
[
−
(
∂
∂y+
∂
∂y′
)
∂
∂ξcψζ(y,y′,ξ)
]
∣
∣
∣
∣
∣
y′=y,ξ=0
+ g(y) + ν∂2
∂y2cζ(y), (6)
together with
∂
∂tcζζ =
∂
∂ycψ(y)
∂
∂ξcζζ(y, y
′, ξ)
−∂
∂y(cζ(y))
∂
∂ξcψζ(y, y
′, ξ)
−∂
∂y′cψ(y
′)∂
∂ξcζζ(y, y
′, ξ)
−∂
∂y′(cζ(y
′))∂
∂ξcζψ(y, y
′, ξ)
+ ν
(
2∂2
∂ξ2+
∂2
∂y2+
∂2
∂y′2
)
cζζ . (7)
In the limit of no forcing or dissipation, equations (6–7)
conserve energy, enstrophy and the Kelvin impulse;thus the CE2 is a
conservative approximation6,19. Theequations are integrated forward
in time numerically us-ing a pseudospectral code with typical
resolutions of(ny, ny′ , nξ) = (64, 64, 16) until the statistics
have set-tled down to a statistically steady state and means
andtwo-point correlation functions are averaged in time.
IV. COMPARISON OF DNS AND DSS
The solutions from the cumulant equations are com-pared with the
corresponding statistics obtained fromDNS by averaging in x and
time. In all cases the DNS so-lutions are averaged over the final
third of the evolutionand the statistics are well converged. The
DSS solutionswere averaged over the final 10% of the evolution,
thoughthe averages rapidly converge in all cases.Figure 5 shows the
comparison between DNS and CE2
for a domain of length 2π and decreasing viscosity. Asthe
viscosity is decreased the mean vorticity amplitude〈ζ〉 and the mean
zonal flow 〈u〉 increase in amplitudeas expected. It is clear that
the comparison of the meanflows between CE2 and DNS is excellent.
CE2 has a ten-dency to emphasise turning points in the vorticity
thatare washed out by eddy + eddy → eddy interactions inthe DNS 6.
However the agreement in the amplitude andform of the solution is
very good. One might expect CE2
-
DSS of Jets to Vortices 4
(a) (b) (c)
FIG. 2. Snapshots of vorticity for (a) ν = 0.03 (b) ν = 0.022
(c) ν = 0.02, with Lx = Ly = 2π. For these figures the coloursare
scaled between (a) [−61, 34], (b) [−67, 46], (c) [−75, 52]. Movies
showing the dynamics for these parameters are containedin the
supplementary material (multimedia view).
0 200 400 600 800 1000
t
0
100
200
300
400
500
600
ζ2
Xmx = 2π
Xmx = 4π
(a)
0 200 400 600 800 1000
t
0
100
200
300
400
500
u2+v2
Xmx = 2π
Xmx = 4π
(b)
FIG. 3. Enstrophy and Kinetic Energy timeseries for the
so-lutions at ν = 0.03 and Lx = 2π and Lx = 4π. The solutionsfor
the larger aspect ratio are significantly weaker in energyand have
less enstrophy.
to improve as the ratio of energy in the zonal mean flowto that
in the fluctuations increases. This expectation islargely met,
though in all cases agreement is good.
What is remarkable is that CE2 is capable of capturingthe
transition to vortices as shown in Figure 6. When thezonal flow is
switched off in DNS by changing the aspectratio, CE2 predicts the
same behaviour. Although CE2does not get the form of the weak zonal
flow completelycorrect, it predicts the amplitude very well. This
is un-expected since zonally averaged CE2 is expected to workpoorly
in a case where the zonal means are small andsubdominant to the
fluctuations.
A more stringent test of the accuracy of statistical
rep-resentation involves a comparison of not only the zonalmeans
(first cumulants) but also the two-point correla-tion functions
(second cumulants). These are given inFigures 7 and 8. These show
the two-point correlationfunction cζζ(y, 3π/4, ξ), i.e. the
correlation in space withthe point three-eighths of the way up on
the left handside of each plot. In all the cases for the jet
solutionsthis is dominated by a kx = 1, ky = 2 solution, thoughthis
is modulated in y. DNS and CE2 can be seen tobe in good agreement
here as they should be for flowswith such strong mean. Figure 8
shows less good agree-ment between DNS and CE2, with CE2 failing to
matchboth the amplitude and spatial form of the second cu-mulant
(underestimating the ky = 0 component) for thecase where the flow
is dominated by strong vortices —the DNS velocity correlation
function has a near k = 0symmetry in the meridional direction and a
near reflec-tion symmetry in the zonal direction as the solution
isdominated by a kx = 1, ky = 0 vortex mode. This isclearly picked
up by the correlation function, which isalso dominated by these
wavenumbers. The failure ofCE2 to match the two-point correlation
function for thevortex state is not surprising as the form of this
correla-tion function is presumably determined by eddy + eddy→ eddy
interactions that are discarded from the quasilin-ear CE2
description. The importance of cubic terms for
-
DSS of Jets to Vortices 5
FIG. 4. Snapshot of vorticity for ν = 0.03 (b) Lx = 4π and Ly =
2π. For this figures the colours are scaled between [−23, 18].A
movie showing the dynamics for these parameters is contained in the
supplementary material (multimedia view).
the form of the solution is determined by the amplitudeof their
projection onto the second cumulant. This canbe seen by comparing
the second cumulants from CE2and DNS. What is clear is that this
projection is smallfor the cases with a significant zonal mean and
large forthe case of the vortex We discuss possible strategies
forimproving the agreement between DNS and DSS for thiscase next in
the Discussion.
V. DISCUSSION
We have shown that an expansion in equal-time andzonally
averaged cumulants, truncated at second order,is able to describe
both the jet- and vortex-dominatedphases of a two-dimensional flow
driven by deterministicKolmogorov forcing. Zonal mean flows in both
phasesare accurately reproduced, and two-point correlations ofthe
vorticity are also captured in the jet phase, but notin the
vortex-dominated phase.
It is remarkable that DSS with such a simple closurecan capture
much of the behavior exhibited by DNS. Weexpect that more
sophisticated closures will be able todescribe the vortex phase
accurately. Higher-order clo-sures such as CE3∗ and CE2.5 (see Ref.
19) include eddy+ eddy → eddy interactions and can improve
qualita-tive agreement in two-point correlations found by DSSin
comparison to DNS. The replacement of zonal aver-ages with ensemble
averages has been demonstrated todescribe non-zonal structures22,
likely including the vor-tices seen here at aspect ratio Ly/Lx = 2.
Finally thegeneralized quasi-linear approximation (GQL)24,25 andits
associated generalized cumulant expansion (GCE2),by allowing
long-wavelength non-zonal structures to in-teract fully
nonlinearly, should also be able to describethe vortex-dominated
phase. Each of these variantsis more computationally demanding than
simple zonal-
average CE2, but we plan to test these other forms ofDSS for the
Kolmogorov forced model.All of these different forms of DSS respect
the real-
izability inequalities studied by Kraichnan26,27. Theygeneralize
the program of understanding the statisticsof turbulence, greatly
advanced by Kraichnan, to encom-pass anisotropy and heterogeneity.
Statistical theoriesof turbulence thus continue to extend their
reach, per-mitting a deeper understanding of fluid flows that
maysomeday allow us to access regimes not currently reach-able by
DNS.
ACKNOWLEDGMENTS
We wish to acknowledge the help of Mark Dixon of TheUniversity
of Leeds HPC facility team. All calculationswere performed on the
Arc2 machine at The Universityof Leeds.
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DSS of Jets to Vortices 6
−30 −20 −10 0 10 20 30 40〈ζ〉
0
1
2
3
4
5
6
y
DNS
CE2
−40 −30 −20 −10 0 10 20 30 40〈u〉
0
1
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DNS
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(a) (b)
−60 −40 −20 0 20 40 60〈ζ〉
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DNS
CE2
−60 −40 −20 0 20 40 60〈u〉
0
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DNS
CE2
(c) (d)
−40 −30 −20 −10 0 10 20 30 40 50〈ζ〉
0
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y
DNS
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−60 −40 −20 0 20 40 60〈u〉
0
1
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DNS
CE2
(e) (f)
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-
DSS of Jets to Vortices 7
−30 −20 −10 0 10 20 30 40〈ζ〉
0
1
2
3
4
5
6
y
DNS
CE2
DNS
CE2
−40 −30 −20 −10 0 10 20 30 40〈u〉
0
1
2
3
4
5
6
y
DNS
CE2
DNS
CE2
(a) (b)
FIG. 6. Comparison of DNS and CE2. Mean vorticity and zonal flow
for ν = 0.03 and Lx = 2π (red and cyan) and Lx = 4π(blue and
green).
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26R. H. Kraichnan, “Realizability Inequalities and Closed
MomentEquations,” Annals of the New York Academy of Sciences
357,37–46 (1980).
27R. H. Kraichnan, “Decimated Amplitude Equations in Tur-bulence
Dynamics,” in Theoretical Approaches to Turbulence(Springer New
York, New York, NY, 1985) pp. 91–135.
-
DSS of Jets to Vortices 8
0 1 2 3 4 5 6
ξ
0
1
2
3
4
5
6
y
−72
−54
−36
−18
0
18
36
54
72
0 1 2 3 4 5 6
ξ
0
1
2
3
4
5
6
y
−102.0
−76.5
−51.0
−25.5
0.0
25.5
51.0
76.5
102.0
(a) (b)
0 1 2 3 4 5 6
ξ
0
1
2
3
4
5
6
y
−84
−63
−42
−21
0
21
42
63
84
0 1 2 3 4 5 6
ξ
0
1
2
3
4
5
6
y
−180
−135
−90
−45
0
45
90
135
180
(c) (d)
0 1 2 3 4 5 6
ξ
0
1
2
3
4
5
6
y
−96
−72
−48
−24
0
24
48
72
96
0 1 2 3 4 5 6
ξ
0
1
2
3
4
5
6
y
−180
−135
−90
−45
0
45
90
135
180
(e) (f)
FIG. 7. cζζ(y, 3π/4, ξ) for DNS (left) and CE2 (right) for the
case ν = 0.03 (top row) ν = 0.022 (middle row) and ν = 0.02(bottom
row). In all cases Lx = 2π.
-
DSS of Jets to Vortices 9
0 2 4 6 8 10 12
ξ
0
1
2
3
4
5
6
y
−34.0
−27.2
−20.4
−13.6
−6.8
0.0
6.8
13.6
20.4
27.2
0 2 4 6 8 10 12
ξ
0
1
2
3
4
5
6
y
−3.40
−2.55
−1.70
−0.85
0.00
0.85
1.70
2.55
3.40
(a) (b)
FIG. 8. cζζ(y, 3π/4, ξ) for DNS (left) and CE2 (right) for the
case ν = 0.03, Lx = 4π.
