Download Infrared Dynamics of Decaying Two

Journal of the Physical Society of Japan
Vol. 70, No. 2, February, 2001, pp. 376-386
Infrared Dynamics of Decaying Two-Dimensional Turbulence Governed
by the Charney-Hasegawa-Mima Equation
Takahiro Iwayama1,∗ , Takeshi Watanabe2,∗∗ and Theodore G. Shepherd1,∗∗∗
1 Department
2 Division
of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
of Global Development Science, Graduate School of Science and Technology, Kobe University,
Kobe 657-8501
(Received July 27, 2000)
The low wave number range of decaying turbulence governed by the Charney-Hasegawa-Mima
(CHM) equation is examined theoretically and by direct numerical simulation. Here, the low
wave number range is defined as values of the wave number k below the wave number kE
corresponding to the peak of the energy spectrum, or alternatively the centroid wave number of
the energy spectrum. The energy spectrum in the low wave number range in the infrared regime
(k → 0) is theoretically derived to be E(k) ∼ k5 , using a quasinormal Markovianized model of
the CHM equation. This result is verified by direct numerical simulation of the CHM equation.
The wave number triads (k, p, q) responsible for the formation of the low wave number spectrum
are also examined. It is found that the energy flux Π(k) for k < kE can be entirely expressed by
Π(−) (k), which is the total net input of energy to wave numbers < k arising from interactions
with wave numbers p, q > k. Furthermore, the contribution of nonlocal triad interactions to
the energy flux is found to be predominant in the range log(k/kE ) < −0.5, where the nonlocal
interactions are defined to be those triad interactions for which the ratio of the largest leg of the
triad to the smallest leg is larger than four.
KEYWORDS: energy spectrum, infrared range, low wave number range, nonlocal interaction, two-dimensional
turbulence, Charney-Hasegawa-Mima equation
§1.
case.
Equation (1.1) contains two characteristic regimes.
The first is the 2D Navier-Stokes (NS) regime that is
obtained when λ → 0. The governing equation in this
regime is the well-known 2D vorticity equation,
Introduction
The Charney-Hasegawa-Mima (CHM) equation, a
two-dimensional (2D) turbulent system, has been actively studied both theoretically and numerically over
the past decade.1-5) The equation describes the temporal evolution of quasi-2D fluctuations of the electrostatic
field on the plane perpendicular to a strong magnetic
field uniformly applied to a plasma.6) It also describes
the temporal evolution of geostrophic motion in geophysical fluids and is called the quasi-geostrophic potential
vorticity equation.7) The CHM equation in the strong
turbulent state neglecting the effects of waves can be
written as follows:
¢
∂ ¡ 2
∇ φ − λ2 φ + J(φ, ∇2 φ) = ν∇4 φ,
(1.1)
∂t
where all quantities are made nondimensional and
J(•, •) is the Jacobian operator, ∇2 the 2D Laplacian,
and φ(x, y) the electrostatic potential for the plasma case
or the variable part of the free surface of the fluid for the
geophysical case. The damping coefficient ν is the reciprocal of the Reynolds number. The constant λ is either
the ratio of the horizontal length scale of interest L to
the ion Larmor radius in the plasma case, or the ratio of
L to the Rossby deformation radius in the geophysical
∂∇2 φ
+ J(φ, ∇2 φ) = ν∇4 φ.
(1.2)
∂t
The other regime is obtained asymptotically for λ → ∞.
The governing equation is
∂φ
+ J(∇2 φ, φ) = −ν∇4 φ,
∂T
(1.3)
where T = t/λ2 is a rescaled time. Since eq. (1.3) has
been called the asymptotic model (AM),1) we shall call
this regime the AM regime. Although the AM regime
is characteristic of the CHM equation, there are resemblances between turbulent solutions of (1.2) and those
of (1.3). It is well known that the 2D vorticity equation has two quadratic inviscid invariants,
the kinetic
R
−1
2
=
(2A)
(∇φ)
dxdy =
energy
per
unit
area,
E
K
A
R∞
E
(k)dk,
and
the
enstrophy
per
unit
area,
Zr =
K
0
R∞
R
(2A)−1 A (∇2 φ)2 dxdy = 0 k2 EK (k)dk, where A is the
area in which the field φ is determined and EK (k) is the
kinetic energy spectrum. The existence of two quadratic
inviscid invariants causes the dual cascade; i.e., the kinetic energy is transported to the small wave number
side (inverse cascade) and the enstrophy is transported
to the large wave number side (direct cascade). This
fact yields two types of energy spectra, EK (k) ∼ k−5/3
in the energy inertial range and EK (k) ∼ k−3 in the
∗
On leave from Division of Global Development Science, Graduate School of Science and Technology, Kobe University, Kobe 6578501. E-mail: [email protected]
∗∗ E-mail: [email protected]
∗∗∗ E-mail: [email protected]
376
2001)
Infrared Dynamics of Decaying Two-Dimensional Turbulence Governed. . . .
enstrophy inertial range.8) Similar to the 2D vorticity
equation, eq. (1.1) also has two quadratic inviscid invariants: the
R ∞E = EK + Eλ =
R total energy per unit area,
(2A)−1 A [(∇φ)2 + λ2 φ2 ] dx dy = 0 E(k)dk and the
potential
= Zr + Zλ =
R enstrophy per unit area, ZR ∞
(2A)−1 A [(∇2 φ)2 + λ2 (∇φ)2 ] dx dy = 0 k2 E(k)dk. In
the AM regime, one thus obtains the invariants Eλ and
Zλ . Moreover, the energy spectra E(k) ∼ k−11/3 in the
energy inertial range and E(k) ∼ k−5 in the enstrophy
inertial range are also derived by dimensional arguments
similar to those for 2D NS turbulence.1, 2, 9)
Recently, Watanabe et al.5) derived scaling laws for
the temporal
evolution of a characteristic wave number
R
k̄ ≡ E −1 k E(k) dk and of the energy spectral density
at the scale k̄ for decaying CHM turbulence in the AM
regime. Their discussion is similar to Batchelor’s analysis
for decaying 2D NS turbulence.10) That is, provided that
the energy spectrum evolves in a self-similar way and the
energy E is an invariant in the limit of high Reynolds
number (ν → 0), the energy spectrum can be expressed
as
³
´
E(k) = E 9/8 t1/4 h k E 1/8 t1/4 ,
(1.4)
where h(x) is a function of universal form.11) This argument yields a temporal evolution of the characteristic
wave number,
k̄ ∼ t−1/4 ,
and of the energy spectral density at the scale k̄,
¡ ¢
E k̄ ∼ t1/4 .
(1.5)
(1.6)
The same type of similarity spectrum (1.4) was also derived by Yanase and Yamada,12) but it was not confirmed
by numerical simulations of the modified zero-fourth cumulant approximated equation of the CHM equation due
to the insufficient calculation time. On the other hand,
direct numerical simulation of decaying turbulence in
(1.1) confirmed the scaling laws (1.5) and (1.6).5) Moreover, the self-similarity assumption was well satisfied.
From the numerical simulation, the asymptotic form of
h(x) was obtained as follows: h(x) ∼ x−6 for the high
wave number range x > 1 and h(x) ∼ x4 for the low wave
number range x < 1, where x ≡ k/k̄. However, theoretical explanation of the asymptotic form of h(x) was left as
an unsolved problem. In this study we thus consider the
asymptotic form of h(x). The asymptotic form of h(x)
for x > 1 is similar to the spectrum in the enstrophy inertial range, h(x) ∼ x−5 , but slightly steeper. Although
the reason why the spectrum in this range deviates from
the prediction by the dimensional arguments remains unknown, we focus our attention on the asymptotic form of
h(x) in the low wave number range, i.e., h(x) for x < 1.
For theoretical analysis of h(x) in the low wave number range, we follow the study of Basdevant et al.
(henceforth BLS78).13) They derived the energy spectrum EK (k) ∼ k3 in the limit k → 0 (infrared range) of
2D NS turbulence by using a quasinormal Markovianized
closure theory. Existence of the spectrum EK (k) ∼ k3
was noted in direct numerical simulations by Frisch and
Sulem14) and Chasnov.15) On the other hand, Yanase
377
and Yamada12) derived the energy spectrum in the infrared range of CHM turbulence, EK (k) ∼ k7 , but their
result was not numerically confirmed in their paper as
mentioned above. Indeed, there are no numerical studies focusing on the low wave number spectrum of CHM
turbulence.
In this paper, we derive the power-law exponent of
the energy spectrum in the infrared range of CHM turbulence theoretically by using the method proposed by
BLS78, and confirm our results by performing direct numerical simulations of (1.1). Moreover, the energy flux
and the triad interactions in wave number space responsible for the formation of the energy spectrum in the low
wave number range are also examined. The derivation of
the infrared energy spectrum proposed by BSL78 relies
on nonlocality of the triad interactions.13) Nonlocality
of the triad interactions in the enstrophy inertial range
of 2D NS turbulence is well known16, 17) and has been
studied numerically by many researchers.18-21) However,
there are no studies on the triad interactions in the low
wave number range even for 2D NS turbulence. In this
article, we focus only on the triad interactions responsible for the formation of the energy spectrum in the low
wave number range of CHM turbulence. However, this
study may contribute to the study of the low wave number range of 2D NS turbulence because the two systems
have similar triad interaction terms.
This paper is organized as follows. In §2, we introduce
a quasinormal Markovianized model of the CHM equation. In §3, we derive the power-law exponent of the
energy spectrum in the infrared range of decaying CHM
turbulence using the method proposed by BSL78. In
§4, we present results from direct numerical simulations
of the CHM equation, including analysis of the energy
transfer function and the triad interactions responsible
for the formation of the energy spectrum in the low wave
number range. Finally, we summarize the results in §5.
§2.
Formulation
2.1
Spectral form and energy equation of the CHM
equation
We consider a system which is confined within the
square domain [0, L]2 and adopt doubly periodic boundary conditions. Then the field φ is expanded as
X
φ̂(k) exp(ik · r),
(2.1)
φ(r) =
k
and eq. (1.1) is rewritten as
¶
µ
νk4
∂
+
φ̂(k)
∂t k2 + λ2
=
∆
X
1
q 2 − p2
(p × q)z 2
φ̂(p) φ̂(q), (2.2a)
2
k + λ2
k =p + q
∆
X
k =p + q
≡
X
δ k , p +q .
(2.2b)
p, q
Here, k = 2πn/L is the wave vector and summation is
taken over the integer vector n = (nx , ny ), k = |k|, p =
|p|, q = |q|. The time argument is omitted for brevity.
378
Takahiro Iwayama, Takeshi Watanabe and Theodore G. Shepherd
The temporal evolution of the energy spectrum, which is
defined as
X
E(k),
(2.3a)
E=
k
E(k) =
X0 1
2
k0
is governed by
µ
D
E
(k02 + λ2 ) |φ̂(k0 )|2 ,
2νk4
∂
+ 2
∂t k + λ2
¶
E(k) = T (k),
(2.4)
P0
0
where k0 is the shell summation in k − ∆k
2 ≤ |k | <
∆k
k + 2 , ∆k = 2π/L, T (k) the energy transfer function,
and the angle bracket denotes the ensemble average. The
energy transfer function T (k) can be expressed in terms
of the triad energy transfer function as
X1
T (k, p, q),
(2.5a)
T (k) =
2
p,q
X0
T (k, p, q) =
1 0
(p × q 0 )z (q 02 − p02 ) δk0 , p0 +q 0
2
k 0 , p0 , q 0
0
0
0
Π(k) = Π
(k) − Π
(−)
(k),
(2.7a)
X
T (k0 , p, q).
(2.7c)
p>k k<q<p
2.2
A quasinormal Markovianized model of the CHM
equation
In the limit L → ∞, eqs. (2.3a), (2.5a), (2.6), (2.7b),
and (2.7c) are represented in terms of integrals, i.e.,
Z ∞
E(k)dk,
(2.8)
E=
0
1
2
Z
Z
∞Z ∞
T (k, p, q) dp dq,
0
∞
Π(k) =
k
Z
Π(+) (k) =
∞
k
Z
0
can be divided into two parts,16)
(2.7b)
Π(+) (k) is the net energy input into all wave numbers
> k from interactions with p and q both < k, while
Π(−) (k) is the net energy input into all wave numbers
< k from interactions with p and q both > k. If we set
λ → 0, eq. (2.4) reduces to the evolution equation for the
energy spectrum of the 2D NS equation. However, the
energy transfer function (2.5) is always equivalent to that
for the 2D NS equation, because (2.5) is independent of
λ.
Π(−) (k) =
k0 >k
T (k0 , p, q),
p<k q<p
X X
Π(−) (k) =
(2.5b)
Since T (k, p, q) satisfies a detailed balance for wave numbers which form the triangle k = p+q, and is symmetric
with respect to p and q, the energy flux
X
T (k0 )
(2.6)
Π(k) =
(+)
k0 >k
T (k) =
×[hφ̂(−k )φ̂(p )φ̂(q )i
+hφ̂(k0 )φ̂(−p0 )φ̂(−q 0 )i].
X X X
Π(+) (k) =
k0 <k
(2.3b)
(Vol. 70,
k
(2.9)
0
T (k0 ) dk0 ,
Z
dk0
Z
dk0
(2.10)
Z
k
p
dp
0
dq T (k0 , p, q), (2.11)
0
Z
∞
p
dp
k
dq T (k0 , p, q). (2.12)
k
Using a quasinormal Markovianized approximation,22)
eq. (2.5b) is reduced to
T (k, p, q)
=
k 2 p2 q 2
2k2
θkpq 2
[2a2 (k, p, q) k E(p) E(q)−b2 (k, p, q) p E(q) E(k)−b2 (k, q, p) q E(k) E(p)] ,
πpq
(k + λ2 )(p2 + λ2 )(q 2 + λ2 )
(2.13)
where T (k, p, q) = 0 outside of the domain in the (p, q)
plane such that k, p, q can be the sides of the triangle
k = p + q (Fig. 1),
b2 (k, p, q) + b2 (k, q, p)
,
2
(z 2 − x2 )(z 2 − y 2 )
,
b2 (k, p, q) = 2
(1 − x2 )3/2
a2 (k, p, q) =
b2 (k, q, p) = 2
(y 2 − x2 )(y 2 − z 2 )
,
(1 − x2 )3/2
(2.14a)
(2.14b)
(2.14c)
are the geometrical coefficients, and x, y, z refer to the
cosines of the interior angles of the triangle facing respectively the sides k, p, q. The function θkpq is the
relaxation time of the third order moments associated
with the triad (k, p, q), the functional form of which is
different for different theories. If θkpq is chosen as
θkpq =
1 − exp(−νkpq t)
,
νkpq
(2.15a)
νk4
, (2.15b)
+ λ2
that is equivalent to the fourth-order cumulant being
zero, and eq. (2.4) is reduced to the modified zero-fourth
cumulant approximated equation derived by Yanase and
Yamada.12) It is well known that this approximation includes only viscous damping and cannot generally lead
to the inertial energy spectra that are derived by dimensional arguments, although the enstrophy inertial
range spectrum of 2D NS turbulence, EK (k) ∝ k−3 ,
was derived by numerical simulations of the modified
zero-fourth cumulant approximated equation.23) On the
other hand, there is an approximation that introduces
eddy damping effects µkpq in θkpq , referred to as the
Eddy-Damped Quasi-Normal Markovianized approximation (E.D.Q.N.M.).24) In this approximation, θkpq can be
expressed as follows:
νkpq = νk + νp + νq ,
νk =
k2
2001)
Infrared Dynamics of Decaying Two-Dimensional Turbulence Governed. . . .
q=
p-
k
q=
p+
k
q
k
0
k
p
Fig. 1. Domain in the (p, q) plane such that k, p, q can be the
sides of the triangle k = p + q. The triad transfer function
T (k, p, q) has a non zero value within the hatched region.
θkpq =
1 − exp[−(νkpq + µkpq )t]
,
νkpq + µkpq
µkpq = µk + µp + µq ,
£
¤1/2
.
µk ∼ λ−6 k9 E(k)g(k/λ)−3
(2.16a)
(2.16b)
(2.16c)
Equation (2.16c) is derived by eliminating the energy
transfer rate ² or the enstrophy transfer rate η from
eqs. (9) and (10) in Watanabe et al.4) Here, g(x) is a
dimensionless function, which is finite for x ¿ 1 and
approaches g(x) = x2 for x À 1. Therefore µk ∼
¤1/2
£ 9
in the AM regime. The proportionality conk E(k)
stant in eq. (2.16c) has not yet been determined. However, as will be seen in the next section, the derivation
of the energy spectrum in the infrared range proposed
by BSL78 is independent of the functional form of θkpq .
Thus we do not mention the functional form of θkpq hereafter. In the AM regime, eq. (2.13) reduces to
T (k, p, q) =
2k4 pq
θkpq [2a2 (k, p, q) k E(p) E(q)
πλ6
−b2 (k, p, q) p E(q) E(k)
−b2 (k, q, p) q E(k) E(p)] .
(2.17)
Equations (2.3b), (2.5)–(2.7) are used in the analysis of the direct numerical simulations of (1.1), while
eqs. (2.9)–(2.12), (2.14), and (2.17) are used in the theoretical derivation of the energy spectrum in the infrared
range of CHM turbulence.
§3.
Derivation of the Energy Spectrum in the
Infrared Range of Decaying CHM Turbulence
In this section, we derive the energy spectrum in the
infrared range of decaying CHM turbulence using the
method proposed by BSL78.13) In their method, the
wave number dependence of the energy transfer function T (k) reflects directly on the functional form of the
energy spectrum in the infrared range. This feature is in
379
contrast to the energy spectrum in the energy and enstrophy inertial ranges, where Kolmogorov-type dimensional
analysis assumes the energy and enstrophy fluxes to be
constant with respect to wave number.8)
The method proposed by BSL7813) is summarized as
follows:22)
1. Interactions among triad wave numbers k, p, q can
be divided into two categories: either all of k, p, q
have comparable magnitude, or one is very small in
comparison with the others. The former refers to
the local interactions and the latter to the nonlocal
interactions. One may then divide the energy transfer function T (k) into two parts: TL (k) constructed
from the local triad interactions, and TN L (k) from
the nonlocal triad interactions.
2. Assume that the nonlocal triad interactions with
elongated triads k ¿ p ' q are dominant in the
infrared range. Then, one introduces the small parameter ² which is the ratio of two interacting wave
numbers, ² ≡ k/p, and expresses TN L (k) in terms of
power series in ².
3. Practical calculation is performed through the energy flux Π(k). Instead of calculating the energy
transfer function TN L (k) directly, one can evaluate
(−)
the nonlocal part of the energy flux −ΠN L (k), which
is equivalent to ΠN L (k) because the elongated triads
k ¿ p ' q cannot contribute to the flux Π(+) (k).
(+)
To say this another way, ΠN L (k) is geometrically
forbidden. Differentiating the resulting energy flux
with respect to wave number, one recovers the nonlocal part of the energy transfer function. Then,
the wave number dependence of the energy transfer function gives the functional form of the energy
spectrum in the infrared range. Note also that generally the characteristic time θkpq is not expanded.
Following the above method, we derive the energy
spectrum in the infrared range of decaying turbulence
of the CHM equation. The nonlocal part of Π(−) (k) is
given by
Z k
Z p
Z ∞
(−)
0
dk
dp
dq T (k0 , p, q), (3.1)
ΠN L (k) ≡
0
sup(k,k0 /²)
p−k0
where ² = k0 /p and the lower limit on the q integral is
a geometrical constraint. By using the cosine law q 2 =
k02 + p2 − 2k0 pz, we transform the variable q in (3.1) to
z. Then, eq. (3.1) is rewritten as
Z k Z ∞
Z 1
k0 p
(−)
T (k0 , p, q),
dp
dz
ΠN L (k) = dk0
q
0
0
0
sup(k,k /²) k /(2p)
(3.2)
Next, we expand the integrand in (3.2) with respect to
the smallness parameter. Since the expansion of the geometrical coefficients and of the energy spectrum are the
same as those for the 2D NS equation, we use the results
of Appendix B of BSL78. The approximate expressions
of the geometrical coefficients and the energy spectrum
are given by
b2 (k0 , p, q)
380
Takahiro Iwayama, Takeshi Watanabe and Theodore G. Shepherd
=2
µ 02 ¶¸
·
k
p2
k0
2 1/2
2
(1
+
2z
(1
−
z
)
)
+
O
−2z
+
,
k02
p
p2
(Vol. 70,
p 2
z (1 − z 2 )1/2 + O(1),
k0
µ 02 ¶
k
∂E
k0
+O
.
E(q) = E(p) − zp
p
∂p
p2
a2 (k0 , p, q) = 4
(3.3)
0
b2 (k , q, p)
µ 02 ¶¸
·
k
k0
p2
,
= 2 02 (1 − z 2 )1/2 2z + (2z 2 − 1) + O
k
p
p2
(3.5)
(3.6)
As a result of substituting these expressions into (2.17)
with k0 in place of k, and keeping the leading order terms,
the integrand of (3.2) is reduced to
(3.4)
T (k0 , p, q)
¾
½
8 2
k0 p
2 1/2
05 3
2
04 4
0 ∂ [p E(p)]
0 pp
'
z
(1
−
z
)
θ
p
E(p)
−
k
p
E(k
)
.
2k
k
q
πλ6
∂p
Since, to the leading order in ²,
Z
Z
1
k0 /(2p)
we obtain
Z
k
dk0 k05
−
Z
∞
dp
sup(k,k0 /²)
0
Z
k
dk0 k04 E(k0 )
Z
Moreover, we obtain the energy transfer function associated with (3.8),
(−)
∂ΠN L (k)
∂k
Z ∞ 3
p
2k4
2
θ
E(p)
dp
−
νT (k) E(k),
' k5
kpp
6
λ2
k/² λ
TN L (k) =
(3.9)
where
Z
∞
νT (k) =
k/²
p4
∂ [p E(p)]
dp.
θkpp
4λ4
∂p
(3.10)
The quantity νT (k) is the AM version of the turbulent viscosity derived by Kraichnan for the 2D NS equation.25) The second term on the right-hand side of (3.9)
is negligible in the infrared range, because E(k) → 0
in the limit k → 0 for decaying turbulence. Moreover,
the viscous damping term, the second term on the lefthand side of (2.4), is also negligible in the infrared range.
Furthermore, the integral in (3.9) is nearly independent
of its lower limit since it is dominated by the energy
∂
E(k) ' TN L (k) ∼ k5
containing scales. Therefore, ∂t
and we obtain the energy spectrum E(k) ∼ k5 in the
infrared range. This is consistent with the energy spectrum EK (k) ∼ k7 derived by Yanase and Yamada using the modified zero-fourth cummulant (quasi-normal
Markovianized) approximation.12)
The analytical result E(k) ∼ k5 is steeper than the
low wave number range spectrum of numerical simulations in Larichev and McWilliams1) and Watanabe et
al.5) In the next section, we perform direct numerical
simulations of decaying turbulence of the CHM equation,
and the resulting energy spectra and energy transfers are
examined.
π
,
16
p3
θk0 pp E(p)2
λ6
∞
dp
sup(k,k0 /²)
0
(−)
z 2 (1 − z 2 )1/2 dz =
0
(−)
ΠN L (k) '
1
z 2 (1 − z 2 )1/2 dz '
(3.7)
§4.
p4
∂ [p E(p)]
.
θk0 pp
6
2λ
∂p
(3.8)
Numerical Simulations and Discussion
In this section, we report direct numerical simulations
of decaying turbulence governed by (1.1) with the hyperviscosity term −ν2 ∇6 φ instead of the normal viscosity term ν∇4 φ. The pseudospectral method is used
in double precision arithmetic and at a resolution N 2 ,
which is the number of grid points in the computational
domain, and the truncation wave number is taken as
kT = [(N − 1)/3]∆k to suppress aliasing errors, where [ ]
denotes the Gaussian symbol. Three simulations (run 1,
run 2 and run 3) are conducted for different resolutions
N = 256, N = 512, N = 1024, with the minimum grid
size held fixed. Initial conditions are made by generating
Gaussian random numbers with a mean value of 0 and a
variance of 2π for the phase of each Fourier component
of φ. We normalize the initial value of the kinetic energy
per unit area to be 0.5, and the initial form of the energy
spectrum is specified by
E(k) ∼
k30
.
(k + k0 )60
(4.1)
These initial condition are the same as those used in
Larichev and McWilliams1) and Watanabe et al.5, 26)
Details of conditions in the simulations are listed in
Table I. Time integration is done by the AdamsBashforth scheme. In the previous work,5) the fourth
order Runge-Kutta scheme for time integration was
adopted. We have checked that the results are independent of the time integration scheme in the case N = 256.
Thus, we adopt the Adams-Bashforth scheme in this
study for computational efficiency.
All numerical simulations are done up to t = 200.
This time corresponds to about 90 times the characteristic advection time τ , which is defined by Larichev and
McWilliams1) as follows:
2001)
Infrared Dynamics of Decaying Two-Dimensional Turbulence Governed. . . .
Table I.
computational domain
boundary condition
space difference
time integration
resolution in physical space
minimum wave number
truncation wave number
∆t
dissipation term
viscosity coefficient ν2
λ
initial kinetic energy
initial energy spectrum
381
Conditions of direct numerical simulations.
run 1
run 2
run 3
2π × 2π
4π × 4π
doubly periodic
pseudospectral method
Adams-Bashforth scheme
5122
0.5
85
2.5 × 10−3
−ν2 ∇6 φ
3.0 × 10−8
50
0.5
k30
E(k) ∼ (k+k
)60
8π × 8π
2562
1
10242
0.25
0
k0
15
Z
τ (t) =
0
t
{2Zr (t0 )}
0
¡
¢2 dt ,
1 + λ/k̄K
R
1/2
k̄K ≡
kEK (k) dk
. (4.2)
EK
When λ = 0, this definition is equivalent to the eddy
turnover time commonly used in studies of 2D NS turbulence.15) Figure 2 shows the temporal evolution of the
characteristic advection time τ (t). As shown in Fig. 2,
τ (t) is almost the same for each simulation. The reason
for the equivalence of τ for each simulation is that the
definition of τ is based on the enstrophy which depends
mainly on information in the large wave numbers. The
evolution of the characteristic wave number k̄ in the three
runs also considerably resemble each other (see Fig. 3).
We note that the wave number kE corresponding to the
peak of the energy spectrum changes discretely due to
the finite resolution of wave number space in the direct
numerical simulations. However, k̄ and kE have nearly
the same values, i.e., they obey the same decay law in
time (figure not shown). Thus we use kE for normalizing
the energy spectrum in this study, although the charac-
Fig. 2. Evolution of the characteristic time τ from (4.2). Broken
line, run 1; dashed and dotted line, run 2; solid line, run 3.
Since the results for run 2 and run 3 are nearly equivalent, the
differences between them are indistinguishable.
teristic wave number k̄ was used for the normalization of
the energy spectrum in the previous study.5)
Figure 4 shows normalized energy spectra (the abscissa
is normalized by kE and the ordinate by E(kE )) at times
t = 0, 4, 10, 20, 50, 100, 150 and 200. It shows that
the energy spectra develop temporally in a self-similar
way. Thus, Fig. 4 manifests the universal function h(x)
in eq. (1.4). The energy spectrum adjusts to h(x) from
the initial condition within τ ' 10. After that, h(x) exists for about 80 times the characteristic advection time.
Therefore we stress that h(x) is not evanescent in our
simulation.
In order to examine the energy spectrum in the low
wave number range in detail, we show compensated
energy spectra, (k/kE )−δ [E(k)/E(kE )], in Fig. 5. If
E(k) ∼ kδ in a range, then the compensated spectrum
has a plateau in the corresponding range. The exponent δ of the energy spectrum in the low wave number
is δ ' 4.1 for run 1, δ ' 4.45 for run 2 and δ ' 4.6
for run 3. Therefore, as one extends the system size
progressively, h(x) in the range x < 1 becomes steeper
and approaches the theoretically predicted infrared en-
Fig. 3. Evolution of the characteristic wave number k̄. Broken
line, run 1; dashed and dotted line, run 2; solid line, run 3.
382
Takahiro Iwayama, Takeshi Watanabe and Theodore G. Shepherd
Fig. 4. Normalized energy spectra for (a) run 1, (b) run 2 and
(c) run 3. The abscissa is normalized by kE , where the energy
spectrum peaks, and the ordinate by E(kE ). Broken line, t = 0;
solid lines, t = 4, 10, 20, 50, 100, 150, 200.
ergy spectrum h(x) ∼ x5 , or E(k) ∼ k5 . This suggests
that the previous studies were under-resolved.
It may be noted in passing, for possible future reference, that the high wave number regime k > kE exhibits
Fig. 5. Compensated
times t = 4, 10, 20,
run 2 and (c) run 3.
solid lines, t = 100,
reference.
(Vol. 70,
energy spectra (k/kE )−δ E(k)/E(kE ) at
50, 100, 150, and 200 for (a) run 1, (b)
Thin solid lines, t = 4, 10, 20, 50; thick
150, 200. The dashed line is a plateau
a power-law scaling of approximately E(k) ∼ k−5.5 in
both run 2 and run 3 (Figs. 5(b) and 5(c)). All wave
numbers shown have k < λ so even the high wave number
dynamics are in the AM regime. The energy spectra in
2001)
Infrared Dynamics of Decaying Two-Dimensional Turbulence Governed. . . .
Fig. 6. Energy fluxes, Π(k), Π(+) (k), and −Π(−) (k) for run 2.
Solid thin line, Π(k) at t = 20; broken thin line, −Π(−) (k) at t =
20; dashed and dotted thin line, Π(+) (k) at t = 20; solid thick
line, Π(k) at t = 100; broken thick line, −Π(−) (k) at t = 100;
dashed and dotted thick line, Π(+) (k) at t = 100. The values of
the energy fluxes at t = 100 are multiplied by a factor of 2.
the high wave number range in this study are somewhat
shallower than those in the previous studies.1, 5)
In the derivation of the infrared energy spectrum
E(k) ∼ k5 , we (or rather BLS78) made the assumption
that the nonlocal interactions with k ¿ p ' q dominated
in the triad interactions in wave number space. We can
check the validity of the above assumption by examining the results of our direct numerical simulations. (Due
to lack of computational resource, the following analyses
are done only for run 1 and run 2.)
Figure 6 shows the energy fluxes Π(k), Π(+) (k) and
−Π(−) (k) for run 2 at t = 20, 100. When Π(+) (k) has a
positive value, this means a downward cascade of energy
due to the triad interactions (k0 , p, q) with p, q < k
and k0 > k. On the other hand, when Π(−) (k) has a
positive value, this means an upward cascade of energy
due to the triad interactions (k0 , p, q) with p, q > k
and k0 < k. In order to redistribute energy within a
triad under the constraint of conserving both the total
energy and the potential enstrophy, only energy flow to
or from the middle wave number among the triad can be
permitted.27, 28) Both Π(+) (k) > 0 and Π(−) (k) > 0 as
shown in Fig. 6, suggesting that energy flow away from
the middle wave number of the triads is statistically dominant. Thus Batchelor’s hypothesis,29) that an initially
narrow spectrum spreads out in time about its centroid,
is supported in this system. As shown in Fig. 6, Π(k) is
entirely expressed by −Π(−) (k) in the low wave number
range k/kE < 1. In contrast, in the range k/kE > 1,
Π(k) is determined by a competition between Π(+) (k)
and −Π(−) (k). That the triad interactions contributing to Π(+) (k), which consist of triads (k0 , p, q) with
p, q < k and k0 > k, are insignificant in the low wave
number range is a geometrical constraint: for sufficiently
small p, q, such triads do not exist. The result that Π(k)
can be approximated by −Π(−) (k) in the low wave num-
383
Fig. 7. The ratio of© the nonlocal
part of the energy flux
£
¤ª to the
total energy flux, Π(k) − Π(+) (k, 2k) − Π(−) (k, 4k) /Π(k),
at some instant, for both run 1 and run 2. Broken thin line, run
1 at t = 20; broken thick line, run 1 at t = 100; dashed and
dotted thin line, run 2 at t = 20; dashed and dotted thick line,
run 2 at t = 100.
ber range was also verified for run 1 at t = 20 and 100.
To determine which triads of wave numbers contribute
to the energy fluxes Π(+) (k) and Π(−) (k) in detail, we
calculate filtered fluxes Π(+) (k, kf ) and Π(−) (k, kf ) as
introduced by Smith and Yakhot,30)
X
X
1 X
Π(+) (k, kf ) =
T (k0 , p, q), (4.3a)
2
0
k<k <kf 0<p<k 0<q<k
1 X
Π(−) (k, kf ) =
2
0
X
X
T (k0 , p, q). (4.3b)
0<k <k k<p<kf k<q<kf
Moreover, following Smith and Yakhot,30) we also define nonlocal interactions as those triad interactions
for which the ratio of the largest leg to the smallest
leg is larger than four. Since the length of one side
of a triangle is smaller than the sum of the lengths
of the other two sides, one can see that Π(+) (k) =
Π(+) (k, 2k). Thus only local triad interactions contribute to Π(+) (k). Figure 7 shows the ratio of the nonlocal
part£ of the energy flux to the
¤ª total energy flux,
©
Π(k) − Π(+) (k, 2k) − Π(−) (k, 4k) /Π(k), at some instant for both run 1 and run 2. The relative contribution of the local and nonlocal interactions to the energy
flux changes dramatically at log(k/kE ) ' −0.5 (k/kE '
0.32). The location of this transition of course depends
on our choice kf = 4k. It is approximately determined
by the relation k/kE ' k/kf , due to the dominance
of the triad interactions between the low wave number
range and the energy containing scale kE , as will be seen
later. For log(k/kE ) < −0.5, the nonlocal interactions
are dominant. On the other hand, in the vicinity of the
energy containing scale log(k/kE ) = 0, the local interactions are entirely dominant. The reason why there
is such a large deviation of the energy spectrum from
the theoretical infrared energy spectrum in the low wave
number range for run 1 is the extreme narrowness of the
384
Takahiro Iwayama, Takeshi Watanabe and Theodore G. Shepherd
(Vol. 70,
wave number range for which nonlocality of the triad interactions is satisfied. In particular, at t ≥ 100 there
are only a few wave numbers which satisfy the condition
k/kE < 0.32. Note that the energy spectra in the low
wave number range in the late stage (t ≥ 100) for run 1
become shallower than those in the early stage (t < 100),
which is further support for this argument.
Next, we directly examine the triad interactions in
wave number space for the direct numerical simulations.
T (k, p, q) does not vanish inside of the domain in the
(p, q) plane such that k, p, q can be the sides of the triangle k = p +√q, and the width of the nonzero region
of T (k, p, q) is 2k (Fig. 1). When we consider the low
wave number range k < kE , it is difficult to represent
T (k, p, q) by a contour plot since the nonzero region of
T (k, p, q) is narrow and data points in the region are not
sufficiently dense. Therefore, we consider the azimuthal
summation of T (k, p, q) in the (p, q) plane,
X
T (k, p, q). (4.4)
T (k, ρ) =
√ 2 2
∆k
∆k
ρ−
2
≤
p +q <ρ+
2
Note that the azimuthally summed
triad transfer func√
tion T (k, ρ) vanishes for ρ < k/ 2. Moreover, the energy
√
containing scale p, q ' kE corresponds to ρ ' 2kE
because if we set q =
p p(1 + ²), where
√ ² is a small pap2 + q 2 = 2p [1 + O(²)]. Figrameter, then ρ =
ure 8 shows T (k, ρ) for various k < kE for run 2 at
t = 100. For wave numbers k ≤ 3 (for which the conthe T (k, ρ) have signifdition k/kE ≤ 0.32 is satisfied),
√
icant values near log[ρ/( 2kE )] ' 0. Thus this result
can be interpreted as demonstrating that interactions
between wave numbers k ≤ 3 and the energy containing scale p, q ' kE are dominant. The above property
of T (k, ρ) was also verified for run 2 at t = 20 and for
run 1 at t = 20 and 100. Therefore, we conclude that
if the system size is expanded in order to realize k → 0
approximately, the nonlocality of the triad interactions
k ¿ p ' q becomes increasingly well satisfied.
We try to interpret these properties of the triad interactions by using the evolution equation (2.2a). We
divide the wave number range into three categories: the
low wave number range denoted by L, the energy containing range E, and the high wave number range H.
Then the evolution equation for the low wave number
modes can be written formally as
∂ φ̂(L)
= CLL φ̂(L)φ̂(L)
∂t
+CEE φ̂(E)φ̂(E) + CHH φ̂(H)φ̂(H). (4.5)
Here, φ̂ are the Fourier coefficients of φ, and C the interaction coefficients; the viscosity term is omitted for
simplicity. Note that only three types of nonlinear terms
are possible because the other types of combination of
modes, e.g. φ̂(H)φ̂(L) and φ̂(E)φ̂(L), cannot form a triad
with φ̂(L), i.e., they are geometrically prohibited. Multiplying (4.5) by φ̂(L), we obtain the evolution equation
for the energy spectrum and the energy transfer function
as follows:
Fig. 8. Azimuthally summed triad transfer functions T (k, ρ) from
(4.4) for (a) k = 1, (b) k = 2 and (c) k = 3 for run 2 at t = 100.
∂ φ̂(L)2
= DLL φ̂(L)φ̂(L)φ̂(L)
∂t
+DEE φ̂(L)φ̂(E)φ̂(E) + DHH φ̂(L)φ̂(H)φ̂(H).
(4.6)
We now estimate the magnitude of each term on the
right-hand side of (4.6). The interaction coefficients
2001)
Infrared Dynamics of Decaying Two-Dimensional Turbulence Governed. . . .
£
¤
D are explicitly written as (p × q)z (q 2 − p2 ) /2 from
(2.5b). Because k is the wave number of interest, it belongs to the low wave number range L and is considered as a fixed value in this calculation. On the other
hand, both p and q belong to the same wave number
range and are treated as variables. Then, we can set
q = p(1 + ²), where ² is a small parameter. It follows that q 2 − p2 is proportional to p. Moreover, by
using the cosine
law k2 = p2 + q 2 − 2pqx, we obtain
p
2
|p × q| = [k − (p − q)2 ] [(p + q)2 − k2 ]/2 ∝ p. As a
result, the interaction coefficient D is proportional to p2 .
Furthermore, in the AM regime, φ̂(p) ∝ [pE(p)]1/2 . Let
the magnitudes of the characteristic wave number in the
range L, E, and H be kL , kE , and kH , respectively. Using
the facts that E(p) ∼ p5 if p is in L and E(p) ∼ p−5.5 if
p in H, the ratios can be estimated as follows:
µ ¶8
kL
DLL φ̂(L)φ̂(L)φ̂(L)
∝
,
kE
DEE φ̂(L)φ̂(E)φ̂(E)
µ ¶−2.5
kH
DHH φ̂(L)φ̂(H)φ̂(H)
∝
.
(4.7)
kE
DEE φ̂(L)φ̂(E)φ̂(E)
Both ratios are considerably smaller than unity provided
kL ¿ kE ¿ kH . Therefore we conclude that the second
term on the right-hand side of (4.6), representing the
triad interactions between the energy containing range
and the low wave number range, is strongly dominant.
This discussion is consistent with Fig. 8. Moreover, the
abrupt change of the ratio of the nonlocal part of the
energy flux to the total energy flux at k/kE ' 0.32 in
Fig. 7 originates from the large exponent in the first relation of (4.7). The triads that contribute to the energy
flux Π(+) (k) in the low wave number range originate entirely from the first term on the right-hand side of (4.6).
Thus the smallness of the first term on the right-hand
side of (4.6) is also consistent with Fig. 6. Note that
the dominance of the interactions between the low wave
number range and the energy containing range is not
trivial. If the energy spectrum in the high wave number
range were shallower than E(p) ∼ p−3 , then the ratio of
the third term to the second term on the right-hand side
of (4.6) would become larger than unity. In that case, the
interaction of the low wave number range and the high
wave number range would be significant. Therefore, the
dominance of the interaction between the low wave number range and the energy containing scale is associated
with the sharpness of the energy spectrum in CHM turbulence. On the other hand, the local triad interactions
in the low wave number range are insignificant as long as
the energy spectrum in the low wave number range has
a positive exponent.
The above discussion can also be applied to turbulence
in the NS regime because the energy transfer function in
the NS regime is equivalent to that in the AM regime
(see eq. (2.5b)). In the NS regime, from the relation
φ̂(p) ∝ [p−1 E(p)]1/2 and the fact that E(p) ∼ p3 if p is
in L13) and E(p) ∼ p−4 if p in H,31) the ratios can be
estimated as follows:
µ ¶4
kL
DLL φ̂(L)φ̂(L)φ̂(L)
,
∝
kE
DEE φ̂(L)φ̂(E)φ̂(E)
DHH φ̂(L)φ̂(H)φ̂(H)
DEE φ̂(L)φ̂(E)φ̂(E)
385
µ
∝
kH
kE
¶−3
.
(4.8)
Therefore, the interaction between the low wave number
range and the energy containing scale is dominant in the
NS regime as well.
§5.
Summary
In this paper, we have examined the energy spectrum in the low wave number range (k < kE ) of decaying turbulence governed by the Charney-HasegawaMima equation (1.1) in the AM regime where k ¿ λ.
The energy spectrum of decaying CHM turbulence develops in a self-similar way, i.e., the universal function
h(x) = E(k)/E(kE ), where x = k/kE , exists. This paper is the first to discuss the functional form of h(x) in
the low wave number range both theoretically and numerically. By using a quasinormal Markovianized model,
we derived the energy spectrum in the infrared range
k → 0: E(k) ∼ k5 as k → 0 (or h(x) ∼ x5 as
x → 0). We also performed direct numerical simulations of eq. (1.1). When we increase the system size from
[0, 2π]2 to [0, 8π]2 , the low wave number spectrum comes
closer to the theoretical infrared spectrum. Moreover,
we examined the triad interactions responsible for the
formation of the low wave number spectrum and found
that the energy flux Π(k) can be entirely expressed as
−Π(−) (k) for k < kE and that nonlocal triad interactions between the low wave number range and the energy
containing scale are dominant.
In decaying 2D NS turbulence, predictions from closure theory or the classical theory of 2D turbulence fail
due to the existence of coherent vortices and the associated intermittency.32) However, in the system considered
here, the prediction from closure theory is in fairly good
agreement with direct numerical simulations. This suggests that intermittency is weak in the AM regime of
CHM turbulence. This is confirmed by the fact that the
flatness of the potential vorticity field q = (∇2 −λ2 )φ calculated from the direct numerical simulations is close to
Fig. 9. Evolution of flatness of the potential vorticity field. Broken line, run 1; dashed and dotted line, run 2; solid line, run
3.
386
Takahiro Iwayama, Takeshi Watanabe and Theodore G. Shepherd
the Gaussian value of 3 (Fig. 9), consistent with Polvani
et al.33) Therefore we can anticipate that the system considered here is one for which the classical theory of 2D
turbulence is applicable. However, in order to verify this
conjecture, we would have to examine higher order moments of the potential vorticity field. This point should
be the subject of a future study.
Acknowledgements
T. I. expresses his deep thanks to Prof. H. Fujisaka of
Kyoto University and Prof. M. Takahashi of the University of Tokyo for helpful comments and encouragement.
He also wishes to thank Prof. T. Nakano of Chuo University, Profs. A. Yoshizawa and F. Hamba, Dr. N. Yokoi
and the other members of the Turbulence Seminar at the
University of Tokyo for many stimulating discussions.
T. W. was supported by a JSPS Research Fellowship
for Young Scientists. This work was partially supported
by the Grant-in-Aid for Scientific Research No. 11740267
from the Ministry of Education, Science, Sports and Culture of Japan and by the Center for Climate System Research, University of Tokyo. T. G. S. is supported by
the Natural Sciences and Engineering Research Council and by the Meteorological Service of Canada. The
GFD-DENNOU Library was used for drawing figures.
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