Journal of the Physical Society of Japan
Vol. 73, No. 12, December, 2004, pp. 3319–3330
#2004 The Physical Society of Japan
Unified Scaling Theory for Local and Non-local Transfers
in Generalized Two-dimensional Turbulence
Takeshi W ATANABE and Takahiro I WAYAMA1 y
Department of Mechanical Engineering, Applied Physics Program,
Nagoya Institute of Technology, Nagoya 466-8555
1
Graduate School of Science and Technology, Kobe University, Kobe 657-8501
(Received June 14, 2004)
The enstrophy inertial range of a family of two-dimensional turbulent flows, so-called -turbulence, is
investigated theoretically and numerically. Introducing the large-scale correction into Kraichnan–Leith–
Batchelor theory, we derive a unified form of the enstrophy spectrum for the local and non-local transfers
in the enstrophy inertial range of -turbulence. An asymptotic scaling behavior of the derived enstrophy
spectrum precisely explains the transition between the local and non-local transfers at ¼ 2 observed in
the recent numerical studies by Pierrehumbert et al. [Chaos, Solitons & Fractals 4 (1994) 1111] and
Schorghofer [Phys. Rev. E 61 (2000) 6572]. This behavior is comprehensively tested by performing
direct numerical simulations of -turbulence. It is also numerically examined the validity of the
phenomenological expression of the enstrophy transfer flux responsible for the derivation of the
transition of scaling behavior. Furthermore, it is found that the physical space structure for the local
transfer is dominated by the small scale vortical structure, while it for the non-local transfer is done by
the smooth and thin striped structures caused by the random straining motions.
KEYWORDS: two-dimensional turbulence, enstrophy cascade, scaling law, direct numerical simulation, non-local
transfer, large-scale correction
DOI: 10.1143/JPSJ.73.3319
1.
Introduction
One of the central issues of the study of two-dimensional
(2d) turbulence is to clarify the scaling properties in the
enstrophy inertial range.1) The classical dimensional arguments by Kraichnan, Leith and Batchelor (KLB)2) suggest
that the enstrophy spectrum QðkÞ in the enstrophy inertial
range obeys the following scaling form
QðkÞ ¼ C2=3 k1 ;
ð1:1Þ
where is the enstrophy dissipation rate, k is the horizontal
wavenumber and C is a non-dimensional constant. Many
studies have been debating on the validity of the KLB
theory. A recent high resolution direct numerical simulation
(DNS) of forced-dissipated 2d Navier–Stokes (NS) turbulence supports the existence of the KLB scaling (1.1).3) In
contrast, it has been reported that the enstrophy spectrum
shows the apparent deviation from the KLB scaling (1.1)4–6)
and QðkÞ is significantly steeper than k1 . Similar results
have also been reported in a series of works for freely
decaying 2d NS turbulence,7) in which it was revealed that
the significant deviation of the enstrophy spectrum from the
KLB scaling originates from the existence of the selforganized coherent vortices. Up to now, many theoretical
and numerical explanations have been proposed for clarifying this spectral behavior.5,8,9)
Concerning forced-dissipated 2d NS turbulence, the
deviation from the KLB scaling is also discussed in
connection with the log-corrected form proposed by
Kraichnan10) as
1=3
k
2=3 1
ln
;
ð1:2Þ
QðkÞ ¼ CK k
k1
E-mail: [email protected]
y
E-mail: [email protected]
where k1 is the smallest wavenumber in the enstrophy
inertial range and CK is a non-dimensional constant. Indeed,
it is pointed out that this log-correction leads to the apparent
scaling law being steeper than (1.1).11,13) The scaling form
(1.2) was recently verified by the DNS studies,4,6,12) the high
resolution closure computation,13) and spectrally reduced
dynamics of the 2d NS equation.14) Moreover, CK was
estimated numerically as CK ’ 1:5{1:74) and CK ’ 1:9.6)
The discussion on the log-corrected form of the enstrophy
spectrum implies that the enstrophy transfer in 2d NS
turbulence is characterized by the non-local interaction
among scales rather than the local interaction.10) This is
different from the fundamental picture of energy transfer in
3d NS turbulence,15) in which the local interaction among
scales is assumed. The existence of the non-local transfer
means that the large-scale natures have directly influence on
the small-scale statistics. In this sence, it is thought that the
small-scale statistics of 2d NS turbulence may be nonuniversal.16)
The local or non-local natures of enstrophy transfer are
also highlighted by investigations of generalized 2d turbulence, which is so-called -turbulence.17) The fundamental
equation for -turbulence is characterized by the following
relation with a real parameter ,
qk ¼ jkj k ;
ð1:3Þ
where qk and k are Fourier coefficients of a scalar function
qðr; tÞ (r ¼ ðx; yÞ) and the stream function ðr; tÞ with
wavevector k, respectively. The equation of motion for
qðr; tÞ is given by
@q
þ Jð; qÞ ¼ dS þ dL þ f ;
@t
ð1:4Þ
where Jða; bÞ ¼ ax by ay bx denotes the Jacobian operator.
The terms dS and dL are for the small-scale and large-scale
dissipations, respectively, and f is an arbitrary external
3319
3320
J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004
T. WATANABE and T. IWAYAMA
force. For some given values of , eq. (1.4) with (1.3) is
reduced to the evolution equation for some well-known 2d
turbulent systems.18)
The statistical dynamics in the inertial range of turbulence is characterized by the transferring processes of
the two inviscid quadratic invariants given by
Z
1
E ¼ q dx dy
ð1:5Þ
2A A
and
Q ¼
1
2A
Z
q2 dx dy;
ð1:6Þ
A
ð1:7Þ
applying the KLB theory to -turbulence.17,19) Although the
theoretical prediction (1.7) is well agree with the results of
DNS of forced-dissipated -turbulence for 0 < < 2, it is
not supported by DNS for > 2. The results of DNS for
> 2 exhibit the enstrophy spectrum
Q ðkÞ k1 ;
2.
Fundamentals
2.1
where A is the area in which the field q is determined.
According to the NS case ( ¼ 2), we shall call E and Q
the energy and the enstrophy throughout this paper,
respectively.
Similarly the spectrum Q ðkÞ defined by Q ¼
R1
Q
ðkÞ
dk
is also called the enstrophy spectrum in this
0
paper. Recently, the DNSs on -turbulence have been
performed to investigate the scaling behavior of the spectra
in the enstrophy inertial range17,19,20) and the energy inertial
range.21) The enstrophy spectrum in the enstrophy inertial
range was theoretically derived to be
Q ðkÞ kð72Þ=3 ;
-turbulence. In §3, we propose a unified scaling theory for
the local and non-local transfers of -turbulence and derive
the new spectral form of the enstrophy spectrum in the
enstrophy inertial range. In §4, we perform DNS of turbulence in order to examine the detailed features of the
transition of scaling behavior and verify the existence of the
large-scale correction predicted from the present theory.
Moreover we discuss the physical space structures for both
the local and non-local transfers. We summarize the results
obtained in this paper in §5.
ð1:8Þ
which is independent of the values of . Although
Pierrehumbert et al.17) and Schorghofer19) pointed out the
importance of the non-local enstrophy transfer responsible
for the failure of (1.7) for > 2 by invoking the analogy
between the enstrophy transfer for > 2 and passive scalar
transfer in the viscous convective range,22) systematic
derivation of (1.8) based on the enstrophy transfer has been
left an unsolved problem. Moreover, the mechanism of
transition between the local and non-local transfers at ¼ 2
cannot be explained by their theories. They use the local
similarity theory by KLB for < 2, while the passive scalar
theory for > 2.
The purpose of this paper is to propose a theory exploring
a systematic derivation of the transition of the scaling
behavior between the local and non-local enstrophy transfers
observed in -turbulence. We derive a new form of
enstrophy spectrum for -turbulence by extending the
KLB theory. An extension of KLB theory has been
discussed by Kraichnan10) and recently re-examined by
Bowman13) for 2d NS turbulence, i.e., ¼ 2. The theory
discussed in this paper is a generalization of these studies to
-turbulence. In this approach, the derived enstrophy
spectrum exhibits the inertial range scaling laws responsible
for the transition of asymptotic scaling behavior at ¼ 2.
These predictions are also re-examined by performing DNS
of -turbulence in detail.
This paper is organized as follows. In §2, we define the
fundamental statistical quantities in the wavenumber space
and discuss some characteristics of the KLB scaling law for
Spectral form of the governing equation for turbulence
Let us consider a system which is confined within the
square domain ½0; L2 and adopt doubly periodic boundary
conditions. Then the 2d scalar field qðr; tÞ is expanded as
X
qðr; tÞ ¼
qk ðtÞ expðik rÞ;
ð2:1Þ
k
where k ¼ 2n=L is the wavevector with n being the integer
vector. Hereafter the time argument is omitted for brevity.
The evolution equation for qk is obtained from eqs. (1.3) and
(1.4) as
X
@qk
S ðl; mÞql qm p k2p qk 0 qk þ fk ; ð2:2Þ
¼
@t
kþlþm¼0
ðl mÞz
ðjmj jlj Þ;
ð2:3Þ
2
where denotes the complex conjugate. The dissipation
terms dS and dL in eq. (1.4) are respectively adopted as the
drag force and the hyperviscosity of degree p. The enstrophy
spectrum Q ðkÞ is defined as
X
Q ¼
Q ðkÞ;
ð2:4Þ
S ðl; mÞ k
Q ðkÞ ¼
shell
X
1
k
2
hjqk j2 i;
ð2:5Þ
P
where shell
means the shell summation over k k=2 k
jkj < k þ k=2 (k ¼ 2=L), and h i denotes the ensemble average. The evolution equation for Q ðkÞ is derived
from eq. (2.2) as
@
Q ðkÞ ¼ TQ ðkÞ 2p k2p Q ðkÞ
@t
ð2:6Þ
20 Q ðkÞ þ F ðkÞ;
where TQ ðkÞ is the enstrophy transfer function, which
originates from the nonlinear term of (2.2), and is defined by
shell
X
X
TQ ðkÞ ¼
S ðl; mÞ<hðqk ql qm Þ i;
ð2:7Þ
k kþlþm¼0
where < denotes the real part. The function F ðkÞ is due to
the external forcing which is assumed to be adopted within
kl jkj kh , otherwise 0. Moreover the coefficient 0 in the
drag force is defined to have finite values in the range jkj kL and zero in jkj > kL . The inequality kL kl is required for
the investigation of inverse cascading inertial range with
kL k kl . These conditions are usual for setting up of
many DNSs of 2d NS turbulence.
The enstrophy transfer flux ðkÞ, which is defind by
J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004
ðkÞ ¼ k
X
TQ ðk0 Þ;
T. WATANABE and T. IWAYAMA
where d ðkÞ is defined by
ð2:8Þ
k0 ¼0
plays an important role in the present study. ð0Þ ¼
ð1Þ ¼ 0 should be satisfied from the definition (2.8) and
the conservation of enstrophy in the nonlinear interactions. If
the system is in the statistically steady state, we can omit the
time dependence in eq. (2.6). Then it reduces to the balance
equation for ðkÞ as follows:
ðkÞ ¼ d ðkÞ L ðkL < k < kl Þ;
ðkÞ ¼ d ðkÞ L þ f ðkh < kÞ;
ð2:9Þ
ð2:10Þ
where the cumulative dissipation function d ðkÞ is defined by
k
X
2p
d ðkÞ ¼ 2p
k0 Q ðk0 Þ:
ð2:11Þ
k0 ¼0
The constants f and L are the input and dissipation rates of
Q by the external
P and drag forces, respectively,
P and are
defined by f ¼ kkh0 ¼kl F ðk0 Þ and L ¼ 20 kkL0 ¼0 Q ðk0 Þ.
One should note that f and L are independent of k by their
definitions, but they depend on . Moreover we define the
enstrophy dissipation rate as
d ð1Þ:
ð2:12Þ
Using the constraint ð1Þ ¼ 0, we can obtain
f ¼ L þ :
ð2:13Þ
Therefore eq. (2.10) reduces to the relation
ðkÞ ¼ d ðkÞ þ ðkh < kÞ:
ð2:14Þ
The range where ðkÞ is asymptotically independent of k,
i.e., ðkÞ ’ (or L ), is called the enstrophy inertial
range because the nonlinear term governs the dynamics of
the system in this wavenumber range.
In the limit L ! 1, the summation with respect to the
wavenumber in the above equations is represented in terms
of the integral, i.e.,
Z1
Q ¼
Q ðkÞ dk;
ð2:15Þ
0
Z
k
ðkÞ ¼ 0
TQ ðk0 Þ dk0 ;
Z
k
2p
k0 Q ðk0 Þ dk0 :
d ðkÞ ¼ 2p
ð2:16Þ
ð2:17Þ
0
The evolution equation for the energy spectrum is also
derived in the same way. The relation between
R 1 the energy
spectrum E ðkÞ, which is defined by E ¼ 0 E ðkÞ dk, and
Q ðkÞ is given by
E ðkÞ ¼ k Q ðkÞ:
ð2:18Þ
In the statistically steady state, the energy transfer flux
ðkÞ, which is defined by
Zk
ðkÞ ¼ k0 TQ ðk0 Þ dk0 ;
ð2:19Þ
0
is related to the energy dissipation rates and L , and the
cumulative energy dissipation function d ðkÞ as
ðkÞ ¼ d ðkÞ L
Z
d ðkÞ ¼ 2p
ðkL < k < kl Þ;
ðkÞ ¼ d ðkÞ L þ f
ðkh < kÞ;
ð2:20Þ
ð2:21Þ
k
k0 2p E ðk0 Þ dk0 :
3321
ð2:22Þ
0
Rk
Rk
The constants f ¼ klh k0 F ðk0 Þ dk0 and L ¼ 20 0 L E
ðk0 Þ dk0 are the input and dissipation rates of E by the
external and drag forces, respectively. The energy dissipation rate d ð1Þ satisfies the relation
f ¼ L þ :
ð2:23Þ
The range where ðkÞ is asymptotically independent of k,
i.e., ðkÞ ’ (or L ), is called the energy inertial range.
2.2 Transferring quantity toward smaller scales
In this subsection, we briefly mention about the transferring natures of both E and Q . One should notice that the
conservation structures of the nonliner term in 2d NS
equation ( ¼ 2) are maintained in the case for eq. (2.2)
with the general case of . Therefore we can extend the
several discussions with respect to the directions of cascade
for 2d NS turbulence to -turbulence. For example, the
application of Fjørtoft’s theorem27) to -turbulence, in which
the conservation laws of E and Q among a single triad
interaction are considered, yields that, for > 0 ( < 0), the
most of Q (E ) must be transferred toward the smaller
scales rather than the larger ones. More details in this
direction have also been investigated by extending the
discussion given by Merilees and Warn28) to -turbulence,20)
in which the all choices of interacting triads are considered.
On the other hand, it has been shown that the centroid
wavenumber of enstrophy spectrum must move toward
larger (smaller) wavenumber for > 0 ( < 0) under the
spreading hypothesis of enstrophy spectrum.21) Thus it is
expected that the enstrophy (or energy) inertial range
responsible for the transferring process toward smaller
scales is established in the case for > 0 (or < 0). We
also present the slightly different discussion from the above
in Appendix.
2.3 The KLB scaling law for -turbulence
In order to clarify the significance of correction to the
KLB scaling at the large-scale, which will be derived in the
later section, we briefly review the KLB theory of 2d NS
turbulence2) extended to -turbulence.17) The important
quantities characterizing the transfers of the energy and
enstrophy in their inertial ranges are the energy transfer flux
ðkÞ and the enstrophy transfer flux ðkÞ; they are
dimensionally
expressed
as
ðkÞ ½kþ2 ½k 3 4 3
2þ2
3
½k ½t and ðkÞ ½k
½k ½k24 ½t3 , where
the stream function k has a dimension ½k ½k2 ½t1 .
According to the classical dimensional arguments of the
KLB theory,2) we assume that ðkÞ and ðkÞ are
independent of k and t in their inertial ranges and equal to
the energy dissipation rate and the enstrophy dissipation
rate , where ð Þ represents the dissipation rate by the
hyperviscosity ( ) or the drag force L (L ), respectively.
Then the scaling laws of characteristic time scales E and Q
are evaluated as
E 1=3 kð4Þ=3 ;
Q 1=3 ð42Þ=3
k
ð2:24Þ
:
ð2:25Þ
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J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004
T. WATANABE and T. IWAYAMA
The enstrophy spectrum, which is dimensionally evaluated
as Q ðkÞ ½k25 ½t2 , is expressed in terms of or and
k by using eqs. (2.24) and (2.25) as follows,
Q ðkÞ 2=3 kð74Þ=3
ð2:26Þ
in the energy inertial range, and
Q ðkÞ 2=3 kð72Þ=3
ð2:27Þ
in the enstrophy inertial range.17)
One should notice a singularity of the scaling of
characteristic time in the enstrophy inertial range [eq.
(2.25)]. If > 2, eq. (2.25) indicates that the characteristic
time grows as k increases. Therefore we expect that the
system for > 2 cannot arrive at the equibilium state
predicted by the KLB theory. This failure suggests that the
appropriate time scale should be defined for the description
of enstrophy transfer in -turbulence. This point is the
subject of the next section.
3.
Unified Scaling Theory for -Turbulence
In this section, we derive the new scaling laws of turbulence in the enstrophy inertial range for > 0 and in
the energy inertial range for < 0 by introducing the largescale correction into the KLB scaling. Large-scale correction
to the KLB scaling for 2d NS turbulence was first discussed
by Kraichnan10) and recently re-examined by Bowman.13)
Here we generalize the study by Bowman13) to -turbulence.
A significant difference between the results of Kraichnan10)
and of Bowman13) is such that the former includes
logarithmic divergence on the enstrophy spectrum at a
wavenumber but the later removes its divergence. This
difference becomes to be very important when we consider
the asymptotic scaling laws of the enstrophy spectrum of the
-turbulence
3.1 Definition of effective rate of shear
The results obtained in the studies by Pierrehumbert et
al.17) and Schorghofer19) indicate that the scaling behavior of
the enstrophy spectrum is closely related to the local or nonlocal natures of transferring dynamics. Moreover, as explained in §2.2, the characteristic time scale for > 2
predicted by the KLB theory, in which the locality of
transferring dynamics is assumed, shows the physically
unacceptable growth as k increases. These facts suggest that
it is required to consider the non-locality of enstrophy
cascade for -turbulence. In order to introduce the non-local
effects into the characteristic time scale in the enstrophy
inertial range of -turbulence, we propose the effective
rate of shear ! ðkÞ acting on the scale k1 from its larger
scales as
Z k
1=2
! ðkÞ Zðk0 Þ dk0
0
ð3:1Þ
Z
1=2
k
k0
¼
0
42
Q ðk0 Þ dk0
;
R1
where ZðkÞ is defined by hðr2 Þ2 i=2 ¼ 0 ZðkÞ dk, r ¼
ð@=@x; @=@yÞ. Thus ! ðkÞ has the dimension of ½t1 . One
should note that the definition of the effective rate of shear
(3.1) is same as that proposed by Kraichnan10) and Bowman.13) Let us discuss the scaling behavior of eq. (3.1) for
> 0. When the KLB scaling (2.27) is used for evaluating
the integration in eq. (3.1), we obtain the following scaling
form
1=2
h
i1=2
3
4ð2Þ=3
4ð2Þ=3
1=3
k
k
; ð3:2Þ
! ðkÞ 1
8 4
where the lower limit of integral in eq. (3.1) is replaced by
k1 for the present. Depending on which term in the square
brackets of eq. (3.2) is dominant, characteristic of transferring nature changes qualitatively. For < 2, the shear
ð42Þ=3
rate takes the asymptotic scaling form ! ðkÞ 1=3
k
in k k1 . This is consistent with the inverse of Q [eq.
(2.25)] which is a characteristic time scale evaluated by the
KLB theory. Thus the system < 2 reveals the spectrally
local nature of enstrophy cascade. For > 2, in contrast,
! ðkÞ is asymptotically independent of k in k k1 , as
ð42Þ=3
! ðkÞ 1=3
. This implies that the effective time
k1
scale in the deep inertial range is predominated by the largescale eddy with k11 scale. This represents the non-local
nature of enstrophy cascade. Therefore, we expect that the
transition between the local and non-local enstrophy transfers occurs at ¼ 2.
Above discussion brings the idea to derive the corrected
form of Q ðkÞ to the KLB scaling of -turbulence by
incorporating the effective rate of shear into the classical
phenomenology by KLB. In the successive subsections, we
will investigate the new scaling form of enstrophy spectrum
for -turbulence and discuss asymptotic scaling behavior of
the resulting spectrum.
3.2 Derivation of new scaling law for -turbulence
We consider the scaling behavior of the enstrophy
spectrum Q ðkÞ for all cases of . As stated in the previous
section, the transferring quantity toward smaller scales
depends on the sign of ; the enstrophy Q (the energy
E ) is transferred toward smaller scales in the case for > 0
( < 0). Therefore, we separate the discussions on the
enstrophy spectrum depending on the sign of .
3.2.1 Positive case
For positive , the enstrophy transfer flux ðkÞ plays an
important role in the scaling theory of enstrophy cascade.
Physical meaning of ðkÞ is such that the enstrophy kQ ðkÞ
is transferred toward smaller scales than k1 with an
effective rate ! ðkÞ. According to the discussion of 2d NS
turbulence ( ¼ 2)10) and the dimensional analysis, we
propose a modeled expression of ðkÞ as
ðkÞ ¼ C kQ ðkÞ! ðkÞ;
ð3:3Þ
where ! ðkÞ is defined by eq. (3.1) and C is a nondimensional constant which depends on . Equation (3.3)
with eq. (3.1) is a naive generalization of Bowman’s
eq. (2.10)13) to -turbulence. Under the assumption that
ðkÞ is independent of k and equals the enstrophy
dissipation rate in the enstrophy inertial range, i.e.,
ðkÞ ¼ , eq. (3.3) reduces to the differential equation
d
3C2
f ðkÞ3 ¼ 2 k32 ;
dk
2
ð3:4Þ
where f ðkÞ kQ ðkÞ. Integrating eq. (3.4) from k1 to k
yields
J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004
Q ðkÞ ¼
þ
ðkÞ
2
3C2
1=3
ð72Þ=3 þ
2=3
½ ðkÞ1=3 ;
k
"
42 #
1
þ k1
1 ¼
;
4 2
k
þ
¼ 1 ð4 2Þþ ;
T. WATANABE and T. IWAYAMA
ð3:5Þ
ð3:6Þ
ð3:7Þ
where the constant þ is determined by the boundary
condition at k ¼ k1 as
þ ¼
22 27
k1
Q ðk1 Þ3 :
3C2
3.2.2 Negative case
In a similar way, we can derive the scaling law for
negative case with the large-scale correction. In this case,
E must be transferred toward smaller scales as mentioned in
the previous section. In order to introduce the large-scale
correction into the KLB scaling for the energy inertial range,
we propose the modeled expression of energy transfer flux
ðkÞ as
ð3:9Þ
where the definition of ! ðkÞ is same as the positive case.
Assuming that ðkÞ is independent of k and equals the
energy dissipation rate , i.e., ðkÞ ¼ , eq. (3.9) yields
the solution
2 1=3 2=3 ð74Þ=3 Q ðkÞ ¼
k
½ ðkÞ1=3 ;
ð3:10Þ
3C2
"
4 #
1
k1
1 ðkÞ ¼
;
ð3:11Þ
4
k
¼ 1 ð4 Þ ;
ð3:12Þ
where is defined by
¼
22 47
k1
Q ðk1 Þ3 :
3C2
The function ðkÞ gives the large-scale correction to the
KLB scaling (2.26). The spectral behavior around k1 can be
evaluated by eqs. (3.10) and (3.11) similar to the case for
> 0, as follows. If is positive (negative), the large-scale
correction leads to the spectra steeper (less steep) than the
KLB scaling laws around k k1 . However, the wavenumber
range where the correction term is significant must be
narrower than that for > 0 because of stronger power-law
42
dependence of ðk1 =kÞ4 in in
ðkÞ ( < 0) than ðk1 =kÞ
þ
ðkÞ ( > 0).
ð3:8Þ
Equation (3.6) should satisfy the inequality þ
ðkÞ > 0 in the
range k k1 for the arbitrary positive . Therefore þ ¼
þ
ðk1 Þ must be positive definite. Indeed, eq. (3.8) satisfies
this condition. The wavenumber k1 should be defined
explicitly by the smallest wavenumber in the range where
the relation ¼ C kQ ðkÞ! ðkÞ is satisfied. Comparing
eq. (3.5) with eq. (2.27), it is found that þ
ðkÞ gives the
large-scale correction to the KLB scaling of -turbulence.
Here we discuss the behavior of derived enstrophy
spectrum (3.5) with (3.6) around k k1 . From the functional
form of eq. (3.6), it is expected that the spectral behavior
around k1 is obviously different from that of the KLB
scaling. In particular, the deviation from eq. (2.27) becomes
to be significant in the limit þ 1=j4 2j. Then the
spectrum asymptotically exhibits Q ðkÞ k1 . Moreover,
the steeper or less steep behavior of the enstrophy spectra
than the KLB scaling (2.27) are determined by the signs
þ
of þ
. For > 0, eqs. (3.5) and (3.6) lead to the spectrum
steeper than eq. (2.27), while for þ
< 0, they lead to the
less steep spectrum than that.
The spectral behavior for k k1 will be investigated in
the later subsection.
ðkÞ ¼ C k1 Q ðkÞ! ðkÞ;
3323
ð3:13Þ
The constant is also positive definite with ¼ ðk1 Þ.
3.3 Asymptotic forms of Q ðkÞ in k k1
One should remark that the functions þ
ðkÞ and ðkÞ
depend on the ratio k=k1 and value of . The wavenumber
dependence of these corrections varies with the choice of or the wavenumber range we consider. Therefore, the
scaling law of the enstrophy spectrum in the inertial range
with k k1 strongly depends on the value of . Here we
derive the asymptotic forms of Q ðkÞ in k k1 and discuss
their several characteristics.
The asymptotic forms of Q ðkÞ in k k1 are divided into
four categories as follows:
(1) < 0
ð74Þ=3
Q ðkÞ ’ K 2=3
;
ð3:14Þ
k
1=3
8 2
K ¼
:
ð3:15Þ
3C2
(2) 0 < < 2
ð72Þ=3
Q ðkÞ ’ K 2=3
;
ð3:16Þ
k
1=3
8 4
K ¼
:
ð3:17Þ
3C2
(3) ¼ 2
1=3
k
2=3 1
Q ðkÞ ¼ K k
ln
;
ð3:18Þ
þ þ
k1
2 1=3
K ¼
:
ð3:19Þ
3C2
(4) > 2
1
Q ðkÞ ’ K 2=3
k ;
4 8 1=3 ð24Þ=3
k1
:
K ¼
3C2 þ
ð3:20Þ
ð3:21Þ
The proportionality constant K depends on . Here we
assumed þ 6¼ 1=ð4 2Þ and 6¼ 1=ð4 Þ. For < 2,
the large-scale correction disappears in k k1 , and we
asymptotically obtain just the KLB scaling, eqs. (2.26) and
(2.27). This means that the enstrophy spectra for < 2 are
locally determined. In the extreme case for þ ’ 1=ð4 2Þ
and ’ 1=ð4 Þ, the wavenumber dependences of
corrected parts are negligible, and then the KLB scaling
asymptotically works up to k k1 . For > 2, on the other
hand, the scaling exponent of Q ðkÞ is 1 which is
independent of . Moreover, a constant K depends on
the quantity at the large-scale k11 . Therefore, it is expected
that K with > 2 is a non-universal constant. These results
mean that the large-scale correction masks the scaling
behavior by the local transfer, and the characteristic in the
large-scale, e.g., the forcing mechanism, governs the
3324
J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004
T. WATANABE and T. IWAYAMA
where urms is the root mean square of velocity field, L is the
integral scale, and lQ is the enstrophy dissipation scale
defined by
0
-1
(2α−7)/ 3
-2
−1
lQ ½3p = 1=ð24þ6pÞ :
ð4:2Þ
4.1 Setting of viscous term in -turbulence
Before performing the comprehensive DNS of -turbulence, it is meaningful to discuss the viscous effects in the
small-scale with respect to the variation of . Here we
consider the role of parameter in the integral scale
Reynolds number RL which measures the relative intensity
of nonlinear advection term to the viscous one. For the
positive , RL is naturally defined as follows,
2pþð24Þ=3
urms L2p1
L
;
ð4:1Þ
RL ¼
lQ
p
4.2 Setup of DNS
We numerically solve eqs. (1.3) and (1.4) with the
hyperviscosity of degree p ¼ 4. In particular, we focus our
attention on the cases for ¼ 1, 2 and 3. The external force
f is defined by fk ðtÞ ¼ A exp½i k ðtÞ, where the amplitude A
is a fixed real number and k ðtÞ denotes the uniform random
numbers with ½0; 2 being independent of the each modes
within the wavenumber band kl jkj kh (kl ¼ 1, kh ¼ 2)
and time steps. The drag force dL ðkÞ ¼ 0 qk at large-scale
is applied to the modes in jkj kL ¼ 2. In the present DNS,
thus, the forcing wavenumber is almost same as the
θ(α)
In this section, we perform DNS of -turbulence in order to
verify the validity of unified scaling theory proposed in the
previous section. We re-examine the scaling behavior of the
enstrophy spectrum for > 0 and discuss several characteristics of transferring process based on the phenomenological
expression of enstrophy transfer flux, i.e., eq. (3.3).
The naive estimations as urms Q =L and Q ð42Þ=3
2=3
are supposed for evaluating eq. (4.1). For the
L
simplicity of discussion, we consider the usual normal
viscosity p ¼ 1, i.e., RL ðL=lQ Þ2ðþ1Þ=3 . In the conventional setting of DNS for the investigation of fully developed
turbulence, the value of ratio L=lQ is taken to be maximized
to achieve the stronger turbulent states under the constraint
Kmax lQ ’ 1, where Kmax is a trancation wavenumber due to
the finite spatial resolution. pUnder
the fixed spatial grid
ffiffiffiffi
points N, although L=lQ N is independent of , RL
depends on as RL N ðþ1Þ=3 . This implies that the smaller
systems have the smaller values of RL than the case for
larger ones. Thus the effect of normal viscosity as the
enstrophy dissipation term strongly varies with the value of
if we perform DNS of -turbulence under the fixed N (or
L=lQ ).
Above nature of -turbulence is actually important for the
present numerical purpose, in which the several statistical
and dynamical properties in the turbulent field will be
studied by varying the value of . In this case, the values of
RL for several cases should be set to be independent of with RL 1 because we would like to quantitatively
compare the scaling behavior of the enstrophy spectra and
discuss their -dependences under the same widths of
enstrophy inertial ranges (or the same degrees of RL ).
Therefore it is not appropriate to discuss the statistical
natures of -turbulence and compare their -dependences
under the fixed N with p ¼ 1. One should notice that the
preceding study19) by the high resolution DNS of turbulence with normal viscosity do not pay attention to
this important -dependence of RL . In order to establish the
above-mentioned state for -turbulence, it may be a good
idea that the hyperviscosity with p > 1 is adopted in the
setting of DNS because it is expected from eq. (4.1) that the
higher degree of hyperviscosity relatively gives the smaller
-dependence of RL . Moreover we have an additional
benefit by making use of hyperviscosity that the much larger
computational resource is not needed for obtaining fully
developed turbulence. Indeed, the hyperviscosity with
satisfying 3p j 2j diminishes the -dependence of
RL , and leads to the same degree of turbulent states for
several cases. In the present DNS, therefore, we use the
higher degree of hyperviscosity to discuss the scaling
properties of -turbulence in the enstrophy inertial range.
-3
-4
(4α−7)/ 3
-5
-6
-7
-3
-2
-1
0
1
2
3
4
5
α
Fig. 1. Asymptotic scaling exponent ðÞ of Q ðkÞ defined by Q ðkÞ k
ðÞ obtained in the present theory with k k1 . The circles at ¼ 0 and
2 represent that ðÞ cannot be defined.
statistics in the small-scale for > 2. This implies that the
interaction among scales is predominated by fully non-local
interaction. The NS case ( ¼ 2) is just the transition point
between the local and fully non-local transfers. Here one
should remark that eq. (3.18) is obtained from eqs. (3.5) and
(3.6) by taking the limit ! 2. Note that eq. (3.18) is
equivalent to the result by Bowman.13) In this case, the largescale correction appears as the weak wavenumber dependence, i.e., lnðk=k1 Þ. It should be noted in addition that
eq. (1.2) is asymptotically satisfied in the extreme limit
k=k1 exp ðþ Þ. Above asymptotics are in good agreement
with the preceding results17,19) and that by the present DNS,
which will be discussed in the next section. In Fig. 1, we
summarize the asymptotic scaling exponent ðÞ defined by
Q ðkÞ k
ðÞ in eqs. (3.14)–(3.21). Since the enstrophy
spectrum for ¼ 2 does not obey the power-law scaling,
ðÞ cannot be defined for such case. In addition, the system
for ¼ 0 has no nonlinear term so that turbulent state of
such system is not realized. Therefore, ðÞ for ¼ 0 cannot
be defined.
4.
Direct Numerical Simulations and Discussions
J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004
T. WATANABE and T. IWAYAMA
1
2
3
4
A
1 1018
0:2
2 1019
0:2
2 1019
0:2
0
0:1
0:1
0:1
E
1:08
0:69
0:56
Q
1:71
1:90
2:65
1:41 104
1:17 106
9:84 109
0:032
0:054
0:081
kd
244
193
131
ku
ki
229
1:22
215
1:17
202
1:19
kf
1:38
1:34
1:34
wavenumber on which the drag force acts. This setting gives
the wider enstrophy inertial range than that of the conventional setting by the other studies of 2d turbulence, in which
the inverse cascading inertial range with the finite width
exists in the regime kL k kl . The pseudo-spectral
method is used in a square area with the doubly periodic
boundary condition. The system size is L L with L ¼ 2,
and the numbers of the spatial grid points N N ¼ 10242 .
The trancation wavenumber is taken as kT ¼ ½N=3 to
suppress the aliasing errors, where ½ denotes the
Gaussian symbol. The time integration is performed by the
fourth-order Runge–Kutta scheme. After discarding the
transient time, the statistical average for evaluating several
statistical quantities were taken as temporal average during
the steady state in place of the ensemble average. The values
of all parameters introduced above and some fundamental
statistical quantities calculated by DNS are summarized in
Table I.
First we mention to the values of characteristic wavenumbers introduced in the present DNS. Making use of the
hyperviscosity in DNS, the ultraviolet cutoff wavenumber ku
defined by
1=
ð4:3Þ
ku is relevant to scale physical quantities in the small-scale.4)
The values of ku are listed in Table I. We can observe that ku
is almost independent of . This feature is contrast to the
behavior of the enstrophy dissipation wavenumber kd ¼
1=lQ ; the value of kd becomes to be smaller as increases in
the present setting of DNS. In the large-scale, on the other
hand, the infrared cutoff wavenumber ki defined by
1=
L
ki ð4:4Þ
L
determines the large-scale characteristic scale. Moreover the
forcing wavenumber kf , which represents the input scale of
external fluctuation, is also defined by
1=
f
kf :
ð4:5Þ
f
The values of ki and kf calculated from DNS are also listed
in Table I. As expected from the present setting of DNS, we
can observe kf ki for all cases of and the values of kf are
also close to each other. From these results, we can conclude
that the width of the wavenumber range dominated by the
inertial term is almost same for each cases, where the
values of ratio ku =kf are calculated as 166, 160 and 151 for
¼ 1, 2 and 3. Therefore the present setting of DNS is
meaningful for discussing the -dependence of the statistical
quantities in the enstrophy inertial range.
4.3 Results of spectral behavior
In order to define the enstrophy inertial range more
precisely, we observe the enstrophy transfer flux ðkÞ
defined by eq. (2.8). Figure 2 shows the variation of ðkÞ
against the wavenumber normalized by ku for ¼ 1, 2 and 3.
It is apparently observed that ðkÞ remains constant in the
wavenumber range 0:01 < k=ku < 0:6. These values are
equal to their enstrophy dissipation rates . This fact means
that the statistically steady states with the significant width
of enstrophy inertial ranges are well established in the
present DNS. Moreover, Fig. 2 manifests that the largest
values of k=ku in each plateau regimes are almost independent of , where k=ku ’ 0:6. Therefore when the wavenumber
is scaled by ku , the location of the transition wavenumber
from the inertial range to viscous one is approximately
independent of , as expected from the discussion in the
previous subsection.
The corresponding enstrophy spectra Q ðkÞ=2=3
are
shown in Fig. 3. Hereafter we use the following definition
of Q ðkÞ in place of eq. (2.5), as
Q ðkÞ khjqk j2 is ;
ð4:6Þ
where h is denotes the temporal and shell averages over
k 1=2 < jkj k þ 1=2. This form of the enstrophy spectrum removes the zig-zag fluctuation of Q ðkÞ due to a few
grid points within a shell in the low wavenumber range.
(One should note that the enstrophy Q evaluated by the
summation of eq. (4.6) is slightly different from the original
value listed in Table I. However, the difference between
them does not affect the main conclusion of the present
study.) We can observe the power-law decays of Q ðkÞ in the
wavenumber range where the transfer fluxes ðkÞ are
independent of k. The spectral slopes are close to the
asymptotic values predicted by our unified scaling theory
0.1
0.08
0.06
Λα(k)
Table I. Numerical parameters and fundamental statistical quantities in
the present DNS of -turbulence. Please see the details of numerical
condition in the text.
3325
0.04
0.02
α=1
α=2
α=3
0
-0.02
0.01
0.1
k/ku
1
Fig. 2. Enstrophy transfer flux obtained from the DNS of eqs. (1.3) and
(1.4) for ¼ 1, 2 and 3. The horizontal lines in the figure represent the
values of enstrophy dissipation rate for each cases, see the Table I.
3326
J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004
T. WATANABE and T. IWAYAMA
10
α=1
α=2
α=3
10 slope 2/ 3
−1
1
10−2
−5/ 3
10−3
10−4
10
−5
10−6
ωα(k)
Qα(k)/η2/3
α
10−1
α=1
α=2
α=3
0.01
1
0.1
k/ku
1
compensated spectra
10
1
α=1
α=2
α=3
slope −0.1
0.01
0.1
0.1
1
k/ku
Fig. 3. Enstrophy spectra obtained from the DNS of eqs. (1.3) and (1.4)
for ¼ 1, 2 and 3.
0.1
0.01
1
k/ku
Fig. 4. Compensated enstrophy spectra for ¼ 1, 2 and 3. The compensated form is 2=3
k5=3 Q ðkÞ for ¼ 1, and 2=3
kQ ðkÞ for ¼ 2 and 3.
derived in the previous section. The scaling behavior of
Q ðkÞ for ¼ 2 and 3 seems to be almost same; the value of
the power-law exponent of Q ðkÞ is about 1.
In order to examine the scaling behavior of the enstrophy
spectrum more precisely, the compensated enstrophy spectra, which are defined by 2=3
k5=3 Q ðkÞ for ¼ 1 and by
2=3
kQ ðkÞ for ¼ 2 and 3, are shown in Fig. 4. The value
of scaling exponent for ¼ 1 seems to be close to the
prediction by the KLB theory (slope 5=3) around k=ku ¼
0:1{0:3 though the width of the scaling range is narrower
than that of the enstrophy inertial range defined by Fig. 2.
Beside this range, we pay our attention to the spectral form
in 0:02 < k=ku < 0:1, where it is thought that the large-scale
correction to the 5=3 law may prevent the compensated
spectrum from remaining constant because this range does
not satisfy the condition k k1 . In contrast, it is recognized
that Q ðkÞ for ¼ 3 has a clear power-law scaling with the
exponent 1. That is, the scaling exponent of Q ðkÞ for
¼ 3 is apparently different from the prediction by the KLB
theory (slope 1=3) rather be in good agreement with
eq. (3.20) obtained in the present theory. Moreover, one
Fig. 5. Variation of the effective rate of shears ! ðkÞ defined by eq. (3.1)
against k=ku for ¼ 1, 2 and 3.
should notice that there is no plateau for ¼ 2 in this
compensated form. The compensated spectrum for ¼ 2
approximately behaves as the power-law decay as kQ ðkÞ k0:1 . As is going to be shown in the later, this behavior also
originates from the existence of the large-scale correction.
In order to investigate further the validity of our
discussion, we directly examine the relation (3.3), because
the asymptotic form of the enstrophy spectrum derived in
§3.3 relies on the validity of eq. (3.3). First, we make the
plot of effective rate of shear ! ðkÞ evaluated by using
eq. (3.1). Figure 5 shows the variation of ! ðkÞ against k=ku .
For ¼ 1, the functional form of ! ðkÞ approximately
behaves as ! ðkÞ k2=3 in 0:05 k=ku 0:3, which is
consistent with the dimensional evaluation by the KLB
theory. For ¼ 3, ! ðkÞ behaves as the constant shear rate
in almost whole wavenumber range, which is expected from
the discussion in the previous section and stems from the
strong non-locality of transferring nature. For ¼ 2, on the
other hand, ! ðkÞ gradually increases as k increases. This
behavior is closely related to the observation in Fig. 4 for
¼ 2. Next we examine the validity of the form (3:3) by
using Q ðkÞ and ! ðkÞ previously obtained. Figure 6 shows
the variation of the enstrophy transfer flux divided by the
dimensionally modeled form (3.3) for ¼ 1, 2 and 3.
Although the plateau of the modeled transfer flux is
narrower than the enstrophy inertial range, eq. (3.3) works
very well. This result means that the existence of the largescale correction obtained in the previous section is verified.
It is interesting to estimate the various constants included
in the spectral form (3.5) with (3.6). The constant C is
given by the value of the plateau in Fig. 6, and the
wavenumber k1 is also estimated by the smallest wavenumber in this plateau. Moreover the constants þ are
calculated by using the value Q ðk1 Þ with C and , and K
is done from eqs. (3.15), (3.17), (3.19) and (3.21). These
results are summarized in Table II. It should be noted that
the value þ ¼ 0:79 for ¼ 1 is larger than 1=2 (or
þ
1 ¼ 0:58 < 0) in the present DNS. From the discussion
in §3.2.1, we expect that the wavenumber dependence of the
large-scale correction yields the scaling behavior of the
enstrophy spectrum less steep than the KLB scaling (slope
J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004
α=2
1.5C 2α(ln(k/k1)+χ+)
1
1
C3=0.226(×3)
C2=0.204
0.1
3327
1.2
α=1
α=2
α=3
η2α(kQα(k))−3
Λα(k)/kQα(k)ωα(k)
10
T. WATANABE and T. IWAYAMA
C1=0.069
0.8
0.6
0.4
0.2
0
0.01
0.01
0.1
k/ku
-2
1
-1
0
1
ln(k/k1)
2
3
Fig. 6. Behaviors of the enstrophy transfer flux divided by its modeled
form (3.3). The horizontal lines denote the constant value C estimated in
the range k1 k 50. The line for ¼ 3 is multiplied by 3 for clarity.
Fig. 7. Verification of the large-scale correction for ¼ 2. The values of
parameter included in the function 1:5C2 ðlnðk=k1 Þ þ þ Þ are from the
results in Table II.
Table II. Various constants included in the new spectral form (3.5) with
(3.6) evaluated from the present DNS. C is evaluated in k1 k 50 for
each values.
raises the question of whether þ is a universal constant for
2d NS turbulence. It is interesting problem to investigate the
value of þ by DNS under the several conditions of external
and drag forces.
Furthermore for 2d NS turbulence ( ¼ 2), we directly
verify the existence of the logarithmic correction by using
the constants listed in Table II, which are evaluated by our
unified scaling theory. Figure 7 shows the plot of
2 ðkQ ðkÞÞ3 againt lnðk=k1 Þ. The solid line indicates the
prediction from our unified scaling theory. This figure shows
that the unified scaling theory works very well in the
enstrophy inertial range. Therefore the result of the present
DNS with ¼ 2 supports the existence of the log-corrected
enstrophy spectrum predicted by Kraichnan10) and Bowman.13)
1
2
3
C
k1
0:069
7
0:204
7
0:226
7
þ
0:79
3:35
58:35
K
6:53
2:52
2:21
þ
þ
þ
5=3). Indeed, the prediction from the present theory is
consistent with the result shown in Fig. 4. For ¼ 3, in
contrast, þ is much larger than unity, i.e., þ
3 1 from
eq. (3.7). This means that the correction (3.6) is approxþ
2
imately represented by þ
3 ðkÞ ’ 3 ðk=k1 Þ =2 for k k1 .
Thus we expect that the large-scale correction with þ 1
leads to the asymptotic spectral form Q ðkÞ k1
[eq. (3.20)] for > 2 even when k k1 . This is consistent
with the observation in Fig. 4; the wide plateau of kQ3 ðkÞ
exists up to k k1 . This fact is also good agreement with our
theoretical prediction of scaling behavior for Q ðkÞ.
Moreover, we should note the value of the proportionality
constant K2 appeared in eq. (3.18), which is equivalent to
CK in eq. (1.2). In the present DNS, CK is estimated as
CK ’ 2:5, which is slightly larger than the values obtained
from the other studies by DNS, e.g., CK ’ 1:5{1:74) and
CK ¼ 1:9.6) This may be originated from the wavenumber
dependence of ! ðkÞ. In fact, the behavior of ! ðkÞ in k k1
significantly depends on the details of the external and drag
forces.30) Therefore we infer that CK is the non-universal
constant, as pointed out in the study of the non-locality of
the triad interaction in the wavenumber space.16) In spite of
this fact, it is noticeable that the result of the present DNS
for 2d NS turbulence ( ¼ 2) shows the value þ ’ 3:4,
which is almost equal to the value þ ’ 3:5 obtained by
Bowman13) in the numerical study of the realizable test-field
model. Though there is an uncertainty with respect to the
determination of the wavenumber k1 , the value of þ seems
to be insensitive to the choice of external or drag forces. This
4.4 Comparison of the physical space structures
Up to now, we have focused our interest on the statistical
nature of -turbulence in the wavenumber space like as the
spectrum and the transfer flux. It will be also desirable to
investigate the statistical properties of -turbulence in the
physical space toward the further understandings of the
difference and universality among several -turbulent
systems. In order to help our understanding of transferring
properties of -turbulence, we visualize the physical space
structure in the scalar field qðr; tÞ and discuss its characteristics for ¼ 1, 2 and 3. Figure 8 shows the instantaneous
snapshots of qðr; tÞ obtained in the present DNS. For ¼ 1,
the field is governed by the small-scale vortices caused by
the filamentary instabilities of the large-scale straining
motions. There are several sizes of small-scale structures
in the scalar field, which reminds us the hierarchical
cascading processes leading to the spectrum of local transfer
(Q ðkÞ k5=3 ) in the wavenumber space. Such a process is
discussed in detail by the numerical study;23) this feature
toward the finer scales is named as curdling cascade. In
contrast, the field for ¼ 3 is dominated by the smooth and
thin striped structures caused by the large-scale straining
random motions. This stems from the strongly non-local
nature of the transferring dynamics with combining the
3328
J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004
T. WATANABE and T. IWAYAMA
external and drag forces. The highly non-local nature of
transfer leading to the spectrum as Q ðkÞ k1 is closely
related to the nature of this spatial structure. The field for
¼ 2 is partly similar to that for ¼ 3, but a few vortices
are observed in the straining and striped structures. The
existence of vortical structure may be the origin of the
logarithmic correction of spectrum in the enstrophy inertial
range for ¼ 2 because these are not observed in the scalar
field for ¼ 3.
5.
(a) α = 1
(b) α = 2
(c) α = 3
Fig. 8. Snapshots of the instantaneous scalar field qðr; tÞ for (a) ¼ 1,
(b) ¼ 2 and (c) ¼ 3.
Summary
In this paper, we have proposed the unified scaling theory
for the local and non-local enstrophy transfers in turbulence on the basis of the phenomenological scaling
theory by KLB with taking in the large-scale corrections.
Moreover we performed DNS of -turbulence for > 0 and
re-examined the transition of scaling behavior of the
enstrophy spectrum at ¼ 2 in detail. The significance of
the large-scale correction was also clarified by investigating
the validity of the modeled enstrophy transfer flux evaluated
by the present DNS. The main results obtained in this paper
are summarized as follows.
i) The new spectral form of Q ðkÞ in the direct cascading
inertial range is derived by considering the large-scale
correction to the KLB scaling. The transition of the scaling
behavior of Q ðkÞ at ¼ 2 is systematically derived in this
framework. This explains the results obtained in the recent
DNSs of -turbulence17,19) and the present DNS.
ii) The mechanism of the transition of scaling behavior in
the enstrophy inertial range was re-examined in detail by
performing the comprehensive DNS of -turbulence. The
further insight into the existence of the large-scale correction
was verified from the examination of the behavior of the
modeled enstrophy transfer flux. Moreover the significant
difference of the physical space structures between the local
and non-local transfers was discussed by visualizing the
scalar field q for each cases. Non-local transfer was
dominated by the thin striped structures caused by the
straining motion, in contrast, the local transfer was done by
the hierarchical vortices.
Finally, we mention to some unsolved problems in the
present study for the future investigations. The width of the
scaling range for ¼ 1 obtained in the present DNS is
especially narrower than the others. Toward the detailed
investigation of the spectral behavior from the large-scale
corrected range to the pure power-law one, we need much
finer spatial resolution than the present one. In this subject, it
will be also challenging problem to investigate the degree of
the deviation from the asymptotic scaling law due to the
intermittency effects in the enstrophy inertial range. The role
of the vortical structures in the transferring process should
be explored in the ¼ 1 system.
Similar state will appear in the study of DNS for the
negative system, which was not studied numerically in the
present paper. Here we refer to the viscous effects in the
negative system, in which the energy E is transferred
toward smaller scales. The evaluation of the integral scale
Reynolds number RL for < 0 yields
2pþð4Þ=3
L
RL ;
ð5:1Þ
lE
J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004
T. WATANABE and T. IWAYAMA
where lE is the energy dissipation wavenumber defined by
lE ½3p = 1=ð4þ6pÞ :
ð5:2Þ
It is fairly interesting to notice a singularity of the power-law
exponent in eq. (5.1). If we would like to realize the fully
turbulent state with the energy cascading inertial range, i.e.,
RL 1 as L=lE 1, the exponent must satisfy the
condition 2p þ ð 4Þ=3 > 0. This constraint determines
the confined selection of the degree of hyperviscosity for the
arbitrary system of negative -turbulence. For example, for
¼ 2, i.e., the asymptotic model of the Charney–Hasegawa–Mima (CHM) turbulence,25,26) the degree of hyperviscosity must be p > 1 to ensure the existence of turbulent
state. This nature originates from the definition of lE (5.2).
The choice of ¼ 2 and p ¼ 1 rules out the possibility to
achieve the meaningful transfer of E toward smaller scales.
The normal viscosity (p ¼ 1) for ¼ 2 works as the linear
drag force in the whole wavenumber range. Therefore, we
should use the higher degree of hyperviscosity than p ¼ 1 in
order to obtain the meaningful turbulent state for the CHM
system. This nature also implies the strong -dependence of
RL for < 0 with p ¼ 1. In other words, the negative system with the normal viscosity is strongly affected by the
viscous effects rather than the case for > 0 under the
fixed N. Therefore the finer spatial resolution or higher
degree of hyperviscosity may be also needed for the study of
DNS in the negative system. In order to confirm the
validity of the present unified scaling theory in < 0, we
need to investigate the energy transferring process in the
statistically steady negative -turbulence. However, such a
subject is beyond the scope of the present paper.
Acknowledgements
The authors thank Professor T. Gotoh for valuable
comments and discussions. T.I. is supported by the Grantin-Aid for Scientific Research No. 15740293 and ‘‘The 21st
Century COE program of Origin and Evolution of Planetary
Systems’’ in the Ministry of Education, Calture, Sports,
Science and Technology of Japan.
Appendix: Cascade Directions for -Turbulence
In this appendix, we discuss the directions of cascade of
both E and Q in the framework of §2.1. The energy E and
enstrophy Q injected by the external forcing at the rates of
f and f are transferred toward the scales larger and smaller
than the forcing wavenumber kf , respectively, which is
defined by kf ¼ ðf =f Þ1= [eq. (4.5)]. As introduced in §2.1,
the input rates f and f balance with the dissipation rates
originated from the hyperviscosity ð ; Þ and drag force
ðL ; L Þ as f ¼ L þ [eq. (2.23)] and f ¼ L þ [eq. (2.13)], respectively. At the high Reynolds number,
we make an assumption for the spectral distribution of the
energy and enstrophy dissipations. For some given values of
and p with p , we suppose that the bulks of and are in
the smaller scales than kf1 , respectively. Then the wavenumber ku defined by eq. (4.3) represents the characteristic
wavenumber in the small-scale, i.e., the ultraviolet cutoff
wavenumber. Whereas the bulks of L and L are within the
larger scales than kf1 if we adopt the definitions of L and L
introduced in §2.1. Thus the characteristic wavenumber in
the large-scale, i.e., the infrared cutoff wavenumber, is
3329
represented by ki given by eq. (4.4).
Now let us consider the behavior of ku =kf and kf =ki in the
high Reynolds number limit. Then we can expect ku =kf 1
and kf =ki 1. Equations (4.3)–(4.5) yield
ku
f = 1=
¼
;
ðA:1Þ
kf
f =
1=
kf
f = 1 1=
f =
¼
:
ðA:2Þ
f =
ki
f = 1
For > 0, ku =kf 1 is established as f = 1 and
f = ¼ 1 þ L = 1. In addition, f = ’ 1, i.e.,
L = 1, leads to kf =ki 1. This means that the enstrophy (energy) transfer dominates the cascading process in
kf k ku (ki k kf ) rather than that of the energy
(enstrophy) transfer. For < 0, in contrast, the energy
(enstrophy) transfer dominates the cascading process in
kf k ku (ki k kf ) because ku =kf 1 and kf =ki 1 means f = 1 and f = ¼ 1 þ L = ’ 1. This is
same as the preceding conclusions given in refs. 20 and 21.
In fact, we do not know exactly whether the assumption
introduced above is correct for all values of because it is
known that the nature of spectral distribution for the energy
and enstrophy dissipations depends on the details of
governing equation of motion; the details of external and
drag forces constrain the spectral distribution of energy and
enstrophy dissipations for 2d NS turbulence ( ¼ 2).29)
Although, in the case for ¼ 2, the above assumption is
acceptable for the setting of external and drag forces
introduced in §2.1,29) the detailed discussion on this subject
for general case of must be explored to complete the above
result.
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