Brief review of phenomenology in tbe decaying 2D-NS turbulence 53

Thus the dynamical scaling laws

(Eqs.(7.5)- (7.7))

are expressed by only the scaling exponent

�·

The existence of these scaling laws is verified by DNS of the 2D-NS equation or the numerical simulations by a particular model of point vortex dynamics that allow vortex merging (modified point vortex model)

[51, 52],

and the laboratory experiments of the 2D electrolyte fluid flow

[55, 56, 57].

From these results, the value of scaling exponent (has been obtained as

0.7""' 0.75.

The theoretical determination of the scaling exponent

�

i a challenging problem in the decaying 2D-NS turbulence. However the scaling exponent � obtained from the numerical simulation seems to depend on the initial condition because the physical quantitie related to the s If­

organized coherent vortices in the first stage fluctuates and depends on the initial form of the energy spectrum

(48].

The scaling theory above discussed is the mean field theory

[54]

which is composed by the mean value related to the coherent vortice . So the value obtained by th numerical simulations contains the effects of fluctuation of physical quantitie , espe ially the fluctuation of

ra

is significant. We must pay attention to choose the initial condition to obtain the scaling exponent in the numerical simulation to be comparable to the theory. Concerning with this point, some investigations show that the relatively broad-band energy spectrum gives rise to a population with a wide distribution of vortex size, while narrow band energy spectrum produces a narrow distribution

[48].

The vortex properties with a narrow distribution must be meaningful for the investigation of the scaling properties of the average quantitie r lat d to the coherent vortices.

Standing the above-mentioned viewpoint, the further insight of the scaling theory was given in the series of work by Iwayama et al.

[53, 54].

They noticed that the succe s of the scaling theory is in the expression of the total kinetic energy E

(Eq.(7.3)).

It is ^as erted in

[53]

that the right hand side of

Eq.(7.3)

is not the total kinetic energy, but an integral of another xpres ion of the energy density ^-

w

'l/J/2 over regions of coherent vortices, which is a quantity corre ponding to the Hamiltonian of the point vortices. So they concluded from the theoretical and numerical investigation that the integral of

-w'l/J

/2 over the coherent vortex region is the Hamiltonian of the collective system of coherent vortices, which is a constant of motion. In th same way, the integral of

w2

/2 over the coherent vortex region is the intrinsic enstrophy related to the coherent vortices in the scaling theory. It is shown from theoretical and numerical investigation that the temporal scaling behavior of the total enstrophy should be generally different from that of the integral of

w2

/2 over the coherent vortices region.

The discussion of conservation quantities related to the coherent vortex region is crucial to construct the statistical theory for determining the scaling exponent �· The attempt to the theoretical determination of the scaling exponent� will be discussed in Sec.7.4.

54 7 PHENOMENOLOGY OF COLLECTIVE MOTION OF COHERENT VORTICES

7.2 Conservation properties in the coherent vortex region

For the decaying CHM turbulence, it was discussed in Sec.5.3 that the coherent vortices self­

organize from a random initial condition, develops into larger vortices through the mutual advection and merging among vortices. However, the dynamics is quite different from that in the decaying 2D-NS turbulence, i.e. ,\ ⁼ 0. In this section, we discuss how the phenomenology of the collective motion of the coherent vortices in the decaying 2D-NS turbulence can be applied to the decaying CHM turbulence. In addition, we discuss its unique statistical properties different from the decaying 2D-NS turbulence.

First, in order to extract the coherent vortex region from the potential vorticity field in Fig.5.4, we define the coherent vortex regions as that where the following two conditions are simultaneously satisfied : (i) the Gaussian curvature Q of¢, defined as

Q =

(�)2- (82¢) (82¢)

8x8y 8x2 8y2 '

^(7.8)

is negative and (ii) the absolute value of the potential vorticity

JqJ

is larger than

(JqJ)

which is the average value of

JqJ

taken over the total area of the system, i.e.

(JqJ)

⁼ (1/

£2) JL2 JqJ

dr.

The reason for the definition of the above conditions is as follows. The coherent region should have the stable structure and Q is a standard measure of the stability of the Lagrangian particle [46, 47, 53]. In addition, we expect that the coherent vortex region has a large absolute value of the potential vorticity. We write the area of coherent vortex regions as Scv.

Figure 7.1 shows coherent vortex regions defined with the above conditions at t ⁼ 10 and 100, respectively. Comparing Fig. 7.1 with the potential vorticity field in Fig.5.4, we can rec­

ognize that the regions of the axis-symmetrical vortices are extracted. The temporal evolution of the amount of area of the coherent vortex region, normalized by the total area, is shown in Fig.7.2. It seems that the area is almost constant with time after forming coherent vortices and has about 25 percent for the initial condition used in this simulation. This result indicates that the region of coherent vortices is approximately conserved with time and that is an important characteristic of coh rent vortices in the decaying CHM turbulence.

Second, we consider the behavior of the energy and the potential enstrophy for coherent vortices defined above. As mentioned in the previous section, the integral values of the two expr ssions of the energy density,

('V ¢ )2

/2 and -

¢

'\7

2¢

/2, over the coherent region are different from each other. By inferring from the Hamiltonian dynamics of the point vortex system, the latter is the Hamiltonian for coherent vortices and is the fundamental conserved quantity in the decaying 2D-NS turbulence [53]. In the CHM system, it is expected from the analogy to the 2D-NS system that the integral value of -¢q/2 over the coherent region is the Hamiltonian for the coherent region. In order to check this point for the coherent vortices obtained in the CHM turbulence, we show in Fig.7.3 (a) the time evolutions of the total energy E, the two

7.2 Conservation properties in the coherent vortex region

(a)

55

(b)

Figure 7.1: The regions of coherent vortices at (a) t ⁼ 10 and (b) t ⁼ 100 corresponding to the Fig. 5.4.

I I I I

0.8

N 0.6

�

(.)

V"J

^0.4

�

0.2

0 ^{I I I I}

0 20 40

t

^{60 80 100}

Figure 7.2: The temporal evolution of the ratio of the amount of area Scv of the coherent region to the total area

£2.

56 7 PHENOMENOLOGY OF COLLECTIVE MOTION OF COHERENT VORTICES

different expression of energies

E1

and

E2,

defined as

1

1 (

¹

)

¹

1

¹

^{[ 2 2 2} ]

E1

= 2

- ^-¢

q dr = 2

- -¢\1 ¢

+ >..

¢ L Scv

²

^L Scv

²

1

r

¹

[ ^{2 2 2} ]

E2

=

£2 J s cv 2 ^(\1¢)

^+)..

^¢

^dr.

dr, (7.9)

(7.10)

Note that if the integrands in Eqs. (7.9) and (7.10) are integrated over the total area, they coincide with

E.

The quantities

E1

and

E2

temporally evolve in a different way from each other. However, the difference between

E1

and

E2

decreases with time and they approach an almo t constant value. This result originates from the difference between the integral values of

-¢\12¢/2

and

(\1¢)2 /

2 in Eqs.(7.9) and (7.10). However for ).. >> k, these quantities are very small in comparison with the integral value of

>..2¢2 /

2 over the coherent region. Therefore

E1

and

E2

remain almost same after remarkable change of

E1

and

E2

finished [

E d E

= 0.67,

E2/ E

= 0.64 at ^t = 100]. By taking into consideration that the temporal behaviors of

E

and

E1

are almost same and

E1

occupies the most of

E,

and therefore

E1

can be regarded as the characteristi nergy of coherent vortices. In other words, since the energy

E1

concerning the coherent region is clearly a large portion of

E

in the whole time region in spite of the area of oh r nt region being small,

Scv / L2

= 0.25 ,..., 0.3, the system behavior is dominated by the dynamic of th coherent vortices. As will be seen later the difference between

-¢\12¢/2

and

(\1 ¢ )2 /

2 remarkably appears in the time evolution of the potential enstrophy rather than that of th energy.

Figure 7.3

(b)

shows the temporal evolution of the total potential enstrophy

U,

the two exprcs ions of th potential enstrophy ul and

u2

defined as

dr, (7.11)

(7.12)

Not that if th integrands in Eqs.(7.11) and (7.12) are integrated over the total area, they coincide with

U.

Figur 7.3 (b) clearly shows that

U1

occupies the large amount of

U [UJ/U

= 0.81 at ^t = 100] and the temporal evolution of

U1

obeys almost the same power law as

U

, which is ^e timated as Ut

"" t-O.&.

On the other hand,

U2

has a very small value in comparison with

U

or

U1 [U2/U

= 0.33 at

t

₌ 100] and decreases with time faster than

U

or

U1

as

U2 rv ^t-0·7.

Thcs result originate from the difference between

->..2¢\12¢/2

and

>..2(\1¢)2 /

2 in Eqs.(7.11) and (7.12). Particularly, in the c e of>..>> k, the difference is remarkable. The difference is equivalent to that of th first terms of the integrands in Eqs.(7.9) and (7.10). According to the numerical re ult ,

U1

can be regarded as a characteristic potential enstrophy for coherent vortices, becau

U1

has the same order of magnitude

U

and the temporal behavior same as that of

U.

Consequently,

E1

and

U1

are different respectively from

�

and

u2,

and are, as will

7.3 Phenomenological scaling theory of coherent vortices 57

be shown in the next sub-section, connected with th vortex dynamics. Hereafter we write

E1

and

U1

as

Ecv

and

Ucv

respectively.

7.3 Phenomenological scaling theory of coherent vortices

In this section, by considering the analogy with the scaling theory developed in the second tage of the freely decaying 2D-NS turbulence [51], we propose a new scaling theory which de crib the time evolution of average quantities related to coherent vortices for the decaying CHM turbulence. Our standing point is that coherent vortices self-organized from a random initial condition dominate the dynamics of the system, and that the energy

E v

and the potential enstrophy

Ucv

related to the coherent vortices are therefore expre sed in term of characteri ti quantities of vortices.

First, we assume that the potential vorticity q is approximately written as

(7.13)

for k << >... The extent of the approximation Eq.(7.13) is measurable by comparing the time

evolution of the following three quantities with each other:

1

r 2

no=-

J. ,

^{q dr,}

Scv Scv

1

r 2 2

nl ⁼

s

^CV

J. Scv , ^{(\1 ¢)}

^dr,

1

r 2 2

n2 =

- ^Scv J. Scv ,

⁽

^-

^)..

^¢)

^{dr. (7.14)}

Numerical results are shown in Fig.7.4.

f!1

is larger than

02

in the early stage, and decreas with time as

01 ""t-1.

On the other hand, n2, gradually increasing, approache 00 and even­

tually becomes constant. These results support the asymptotic behavior Eq.(7.13). Moreover we can conclude that the average value of

¢2

in the coherent region is conserved.

By employing Eq.(7.13),

Ecv

and

Ucv

can be expressed in terms of the total number N of coherent vortices, their average radius ra and the average potential vorticity qa of vortex centers as

(7.15)

(7.16)

If

Ecv

and qa are regarded as conserved quantities of the system in the high Reynolds number limit, Eq. (7.15) yields

(7.17)

Equation (7.17) clearly indicates that the area of the coherent region is constant with time.

This is consistent with the numerical results shown in Fig.7.2.

If the total number of vortices N decreases algebraically with time as

N(t) "' t-X,

(7.18)

58 7 PHENOMENOLOGY OF COLLECTIVE MOTION OF COHERENT VORTICES

0.5 ,....---.---, 0.4

0.3

0

0.2 •• . .. -0.05

-···

••

....

0 o o oo

0.1

(a)

100 101 102

t

103

u ⁰ u1 ^• u2 ⁰ 0

N • -1/2

�

" 102

� 0

�

^{' ' ' 0 0}

" '

�

^{' ' '}

101

(b)

100 10' 102

t

Figure 7.3: The t mporal evolutions of (a) E, E1, and E2, (b)

U, U1,

and

U2

defined in the Sec. 7.1. The slope -0.05 in (a) indicates that the energy slightly decreases with time to the finite Reynolds number effect. The slope -1/2 in (b) is the theoretical value derived in Eq.

(6.16) and in Sec.7.3 and the value -0.55 may be due to the effect of finite Reynolds number.

7.3 Phenomenological scaling theory of coherent vortices

104

0 oo

N 0 0 oo

�

¹⁰³ •

... •

�

�

^{• •}

d

₁₀₂

�

t

59

.ao

⁰ .Q1 ^• .Q2 ⁰

Figure 7.4: The time evolution of

no, n1,

and

n2

defined in Eqs.(7.14). After time sufficiently goes on,

no� n2 "'t0

and

no

>>

n1.

we can derive the scaling laws for

ra(t)

and

Ucv(t)

from Eqs.(7.15) and (7.16) as

(7.19)

(7.20)

Moreover, the average distance

la

between vortices with same sign of circulation and the average circulation par vortex are evaluated respectively as

l a t ( )

^"-'

y'N(t} 1

""

t ^x./2 ,

(7.21)

Namely, the exponents of dynamical scaling of the quantities related to coherent vortices are expressed only by X·

Next, we check the prediction of the scaling theory by investigating the scaling behavior obtained in the numerical simulation. In order to check the validity of the assumption Eq.(7.18), we must count the number of coherent vortices in each time step. However it is so complicated to identify ' coherent' region for each vortex. For example, the ' vortex census' carried out by McWilliams in (45] contains several arbitrary criteria. In order to avoid the complexity

60 7 PHENOMENOLOGY OF COLLECTIVE MOTION OF COHERENT VORTICES

Figure 7.5: The extremum positions of the

lql

evaluated in the potential vorticity field at t = 100. The solid (dashed) contour line represents positive (negative) values of the potential vorticity.

for th det rmination of the vortex number, we simply count the number N of extrema of the pot ntial vorticity field in the coherent vortex regions defined in the previous section. The xample of the vortex extrema evaluated in the coherent region is shown in Fig. 7.5. Moreover we calculate the average value of the potential vorticity

qa

over the positions of vortex extrema.

The scaling behaviors of these quantities are shown in Fig.7.6. We can observe the scaling behavior of t mporal evolution of the number N and the conservation of qa. The area

Scv

of coher nt vortex regions is also conserved as predicted in the scaling theory. These results clearly indicate the validity of the scaling theory.

In Sec.6.3, we found that th temporal evolution of the characteristic wavenumber

k(t)

corresponding to the peak position of the power spectrum of

q

obeys the scaling

k(t)

rv

E-l/B

>..314t-114.

If the inverse of

k(t)

is regarded as the average distance

la

among vortices with same ign of circulation, the scaling law of

k(t)

yields the scaling law of

la(t)

_as

) 1 1/4

la(t

rv

k(t) "-' t .

_(7.22)

7.3 Phenomenological scaling theory of coherent vortices

1000

100

10

1

0.1

�lila -1/4

+ + + + + +++t I 111111111111111111111111111111111111111111111111111

S

cv

fL

2

0 0 o o oooocx:xc:ooXJorn• ·•oowi!Bl•m _.. mwmwm" ---•

E

cv

10

t

¹⁰⁰

61

Figure 7.6: The temporal evolutions of the quantities related to the coherent vortices. N is the total number of the extrema of the potential vorticity field within the coherent vortex region and

qa

is the average value of the potential vorticity at extremum. Here

Ecv

⁼

E1, Ucv

⁼

U1, 1/la

=

k(t),

and

Scv

is the area of coherent vortex region.

By combining Eqs.(7.21) and (7.22), the scaling exponent is thus determined as

x-­-2'

1

(7.23)

Consequently, the dynamical scaling of the other quantities related to the vortices are evaluated as

(7.24)

In the numerical simulation, the power law of

Ucv ( t)

can be easily compared with the above result. As shown in Fig.7.3 (b) or Fig.7.6, the scaling

Ucv(t)

rv

t-112

is in agreement with the numerical result, although the theoretical exponent is somewhat smaller than the experimental value. This deviation is discussed by considering the effect of the finite Reynolds number

62 7 PHENOMENOLOGY OF COLLECTIVE MOTION OF COHERENT VORTICES

on the scaling exponent evaluated from the scaling theory. The scaling theory of the energy spectrum in the wavenumber space and that of quantities related to coherent vortices in the physical space are valid in the limit of infinite Reynolds number. The energy is not conserved completely and slightly decreases with time on account of the finite Reynolds number effect.

From Fig.7.3 (a), we suppose that the temporal decrease of

E

_and

Ecv (

=

El)

behaves as (7.25)

although the area of the coherent region is conserved. Substituting Eq.(7.25) into Eqs.(6.11) and (6.12), we obtain the correction of the scaling law as

(7.26) (7.27)

Moreover, by substituting

Qa2 ""c0,

which is the result from

Ecv ""t-0

and

Nra2 ""t0,

into Eq.(7.16) yield

Ucv(t) ""t-(1+20)/2.

(7.28)

If w e timate () � 0.05 from Fig.7.3(a), Eqs.(7.26), (7.27) and (7.28) are evaluated as,

(7.29)

A hown in Figs.6.3(a) and 7.3(b), the correction of the scaling exponents for

Smax(t)

and

Ucv (

=

U1 ( t))

are in good agreement with the results of our simulation. The extent of correction of th s aling exponent for

k(t)

is about 2.5 percent in comparison with the original one, i.e., th caling xponent does not change so much. Consequently, it seems that the correction of the caling exponents for

N

or r a becomes small.

The dynamical scaling law of the potential enstrophy

Ucv

^rv

t-1/2

is consistent with Eq.(6.16) which is the scaling theory in the decaying CHM turbulence for the total poten­

tial enstrophy b ing analogous with that of decaying 2D-NS turbulence proposed by Batchelor [13]. In short, we should notic that the temporal scaling of

Ucv(t)

is almost same as that of the total potential en trophy

U(t).

This point will be quite different from that in the decaying 2D-NS turbulen . Moreover the scaling laws in the physical space coincide with those in the wavenumber pace. This is the most interesting point of the CHM turbulence in comparison with th 2D-NS turbulence.

Here, we discuss th implication that

ra(t)

and

la(t)

have the same scaling exponents. In the physical space, coherent vortice with arne sign of circulation develop into larger ones through the vortex merging, con erving the area of coherent region. The fact that the growing laws of

ra(t)

and

la(t)

are same in thi coagulation process indicates that the potential vorticity field q temporally develop in a elf-similar way. Figure 7.7 shows the contour plot of the potential

7.4 Phenomenological determination of the scaling exponent 63

vorticity field at

t

⁼ 20 and

t

⁼ 100. The figure fort ⁼ 20 is the magnification of a portion of the total potential vorticity. The ratio of magnification is defin d by the ratio of the characteristic scale k(100)/k(20) from the data in Fig.6.3 (a). From this figure, we clearly recognize that the potential vorticity field develops with time in self- imilar way. Consequ ntly we conclude that this is the reason of the existence of the dynamical scaling law in the wavenumber space. On the other hand, in the scaling theory for coherent vortices in the 2D decaying NS turbulence, the scaling laws

ofra(t)

and

la(t)

are represented in Eqs.(7.6) and (7.7) as

ra(t)

^rv

t{/4, la(t)

^rv

t{/2.

This implies that coherent vortices gradually become thinner as time passe becau e the growth rate of

la(t)

is larger than that of

ra(t),

which is the essential difference of the colle tive motion of coherent vortices between the 2D-NS and CHM turbulence.

7.4 Phenomenological determination of the scaling exponent

In this section, we discuss the theoretical determination of the scaling exponent characterizing the scaling theory by considering the advection velocity of coherent vortice as the Hamiltonian point vortex advection mentioned in Sec.7.1.

First, we review the case of the decaying 2D-NS turbulence [54). The main point of the theory is the estimation of the scaling of the average advection velocity

Ua

in th vortex center.

Carnevale et al. [51) estimated

Ua

^rv r

a/ la

^rv

t0

and concluded that this scaling is elf- onsistent supposing the total kinetic energy

E

rv ui is conserved. However this estimation i invalid from the following points: 1) The advection velocity of coherent vortices decreases with time as developing for larger vortices, and 2)

Ua "" t0

obtained by Carnevale et al. is based on the assumption of the conservation of the total kinetic energy. However the representation of th energy

Nwir!

is not the total kinetic energy as reviewed in the Sec.7.1. Other r pre entation of the advection velocity of the coherent vortices should be considered by the point vortex model because the coherent vortices move like point vortices in the second stage [46, 47]. Therefore, the velocity

Ua

can be evaluated with the advection velocity of point vortex model (Eq.(7.1)) as

(7.30)

where the Hamiltonian H corresponds to the conservation quantity

N wir!

for coherent vortices.

By using Eq.(7. 7) and noting H""

t0,

Eq.(7.30) yields

(7.31)

This result clearly indicates that

Ua.

decreases with time, which is consistent with the picture of numerical result. On the other hand, it is known that the motion of point vortices is chaotic provided that the system includes more than four vortices. Since there are many coherent vortices in the second stage of the decaying 2D-NS turbulence, the mixing property due to the

64 7 PHENOMENOLOGY OF COLLECTIVE MOTION OF COHERENT VORTICES

Figure 7.7: Th contour plot of the potential vorticity field at

t

= 20 (upper) and

t

= 100 (lower) respectively. Po itive ontour are indicated by solid lines, and negative contours by dashed line . The z ro level contour is omitted. One should notice the difference of the scale between upp r and lower figure.

7.4 Phenomenological determination of the scaling exponent 65

chaotic motion of vortices must realize the isotropic flow. This nature leads to the fact that

Ua

due to the advection by vortices is of the same order as the relative velocity

dla/ dt

among vortices, which yields the scaling law

dla �/2-1 Ua f",J -;}j ""t .

Combining Eq.(7.31) and Eq.(7.32) lead to

(7.32)

(7.33)

This is in good agreement with the numerical and experimental results as 0. 7

""

0. 75. Thus we obtain

ra f",J t1/6, Q f",J t-l/3, la "" t1/3, fa ""t1/3

and

Ua

rv

t-2/3.

The analogy of the above-mentioned picture can be applied to the coherent vortices in the decaying CHM turbulence. The advection velocity

Ua

of the vortex center is estimated by Eq.

(7.30), where H is the Hamiltonian for the coherent vortices in the decaying CHM turbul nee.

Because we have obtained that

Nra2

must be conserved actually a compared with concerning

Nra

4 for the Hamiltonian H, we assume that Ecv is the Hamiltonian H in the CHM turbulence, whose value is evaluated asH""

Nra2>.-2qa2.

Substituting

fa"" tX

and

la"" tXI2

into Eq.(7.30) leads the scaling law

(7.34)

This result indicates that the advection velocity of vortices slows down with tim Mor ov r, if we assume that the advection velocity is evaluated as the time variation of the distance among vortices (the relative velocity of vortices), then we find

dla x/2-1 Ua "" dt rv t .

By combining Eq.(7.34) and Eq.(7.35), the scaling exponent turns out be x=-1 2 .

(7.35)

(7.36)

This is perfectly in agreement with the value (Eq.(7.23)) obtained in the scaling theory of the inverse energy cascade in the wavenumber space. Thus we can notice that the value = 1/2 obtained in the phenomenology of collective motion of coherent vortices quite coincide with that obtained by the Kolmogorov-Batchelor like scaling theory for the inverse energy cascade in the wavenumber space. This nature is an interesting problem because the developing process of coherent vortices corresponds to the inverse energy cascade process not only in the dynamical sense but also in the statistical sense. In addition, we must notice that thi nature is not observed in the decaying 2D-NS turbulence, i.e. ).. -+ 0. The characteristic wavenumber

k(t)

in the decaying 2D-NS turbulence is dimensionally represented by the total energy E and the time

t

under the assumption that E is conserved in time as

(7.37)

66 7 PHENOMENOLOGY OF COLLECTIVE MOTION OF COHERENT VORTICES

If the inverse of

k(t)

is estimated as the mean distance among vortices La, we get the scaling exponent�= 2. However this is not in agreement with the numerical and experimental results

as � = 0.7 "' 0.75 or theoretical result � = 2/3. The failure of this estimation will originate

from the fact that the scale

k(t)-1

is not estimated by the scale La. The physical meaning of the characteristic scale estimated by Eq.(7.37) seems not to be clear in the decaying 2D-NS turbulence. This point will be discussed elsewhere.

Comparing the values of the scaling exponent x = 1/2 with�= 2/3, we can notice that the decreasing rate of the number of coherent vortices in the CHM turbulence is slow in comparison with that in the 2D-NS turbulence. This nature is fully consistent with the numerical results, see Fig .5.4 and 5.5. Shielding effect of mutual advection with

O(A

^-1

)

distance among vortices plays an e sential role in the CHM turbulence, which causes the difference of the theoretical determinations of scaling exponents for two turbulences.

67

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