- Email: [email protected]
On helicity and shell models
and oceans ELSEVIER
Dynamics of Atmospheres and Oceans 27 (1997) 707-714
On helicity and shell models A. W i i n - N i e l s e n Department of Geophysics, Niels Bohr Institute of Astronomy, Physics and Geophysics, Juliane Maries Vej 30, 2100 Copenhagen, Denmark Received 21 December 1995; revised 17 June 1996; accepted 29 July 1996
Abstract Helicity is a conserved quantity in an integrated sense for the unforced, non-dissipative atmospheric equations. Its use in one-dimensional shell models is discussed, and it is recommended that its equivalent should not be used in shell models as it changes sign from one shell to the neighbouring shell. A shell model conserving kinetic energy and the mean value of the velocity is designed. Integrations show that if the model starts from a velocity slope of - I (corresponding to a kinetic energy slope of - ~), it will deviate a little from this slope during the integration. The model shows intermittency on the smallest scale. © 1997 Elsevier Science B.V.
1. Introduction Recently, a new interest in the field of turbulence has appeared. The interest comes mostly from what we might call numerical physics. Away from boundaries, turbulence appears to be a truly three-dimensional phenomenon. Numerical simulations of three-dimensional turbulence have been carried out by Kerr (1985, 1990) and by Vincent and Meneguzzi (1991) by an integration of the Navier-Stokes equations in a box. The theoretical framework originates with Kolmogorov (1941) and the refinements of the original ideas by the same worker (Kolmogorov, 1962). Almost all research deals with homogeneous turbulence, and many researchers use the so-called shell models to reduce a three-dimensional problem to one of only one dimension. This is done by working in wavenumber space and disregarding all references to the phase, and dealing only with the modulus of the wavenumber vector. In addition, the wavenumber space is subdivided in a finite number of shells of increasing thickness. A given shell contains all moduli between a lower and an upper limit. With k o being a scaling wavenumber, the lower and upper limits of shell number n are koA n and k o An+ 1, where h is to be specified and in many cases has the value two (see, e.g. Jensen et al., 1991). Shell n is 0377-0265/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 0 2 6 5 ( 9 7 ) 0 0 0 3 9 - 0
708
A. Wiin-Nielsen/ Dynamics of Atmospheres and Oceans 27 (1997) 707-714
characterized by the single velocity u(n), which is considered as an averaged velocity over the shell. A major problem in shell models is to specify the nonlinear interactions between a given shell and the others. It is well known that a given wavenumber interacts with all pairs of wavenumbers whose sum or difference equals the given wavenumber (Wiin-Nielsen, 1979). Owing to the design of a shell model, it appears impossible to simulate this general rule, and it is replaced by specifying the interactions in an ad hoc fashion. Typical specifications are that a given shell shall interact only with the immediate neighbouring shells. The interaction may involve three or five shells. The coefficients in the interaction terms are found by requiring that integral quantities such as the total kinetic energy shall be conserved in the unforced, non-viscous limit. Although one of the coefficients may be assumed to be unity, because one may rescale the time to obtain this result, it requires normally more than one conservative quantity to determine the coefficients in a unique way. In this regard, the so-called helicity has been invoked (Kadanoff et al., 1995). Without any forces and without dissipation it can be shown that a scalar quantity, being the scalar product of the three-dimensional velocity and the three-dimensional vorticity vector, is conserved provided that the kinetic energy vanishes at the physical boundaries of the fluid or the gas. As helicity is an unfamiliar quantity in the meteorological and oceanographic communities, an elementary demonstration of the conservation of helicity in the general three-dimensional case will be provided (Section 2). Thereafter we shall discuss the use of helicity in shell models. We shall also make use of the fact that the mean velocity is a conserved quantity, as it must be in the unforced, non-viscous case. We shall demonstrate that it is possible to define a shell model in a unique way by using the conservation of the mean velocity and the kinetic energy. This model will be integrated numerically (Section 3). lntermittency is considered to be an essential part of turbulence (Ohkitani and Yamada, 1989; Eggers and Grossman, 1991; Jensen et al., 1991), and it is connected with chaos and multi-fractal structures (Meneveau and Sreenivasan, 1991; Jensen, 1994). We shall also touch on these matters in the paper.
2. Helicity A quantity called the helicity, adapted for use in a one-dimensional model, has been used as a conservative quantity in shell models (Kadanoff et al., 1995). Helicity is in general defined in the following way: he~-V. ~
(1)
Helicity was discussed by Moffatt (1969), who gave a very general discussion of the concept and examples of cases in which helicity is conserved and other cases where the conservation fails. Our interest is essentially in the atmosphere, and it may be appropriate to provide an elementary demonstration of conservation of helicity in an integrated sense, although the proof actually is a part of Moffatt's paper. As usual, we shall use u,
A. Wiin-Nielsen/ Dynamics of Atmospheres and Oceans 27 (1997) 707-714
709
v and w as the three components of the velocity. We recall also that the three components of the vorticity vector are ~1 = ~'2 =
Ow
av
ay au
az Ow
az av
ax au ay
if3 ~--- aX
(2)
In geophysical fluid dynamics, most considerations are based on the vertical component of the vorticity vector, leading in barotropic, two-dimensional flow to the conservation of enstrophy (i.e. the integral over the total area of half the vorticity squared) and the conservation of potential enstrophy in baroclinic quasi-nondivergent flow. It is these conservation laws combined with the conservation of kinetic energy that resulted in the so-called two-dimensional turbulence theory as developed by Kraichnan (1967) and Leith (1967) for the barotropic case. The baroclinic, quasi-geostrophic case can be treated in the same way using the conservation of potential enstrophy combined with the conservation of the sum of available potential energy and kinetic energy. These considerations imply a spectral slope of - 3 for large values of the wavenumber for both the kinetic and the available potential energy, and these facts have implications for the limited predictability of the atmosphere as discussed, for example, by Wiin-Nielsen (1991). Here we shall use the total vorticity vector. It is assumed that the fluid is homogeneous such that the three-dimensional divergence vanishes. By cross differentiation of the Navier-Stokes equations excluding all forces we obtain three vorticity equations that may be written in the following form: -- +v.V~l-Vu.i;=O at aft2 at
--
+ v . V~2 - V v .
~ = 0
(3)
a~"3 + v . V ~ 3 - X7w.i;= 0 at The conservation theorem says then that d
I= - ~ t f v ' ~ d V =
-~t-t " ~'+ v • a~']dV= 0 (4) at 1 By using the component equations and the fact that the three-dimensional divergence is zero one may write the above integral in the form I = - f f l v - ( ~ , u v ) + V" ( ~2vv) + V" ( ¢3wv) - Vk" ~ ]dV = J~f,[v" (k~') - kV. (V x v ) ] d V = O, k = 1 / 2 v . v
(5)
710
A. Wiin-Nielsen / Dynamics of Atmospheres and Oceans 27 (1997) 707- 714
It is thus seen that 1 = 0 if the kinetic energy goes to zero at the boundary of V, as the last term in I is identically zero. The helicity is thus conserved if this condition is satisfied. The helicity is a rather peculiar conserved quantity because it is not always positive as, for example, is the kinetic energy. In addition, for the large-scale, quasi-nondivergent flow in the atmosphere, where the vertical velocity is about three orders of magnitude smaller than the horizontal velocity, the helicity reduces to a quantity proportional to the temperature advection, which can be seen by using the thermal wind equation h:
g ---r.V(InT)
(6)
L, and it is thus seen that helicity vanishes when integrated over the globe. This argument does not apply for the small-scale flow. The definition of helicity in a shell model is far from obvious. It turns out that one has to be satisfied by a quantity that is of the correct dimension. By definition it becomes N
/4,, = E
(7)
n=l
where N is the number of shells as derived by Ditlevsen and Mogensen (1996). This definition is rather unfortunate because it means that the local contribution to helicity changes sign from shell to shell. In other words, the helicity will cause a production also in the dissipation range. It appears therefore to be a more sound approach to avoid helicity and use the conservation theorems of kinetic energy and mean velocity to formulate the shell model. This will be done in the next section.
3. A n e w shell m o d e l
We shall formulate a shell model based on the conservation of the mean velocity and the kinetic energy. When we consider the one-dimensional advection equation without forcing and dissipation, it is easy to see that the mean value is conserved by integrating over the total length with periodic boundary conditions as we would when we consider the advection equation along a given latitude circle. The same result is obtained when the dependent variable is written as a Fourier series where the equation is transformed to a finite number of ordinary equations for the time-dependent Fourier coefficients. In a shell model we make a local approximation to the advection term by restricting the nonlinear interactions to the nearest neighbour shells. It is then natural to formulate these approximations in such a way that the mean value is conserved. It turns out that this is an impossible task if we restrict the interactions to the immediate two neighbouring shells. We try therefore to obtain the required result with
A. Wiin-Nielsen / Dynamics of Atmospheres and Oceans 27 (1997) 707-714
711
five shells as in the GOY (Gledzer-Ohkitani-Yamada) model, but of course with a different choice of interaction terms. The GOY model does not conserve the mean value. This is due to the fact that it includes the interaction involving the terms u n_ ~ and u~ + 1 as the only term where the counter increases by two. Therefore, such a term can never cancel. The following model is proposed: du n dt = ak"U~+ lUn+ 2 + b k " - 3u"- 2u"- I + c k , _ lU, Un+ n + d k , _ 2u ~_ l u . 2 + e k . u .2+ I _{._fkn _ 2u._n
(8)
with the energy equation becoming du2./2 dt
a k . u . u . + lU.+ z + bkn_ 3Un_ 2Un_ lUn + c k . _ lU2.U.+ l + d k . _ 2u . _ llA 2 2
+ ek.u.u]+, + fk._2u._,u .
(9)
The next task is to determine the six constants in such a way that the kinetic energy and the mean value of the velocity are conserved when summed over the total number of shells. Invoking the two conservative quantities we obtain the following equations for the constants: a+b+c+d=O e+f=0 a + b/2 = 0
(lO)
c+f=0 d/2 + e = 0 With a -- 1 obtained by rescaling the time, we obtain b = - 2 , c = - 1, d = 2, e = - 1 and f = 1. The next question is then to determine the fixed-points which may be present in the model. It is assumed that the solution is of the form u[n] ~ k[n] -~. When this is inserted in the steady-state equation and setting z = 2 -~ we obtain a sixth-degree I expression. The roots are determined, and we find the following real values of a : 0, 1 and i. A stability analysis of each of these steady-state solutions shows that all of them are unstable. As in the GOY model, it is also found that only a few of the eigenvalues are numerically large whereas the rest are very small, although all are positive in the 1 proposed model as shown in Fig. 1 for a = 7In the following, we shall refer to this model as the MEKIN model, as it conserves the mean velocity and the kinetic energy in the unforced, non-viscous case.
4. Integrations of the MEKIN model To compare the model with the GOY model, the integrations of the two models have been done in essentially the same way. We present the results using time averages for
A. Wiin-Nielsen/ Dynamics of Atmospheres and Oceans 27 (1997) 707-714
712
Stability 3500
i
i
0
0
of
I
0
0
fixpoint
r
0
with
i
0
">
&
slope
-1/3
t
0
O
3000
i
i
O O
O
O O
2500 D > O1
2000
{1} o
1500
o i000
I
I
I
1
I
I
I
I
l
2
4
6
8
10 [!
12
14
16
18
20
Fig. 1. The cumulative sum for the real eigeuvatues for the fixp~int - ~. The plotted value s[n] is the sum of the eigenvalues from unity to n. MEKIN-mode], -8
R:I0~*6,
init,
i
7 - -
i
and
final i
slopes:
i/3,
0.38
I
i
© © -i0
o -12
--o_
-
o,
-14 v -16
-18
-20
-22
l
-18
t
-16
t
-14
Fig. 2. A d o u b l e l o g a r i t h m i c ~ o t o f the t i m e - a v e r a g e d v a l u e s o f final slope
-0.38;
Reynolds
number
106
.
~
-12 -i0 ln(k{n])
u[n] as
t
l
-8
6
a function of
k[n].
-
Initial slope - ~:
A. Wiin-Nielsen/ Dynamics of Atmospheres and Oceans 27 (1997) 707-714 M e a n vel. 0.06
i
i
& kin. i
en. l
cons., i
713
abs(u[18]) i
i
0.05
0.04
0.03
..Q 0.02
0.01
I
i000
2000
3000
4000
5000 6000 time, sec
7000
8000
9000
10000
Fig. 3. The absolute value of the component u[18] as a function of time. the various components of the velocity. In addition, we present the results in the form of diagrams where we use the logarithmic values for the wavenumbers and the velocity components. 1 Fig. 2 shows a case where the starting distribution is given by a slope of - 7. After an integration of 10000 time steps corresponding to about 42h, and with a Reynolds number of 1013, we obtain a slope of - 0 . 3 8 , which happens to be the same as obtained with the GOY model (Jensen, 1994). Starting from an initial slope of - 0 . 5 , which also represents an unstable fixpoint, it was found that the slope does not change on average. Fig. 3, in which we display the time series of a single component, u[18], shows that the intermittency appears for the wavenumbers at the upper end of the shell model MEKIN, which in this integration, as in the others, was restricted to 19 shells.
5. Conclusions
The MEKIN model is capable of giving a slope comparable with that for the GOY I model if one starts from an initial slope of the velocity components of - ~ corresponding to a kinetic energy slope of - 75 The resulting slope ( - 0 . 3 8 ) after many integrations deviates somewhat from the initial slope. We stress that we are simulating the three-dimensional case. If we were to simulate two-dimensional turbulence we would have to determine the coefficients in such a way that kinetic energy and enstrophy are conserved. This has been done by Ditlevsen and Mogensen (1996).
714
A. Wiin-Nielsen / Dynamics of Atmospheres and Oceans 27 (1997) 707-714
References Ditlevsen, P.D. and Mogensen, I.A., 1996. Cascades and statistical equilibrium in shell models of turbulence. Phys. Rev. E, 53(5): 4785. Eggers, J. and Grossman, S., 1991. Does deterministic chaos imply intermittency in fully developed turbulence? Phys. Fluids, 3: 1958. Jensen, M.H.. 1994. Multifractals and multiscaling. Doctoral Thesis, University of Copenhagen, 177 pp. Jensen, M.H., Paladin, G. and Vulpiani, A., 1991. Intermittency in a cascade model for three dimensional turbulence. Phys. Rev. A, 43: 798. Kadanoff, L., Lohse, D., Wang, J. and Benzi, R., 1995. Scaling and dissipation in the GOY shell model. Phys. Fluids. 7: 617. Kerr, R., 1985. Higher-order derivative correlations and the alignment of small scale structures in isotropic turbulence. J. Fluid Mech., 153: 31. Kerr, R., 1990. Velocity, scalar and transler spectra in numerical turbulence. J. Fluid Mech., 211: 309. Kolmogorov, A.N., 1941. Local structure of turbulence in an incompressible viscous fluid at very large Reynolds numbers. C. R. Acad. Sci. URSS, 30: 299. Kolmogorov, A.N., 1962, A refinement of previous hypothesis concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds numbers. J. Fluid Mech., 13: 82. Kraichnan, R.H., 1967. Inertial subranges in two-dimensional turbulence. Phys. Fluids, 10: 1417-1423. Leith, C.E., 1967. Diffusion approximation for two-dimensional turbulence. Phys. Fluids, 1t: 617-673. Meneveau, C. and Sreenivasan, K.R., 1991. The multifractal nature of turbulent energy dissipation. J. Fluid Mech., 224: 429. Moffatt, H.K., 1969. The degree of knottedness of tangled vortex lines. J. Fluid Mech., 35:117-129. Ohkitani, K. and Yamada, M., 1989. Temporal intermittency in the energy cascade process and the local Lyapunov analysis in fully developed model turbulence. Prog. Theor. Phys., 81: 329. Vincent, A. and Meneguzzi, M., 1991. The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech., 225: I. Wiin-Nielsen, A., 1979. On the asymptotic behavior of a simple stochastic-dynamical system. Geophys. Astropbys. Fluid Dyn., 12: 295. Wiin-Nielsen, A., 1991. On resolution in atmospheric numerical weather prediction models and cascade processes. Atmosfera, 4: 3-22.
Dynamics of Atmospheres and Oceans 27 (1997) 707-714
On helicity and shell models A. W i i n - N i e l s e n Department of Geophysics, Niels Bohr Institute of Astronomy, Physics and Geophysics, Juliane Maries Vej 30, 2100 Copenhagen, Denmark Received 21 December 1995; revised 17 June 1996; accepted 29 July 1996
Abstract Helicity is a conserved quantity in an integrated sense for the unforced, non-dissipative atmospheric equations. Its use in one-dimensional shell models is discussed, and it is recommended that its equivalent should not be used in shell models as it changes sign from one shell to the neighbouring shell. A shell model conserving kinetic energy and the mean value of the velocity is designed. Integrations show that if the model starts from a velocity slope of - I (corresponding to a kinetic energy slope of - ~), it will deviate a little from this slope during the integration. The model shows intermittency on the smallest scale. © 1997 Elsevier Science B.V.
1. Introduction Recently, a new interest in the field of turbulence has appeared. The interest comes mostly from what we might call numerical physics. Away from boundaries, turbulence appears to be a truly three-dimensional phenomenon. Numerical simulations of three-dimensional turbulence have been carried out by Kerr (1985, 1990) and by Vincent and Meneguzzi (1991) by an integration of the Navier-Stokes equations in a box. The theoretical framework originates with Kolmogorov (1941) and the refinements of the original ideas by the same worker (Kolmogorov, 1962). Almost all research deals with homogeneous turbulence, and many researchers use the so-called shell models to reduce a three-dimensional problem to one of only one dimension. This is done by working in wavenumber space and disregarding all references to the phase, and dealing only with the modulus of the wavenumber vector. In addition, the wavenumber space is subdivided in a finite number of shells of increasing thickness. A given shell contains all moduli between a lower and an upper limit. With k o being a scaling wavenumber, the lower and upper limits of shell number n are koA n and k o An+ 1, where h is to be specified and in many cases has the value two (see, e.g. Jensen et al., 1991). Shell n is 0377-0265/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 0 2 6 5 ( 9 7 ) 0 0 0 3 9 - 0
708
A. Wiin-Nielsen/ Dynamics of Atmospheres and Oceans 27 (1997) 707-714
characterized by the single velocity u(n), which is considered as an averaged velocity over the shell. A major problem in shell models is to specify the nonlinear interactions between a given shell and the others. It is well known that a given wavenumber interacts with all pairs of wavenumbers whose sum or difference equals the given wavenumber (Wiin-Nielsen, 1979). Owing to the design of a shell model, it appears impossible to simulate this general rule, and it is replaced by specifying the interactions in an ad hoc fashion. Typical specifications are that a given shell shall interact only with the immediate neighbouring shells. The interaction may involve three or five shells. The coefficients in the interaction terms are found by requiring that integral quantities such as the total kinetic energy shall be conserved in the unforced, non-viscous limit. Although one of the coefficients may be assumed to be unity, because one may rescale the time to obtain this result, it requires normally more than one conservative quantity to determine the coefficients in a unique way. In this regard, the so-called helicity has been invoked (Kadanoff et al., 1995). Without any forces and without dissipation it can be shown that a scalar quantity, being the scalar product of the three-dimensional velocity and the three-dimensional vorticity vector, is conserved provided that the kinetic energy vanishes at the physical boundaries of the fluid or the gas. As helicity is an unfamiliar quantity in the meteorological and oceanographic communities, an elementary demonstration of the conservation of helicity in the general three-dimensional case will be provided (Section 2). Thereafter we shall discuss the use of helicity in shell models. We shall also make use of the fact that the mean velocity is a conserved quantity, as it must be in the unforced, non-viscous case. We shall demonstrate that it is possible to define a shell model in a unique way by using the conservation of the mean velocity and the kinetic energy. This model will be integrated numerically (Section 3). lntermittency is considered to be an essential part of turbulence (Ohkitani and Yamada, 1989; Eggers and Grossman, 1991; Jensen et al., 1991), and it is connected with chaos and multi-fractal structures (Meneveau and Sreenivasan, 1991; Jensen, 1994). We shall also touch on these matters in the paper.
2. Helicity A quantity called the helicity, adapted for use in a one-dimensional model, has been used as a conservative quantity in shell models (Kadanoff et al., 1995). Helicity is in general defined in the following way: he~-V. ~
(1)
Helicity was discussed by Moffatt (1969), who gave a very general discussion of the concept and examples of cases in which helicity is conserved and other cases where the conservation fails. Our interest is essentially in the atmosphere, and it may be appropriate to provide an elementary demonstration of conservation of helicity in an integrated sense, although the proof actually is a part of Moffatt's paper. As usual, we shall use u,
A. Wiin-Nielsen/ Dynamics of Atmospheres and Oceans 27 (1997) 707-714
709
v and w as the three components of the velocity. We recall also that the three components of the vorticity vector are ~1 = ~'2 =
Ow
av
ay au
az Ow
az av
ax au ay
if3 ~--- aX
(2)
In geophysical fluid dynamics, most considerations are based on the vertical component of the vorticity vector, leading in barotropic, two-dimensional flow to the conservation of enstrophy (i.e. the integral over the total area of half the vorticity squared) and the conservation of potential enstrophy in baroclinic quasi-nondivergent flow. It is these conservation laws combined with the conservation of kinetic energy that resulted in the so-called two-dimensional turbulence theory as developed by Kraichnan (1967) and Leith (1967) for the barotropic case. The baroclinic, quasi-geostrophic case can be treated in the same way using the conservation of potential enstrophy combined with the conservation of the sum of available potential energy and kinetic energy. These considerations imply a spectral slope of - 3 for large values of the wavenumber for both the kinetic and the available potential energy, and these facts have implications for the limited predictability of the atmosphere as discussed, for example, by Wiin-Nielsen (1991). Here we shall use the total vorticity vector. It is assumed that the fluid is homogeneous such that the three-dimensional divergence vanishes. By cross differentiation of the Navier-Stokes equations excluding all forces we obtain three vorticity equations that may be written in the following form: -- +v.V~l-Vu.i;=O at aft2 at
--
+ v . V~2 - V v .
~ = 0
(3)
a~"3 + v . V ~ 3 - X7w.i;= 0 at The conservation theorem says then that d
I= - ~ t f v ' ~ d V =
-~t-t " ~'+ v • a~']dV= 0 (4) at 1 By using the component equations and the fact that the three-dimensional divergence is zero one may write the above integral in the form I = - f f l v - ( ~ , u v ) + V" ( ~2vv) + V" ( ¢3wv) - Vk" ~ ]dV = J~f,[v" (k~') - kV. (V x v ) ] d V = O, k = 1 / 2 v . v
(5)
710
A. Wiin-Nielsen / Dynamics of Atmospheres and Oceans 27 (1997) 707- 714
It is thus seen that 1 = 0 if the kinetic energy goes to zero at the boundary of V, as the last term in I is identically zero. The helicity is thus conserved if this condition is satisfied. The helicity is a rather peculiar conserved quantity because it is not always positive as, for example, is the kinetic energy. In addition, for the large-scale, quasi-nondivergent flow in the atmosphere, where the vertical velocity is about three orders of magnitude smaller than the horizontal velocity, the helicity reduces to a quantity proportional to the temperature advection, which can be seen by using the thermal wind equation h:
g ---r.V(InT)
(6)
L, and it is thus seen that helicity vanishes when integrated over the globe. This argument does not apply for the small-scale flow. The definition of helicity in a shell model is far from obvious. It turns out that one has to be satisfied by a quantity that is of the correct dimension. By definition it becomes N
/4,, = E
(7)
n=l
where N is the number of shells as derived by Ditlevsen and Mogensen (1996). This definition is rather unfortunate because it means that the local contribution to helicity changes sign from shell to shell. In other words, the helicity will cause a production also in the dissipation range. It appears therefore to be a more sound approach to avoid helicity and use the conservation theorems of kinetic energy and mean velocity to formulate the shell model. This will be done in the next section.
3. A n e w shell m o d e l
We shall formulate a shell model based on the conservation of the mean velocity and the kinetic energy. When we consider the one-dimensional advection equation without forcing and dissipation, it is easy to see that the mean value is conserved by integrating over the total length with periodic boundary conditions as we would when we consider the advection equation along a given latitude circle. The same result is obtained when the dependent variable is written as a Fourier series where the equation is transformed to a finite number of ordinary equations for the time-dependent Fourier coefficients. In a shell model we make a local approximation to the advection term by restricting the nonlinear interactions to the nearest neighbour shells. It is then natural to formulate these approximations in such a way that the mean value is conserved. It turns out that this is an impossible task if we restrict the interactions to the immediate two neighbouring shells. We try therefore to obtain the required result with
A. Wiin-Nielsen / Dynamics of Atmospheres and Oceans 27 (1997) 707-714
711
five shells as in the GOY (Gledzer-Ohkitani-Yamada) model, but of course with a different choice of interaction terms. The GOY model does not conserve the mean value. This is due to the fact that it includes the interaction involving the terms u n_ ~ and u~ + 1 as the only term where the counter increases by two. Therefore, such a term can never cancel. The following model is proposed: du n dt = ak"U~+ lUn+ 2 + b k " - 3u"- 2u"- I + c k , _ lU, Un+ n + d k , _ 2u ~_ l u . 2 + e k . u .2+ I _{._fkn _ 2u._n
(8)
with the energy equation becoming du2./2 dt
a k . u . u . + lU.+ z + bkn_ 3Un_ 2Un_ lUn + c k . _ lU2.U.+ l + d k . _ 2u . _ llA 2 2
+ ek.u.u]+, + fk._2u._,u .
(9)
The next task is to determine the six constants in such a way that the kinetic energy and the mean value of the velocity are conserved when summed over the total number of shells. Invoking the two conservative quantities we obtain the following equations for the constants: a+b+c+d=O e+f=0 a + b/2 = 0
(lO)
c+f=0 d/2 + e = 0 With a -- 1 obtained by rescaling the time, we obtain b = - 2 , c = - 1, d = 2, e = - 1 and f = 1. The next question is then to determine the fixed-points which may be present in the model. It is assumed that the solution is of the form u[n] ~ k[n] -~. When this is inserted in the steady-state equation and setting z = 2 -~ we obtain a sixth-degree I expression. The roots are determined, and we find the following real values of a : 0, 1 and i. A stability analysis of each of these steady-state solutions shows that all of them are unstable. As in the GOY model, it is also found that only a few of the eigenvalues are numerically large whereas the rest are very small, although all are positive in the 1 proposed model as shown in Fig. 1 for a = 7In the following, we shall refer to this model as the MEKIN model, as it conserves the mean velocity and the kinetic energy in the unforced, non-viscous case.
4. Integrations of the MEKIN model To compare the model with the GOY model, the integrations of the two models have been done in essentially the same way. We present the results using time averages for
A. Wiin-Nielsen/ Dynamics of Atmospheres and Oceans 27 (1997) 707-714
712
Stability 3500
i
i
0
0
of
I
0
0
fixpoint
r
0
with
i
0
">
&
slope
-1/3
t
0
O
3000
i
i
O O
O
O O
2500 D > O1
2000
{1} o
1500
o i000
I
I
I
1
I
I
I
I
l
2
4
6
8
10 [!
12
14
16
18
20
Fig. 1. The cumulative sum for the real eigeuvatues for the fixp~int - ~. The plotted value s[n] is the sum of the eigenvalues from unity to n. MEKIN-mode], -8
R:I0~*6,
init,
i
7 - -
i
and
final i
slopes:
i/3,
0.38
I
i
© © -i0
o -12
--o_
-
o,
-14 v -16
-18
-20
-22
l
-18
t
-16
t
-14
Fig. 2. A d o u b l e l o g a r i t h m i c ~ o t o f the t i m e - a v e r a g e d v a l u e s o f final slope
-0.38;
Reynolds
number
106
.
~
-12 -i0 ln(k{n])
u[n] as
t
l
-8
6
a function of
k[n].
-
Initial slope - ~:
A. Wiin-Nielsen/ Dynamics of Atmospheres and Oceans 27 (1997) 707-714 M e a n vel. 0.06
i
i
& kin. i
en. l
cons., i
713
abs(u[18]) i
i
0.05
0.04
0.03
..Q 0.02
0.01
I
i000
2000
3000
4000
5000 6000 time, sec
7000
8000
9000
10000
Fig. 3. The absolute value of the component u[18] as a function of time. the various components of the velocity. In addition, we present the results in the form of diagrams where we use the logarithmic values for the wavenumbers and the velocity components. 1 Fig. 2 shows a case where the starting distribution is given by a slope of - 7. After an integration of 10000 time steps corresponding to about 42h, and with a Reynolds number of 1013, we obtain a slope of - 0 . 3 8 , which happens to be the same as obtained with the GOY model (Jensen, 1994). Starting from an initial slope of - 0 . 5 , which also represents an unstable fixpoint, it was found that the slope does not change on average. Fig. 3, in which we display the time series of a single component, u[18], shows that the intermittency appears for the wavenumbers at the upper end of the shell model MEKIN, which in this integration, as in the others, was restricted to 19 shells.
5. Conclusions
The MEKIN model is capable of giving a slope comparable with that for the GOY I model if one starts from an initial slope of the velocity components of - ~ corresponding to a kinetic energy slope of - 75 The resulting slope ( - 0 . 3 8 ) after many integrations deviates somewhat from the initial slope. We stress that we are simulating the three-dimensional case. If we were to simulate two-dimensional turbulence we would have to determine the coefficients in such a way that kinetic energy and enstrophy are conserved. This has been done by Ditlevsen and Mogensen (1996).
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A. Wiin-Nielsen / Dynamics of Atmospheres and Oceans 27 (1997) 707-714
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