数理解析研究所講究録
第 892 巻 1995 年 248-262

248

Flow pattern in forced two-dimensional turbulence
on a rotating sphere
Toru Nozawa
Shigeo Yoden$(’\phi(Si*\cdot lE\cdot 1\Psi M)$$E_{\grave{J}}^{\backslash }R$$\frac{*}{\prime J\backslash }fflRF(\prime P\backslash rightarrow\star\cdot lE\cdot l4M)$

Department of Geophysics, Kyoto University, Kyoto 606-01, JAPAN1 Introduction
Large-scale motions in planetary atmospheres are largely turbulent in space and time. Although
the large-scale mrbulent motions are actually three-dimensional, one can consider them to benearly two-dimensional (2D) on a horizontal surface owing to the stratification of atmospheres
and the rotation of planets. For example, synoptic-scale disturbances in the atmosphere haveas shown by Boer&Shepherd (1983) with a global atmosuch quasi-2D mrbulent characteris
spheric data set;energy spectra and transfer functions in the spherical domain have some feamresof an ensffophy inertial subrange for a high-wavenumber region of $n>10$ , where is the total
wavenumber of the spherical harmonics.
A geophysical application of the $2D$ mrbulence theory (Kraichnan 1967; Leith 1968; Batchelor1969) was firstly done by Rhines (1975) with a numericalon a -plane to investigate
the effect of the rotation of planets on the $2D$ turbulence. He showed that the upward energy$\dot{\mathfrak{a}}cs$$n$$m\alpha lel$$\beta$$cmeridiona1gradientoftheCo1iorisparameterf;teconversionfromturbulenceintoRossbywavesascadeceasesrough1yatawavenumberk_{\beta}=\beta\ulcorner/2U,whereUisther.m.s.velocityand\beta the$takes place around the wavenumber . He also found that the flow field becomes anisotropic
emerges owing to the -effect. Malmid&Vallis (1991) numericallyand a zonal bandstudied the forced $2D$ turbulence on a -plane with a high-resolution model (256 or 512 grids)
under recent advanced computing facilities. In their experiments with low-wavenumber forcing,coherent vortices, which were found by Basdevant et al. (1981) and McWilliams (1984) in the
numerical experiment on planar $2D$ turbulence, become weak while anisoffopy of the flow fieldin the enstrophyincreases as the strength of the -effect increases, although energy).
energyAncascading range remains relatively unchanged (steeper thanspecmim nearly
proportional towas also observed in the energy-cascading range in the experiments withhigh-wavenumber forcing.
Williams (1978) reproduced a band smcture similar to that of Jovian atmosphere in a numerical(1994) did a
expenment on the forced $2D$ turbulence on a rotating sphere. Nozawa andseries of numerical experiments with the same framework as Williams aiming at a methodical
sweep in parameter space. When the amplitude of the $stochas\dot{u}c$ vorticity forcing is not verysmall and the latitudinal wavenumber of the forcing is relatively small, a clear band $sOucmre$
which consists of altemating easterly and westerly jets emerges in the flow field. However, the, and the forcing function
numerical model has a cyclic boundary condition in longitudes ofis anisotropic. Thus the obtained band smcture might be influenced by the assumed longimdinal
periodicity and the anisotropic vorticity forcing.
$k_{\beta}$$S\mathfrak{a}uCture$$\beta$$\beta$$\beta$$k^{-3}$soecmm$k^{-5/3}$$Y\propto len$$45^{o}$249

In this paper, we perfomi a series of numerical experiments on the forced $2D$ mrbulence ina full spherical domain with a homogeneous and isotropic forcing at high wavenumbers. The
sensitivity to the rotation rate and that to the choice of the random numbers for the forcing arestudied in detail. The flow field, pmicularly the formation of the band smctuoe, is investigatedas well as the energy spcmim and the transfer function. The numerical $proeedure$ is described in
section 2, and results are given in section 3. Discussion is given in section 4, and conclusions are
given in section 5.

2

Model and experimental procedure
Two-dimensional nondivergent flow on a rotating sphere is govemed by a vorticity equation:$\frac{\partial(}{\partial t}+\frac{1}{a^{2}}J(\psi, \zeta)+\frac{2\Omega}{a^{2}}\frac{\partial\psi}{\partial\lambda}=\nu(\nabla^{2}+\frac{2}{a^{2}})(+F$,(1)

is a streamfunction field, $((\lambda, \mu, t)=\nabla^{2}\psi:vorticity,$ $\lambda:longimde,$ :sine latitude,
where$\nabla^{2}:horizonta1$
Laplacian,:time:horizontal Jacobian, $a;radius$ of the sphere, :rotation
rate of the sphere, $\nu;kinematic$ viscosity coefficient, and:vonicity forcing function. The)
radius of the sphere is set to that of Jupiter;aand the time is measured by Jovian
is the rotation rate ofday; 1 J.day $=2\pi/\Omega_{J}=3.57\cross 10^{4}s$ , where $\Omega_{J}=1.76\cross 10^{-4}$ rad$\nu=5.00\cross 10^{5}m^{2}s^{-1}$
Jupiter. The kinematic viscosity coefficient ofis adopted in this study.
For the forcing function $F$ , a random Markovian $fo$–ulation is used as in Lilly (1969) andWilliams (1978):$\psi(\lambda, \mu, t)$$\mu$$\Omega$$J(\psi, \zeta)$$t$$F(\lambda, \mu, t)$$=7.\alpha$$\cross 10^{7}m$$t$$s^{-1}$$F(\lambda, \mu,j\Delta t)=RF(\lambda, \mu, C-1)\Delta t)+(1-R^{2})^{1/2}\hat{F}(\lambda, \mu,j\Delta t)$,(2)

where is a dimensionless memory coefficient ( $R=0.98$ as in Williams (1978)), and is arandom vorocity source at every time step. Here the random vonicity source function is defined
as follows:(3),$\hat{F}$$R$$\hat{F}(\lambda, \mu,j\Delta t)=\sum_{n=n_{mn}}^{n_{r}}\sum_{m--1}^{n}\hat{F}_{n}^{m}C)P_{n}^{m}(\mu)e^{im\lambda}$

is an expansion coefficient of with spherical haronics, which is deteimined sowherethat it has random amplitude and phase at eveiy time step in order to consmct a homogeneous$\hat{F}_{n}^{m}(j)$$\hat{F}$$j$and isotropic forcing under the resmction of $||\hat{F}(\lambda, \mu,j\Delta t)||=\ovalbox{\tt\small REJECT}_{m=1}^{n}$(con-

stant). The forcing is given in a narrow range of $n_{mm}=77$ and $n_{\max}=81$ and the amplitude is held

constant to $F=7.85\cross 10^{-11}s^{-2}$ for all the experiments in this study. Figuoe 1 shows an exampleof the forcing field at a paticular J.day.
A pseudospectral method with a triangular mncation of T199 $(n\leq 199=N)$ is used forctral transformation are 600 in
the computation of the advection (Jacobian) term;grids for thelongitudes and 300 in latimdes. Equation (1) is integrated ffom an initial condition of zero velocity
field for a period of 1000 J.days. The fourth-order Runge-Kutta-Gill method is used for the time
integrations with $\Delta t=0.05$ J.days. All of the computations are done in double precision.
), $0.25$ ,As an experimental parameter, six values of the rotation rate aoe taken$2.tX)$ , and 4.(X). The sphere has
0.50,
rotations par unit J.day. For each rotation rate,
three mns are done with different random sequences for the amplimde and phase of the vorticity
source function.
$s\varphi$$:\Omega/\Omega_{J}=0.\alpha$$1.\alpha),$$\Omega/\Omega_{J}$250

Fig.1: Vorticity forcing field at $t=1\alpha$) J.days. Contour interval is $1.0\cross 10^{-10}s^{-2}$ and negative
areas are drawn by dotted lines.

3 Results
Figure 2 shows the time variation of the total kinetic energy

ens$\sigma ophy\mathcal{Q}(t)=\sum_{n=2}^{N}Q(n, t)$$\mathcal{E}(t)=\sum_{n=2}^{N}E(n, t)$

and the total

for six values of the rotation rate, which aoe ensemble averages ofthoee $mns$ . Here the energy specmm density $E(n, t)$ and the enstrophy specmm density $Q(n, t)$are, respectively, defined as follows:where$\psi_{n}^{m}(t)$,$E(n,t)$$=$$\frac{n(n+1)}{2}\sum_{m=-n}^{n}|\psi_{n}^{m}(t)|^{2}$$Q(n, t)$$=$$\frac{n^{2}(n+1)^{2}}{2}\sum_{m=-n}^{n}|\psi_{n}^{m}(t)|^{2}$is an expansion coefficient of$\psi$

(4)

,

(5)

with spherical harmonics :$\psi(\lambda, \mu,t)=\sum_{n=2}^{N}\sum_{m=-n}^{n}\psi_{n}^{m}(t)P_{n}^{m}(\mu)e^{im\lambda}$

(6)

251

At the beginning of $t<20$ J.day, both of the total energy and the total enstrophy increase sharply,
growth rates of which aoe almost independent of the rotation rate. As the time goes by, the energy increases gradually while the enstrophy decreases. The energy is smaller for larger,). On the other hand, thealthough it is not very different for the expenments with
enstrophy is larger for larger. Even at the end of time integrations $(t=1\alpha n$ J.day , thetotal energy goes on increasing slightly while the enstrophy becomes nearly constant.
$\Omega/\Omega_{J}$$\Omega/\Omega_{J}\leq 1.\alpha$$\Omega/\Omega_{J}$

$($

O

$)$

$)$

$(\backslash |y)$$e\backslash \in$$\succ$

$(D$

$K$

Ld
$z$
$L\Delta$$\lrcorner$$\triangleleft$$\vdash$$O$$\vdash$$arrow$(b)

T I ME$(J. doy)$

T I ME$(J. doy)$$\sim|y)$$\succ$I$0_{-}$$O$$\not\subset$$\vdash$$\omega$$z$$L\perp\rfloor$$\lrcorner$$\triangleleft$$\vdash$$O$$\vdash$Fig.2: Temporal variation of the total kinetic energy. Ensemble averages of three $mns$ .
six values of$\mathcal{E}(t)(a)$$\Omega/\Omega_{J}$, and the total enstrophy$\mathcal{Q}(t)(b)$

for

252

. These are time averages from
Figure 3 shows the energy spectra for six values of
to ltXn J.days. Three mns are plotted for each rotation rate. A wavenumber
indicates theforcing wavenumber: $n_{f}= \frac{n_{\min}+n_{\max}}{2}=79$ , which corresponds to the wavenumber of energy
in the ens
and enstrophy inputs. The power law of the energy specffum is close torange,
independentcascadingwhich slope isof the choice of the random numkrs for each.
$7be$increases,
althoughrange
gentlealittleinthisitbecomesasrate
therotation
is
yet
slor
) $(f)$ .
steeper thaneven for the mns with
For all the expenments, an upward energy-cascading range is obtained. The energy spectrain the energy-cascading range, and the slope steepens
are nearly proportional toasincreases. Three energyof different random numkrs are very similar in thisrange as well as those in the enstrophy-cascading range. In the case of no rotation (a).the energy
specmm is steeper thanAt the low wavenumbers of $n<5$ owing to the finiteness of thespherical domain. For the experiments with rotation (b-f), on
other hand, the specmim does
not obey the power law at low wavenumbers;the energy-cascading range becomes nanow and theenergy specmm in this range kcomes noisy as
incoeases. Here another wavenumber$\sigmaoduced$is infollowing to Nozawa and Yoden (1994):$\Omega/\Omega_{J}$$8\infty$$n_{f}$$n^{-4}$$\sigma ophy-$$\Omega/\Omega_{J}$$n^{-3}$$\Omega/\Omega_{J}=4.\alpha$$n^{-5/3}$$\Omega/\Omega_{J}$$sligh\ddagger ly$$spec\sigma a$$n^{-5/3}$$\tilde{th}e$$\Omega/\Omega_{J}$$n_{\beta}$$n_{\beta}\equiv a\sqrt{\frac{\{\beta\rangle}{2U}}$,(7)

at $t=1000$ J.day (ensemble averages of three mns
is the r.m. . velocityhere), and. At the horizontal scale ofis the spherical average of, the nonlinear Jacobian term is comparable to the-term” in Eq.(l). Figure 3 shows thatroughly gives the lower bound of the energy-cascading range. The specmm does not show any(b-f), where theclear power law in-term” is larger than the nonlinear term. In thisrange, the energy specmm is very noisy and it depends on the choice of the random numbers. For(e.f), the energy density is relatively small at the lower wavenumbers in this range.
largerThis is why the total energy is relatively small for larger
as seen in Fig.2 (a).
The energy flux function $\Pi(n, t)$ is calculated from the energyfunction $T(n, t)$ :

where

$U$

$(U=\sqrt{2\mathcal{E}})$$s$$\beta:\langle\beta\rangle=\frac{1}{2}\int_{-1}^{1}\beta d\mu=\pi\Omega/2a$$\langle\beta\rangle$$a/n_{\beta}$$\beta$$n_{\beta}$$2\leq n<\sim n_{\beta}$$\beta$$\Omega/\Omega_{J}$$\Omega/\Omega_{J}$$\sigma ansfer$$\Pi(n, t)$$= \sum_{n=2}^{n}T(n’, t)$$T(n, t)$

$=$

,

(8)

$- \sum_{m=-n}^{n}\{\psi_{n}^{m}(t)\}^{*}N_{n}^{m}(t)$,(9)

is the expansion coefficient of the nonlinear Jacobian term, and denotes complex, which are time
conjugate. Figure 4 shows the energy flux functions for six values of
averages from 800 to ) J.day and ensemble averages of the three $mns$ . For the case of norotation, the energy flux is nearly constant in the range of $2\leq n<\sim 20$ , thus the energy is transferred
as far as the lowest wavenumber of $n=2$ . When the rotation rate is small or moderate (0.m, the energy flux function in the range ofis parallel to that of noratation, indicating that the effect of the rotation is weak in this range. The energy flux in thisrange becomes large asincreases kcause the energy density is larger for larger
inthis range as seen in Fig.3 (a-d). In the range of $2\leq n<n_{\beta}$ the energy flux function is notparallel to that of no rotation, but it has a large decline;only a little energy is transfened to this
where

*

$N_{n}^{m}(t)$$\Omega/\Omega_{J}$$1\alpha$

$0$

$<$

$\Omega/\Omega_{J}\leq 1.\alpha))$$n_{\beta}<\sim n\leq n_{f}$$\Omega/\Omega_{J}$$\Omega/\Omega_{J}$253

TOTAL WAVENUMBER$n$TOTAL WAVENUMBER

$n$

. The
averaged from 800 to 1000 J.days for six values ofFig.3: Energy specmm
is also indicated, at which scale
and the wavenumber
forcing wavenumber is indicated byare plotted for eachthe -term” is comparable to the nonlinear Jacobian term. Three
figure. The specmm obtained with the same random numbers is plotted with the same line type.
(above) and, and lines ofA line ofis added in the range of$(\Re 1ow)$ are also added in the range of $n_{f}\leq n\leq N$ .
$\Omega/\Omega_{J}$$\overline{E}(n)$$n_{f}$$n_{\beta}$$spec\sigma a$$\beta$$\overline{E}(n)\propto n^{-5/3}$$n_{\beta}\leq n\leq n_{f}$$n^{-3}$$n^{-4}$

254

range beyond
For large$n_{\beta}$. The process of the energy interchange is largely influenced by the rotation.
, on the other hand, the energy flux function is not parallel to that of no$\Omega/\Omega_{J}(\geq 2.00)$, indicating that the process of the energy interchange
rotation even in the range ofis largely influenced by the rotation even in the energy-cascading range. The energy flux is nearly
for the largest rotation rate $(\Omega/\Omega_{J}=4.m)$ ;little energy
equal to zero in the lower range ofis transferred to this range because the energy upward cascade is suppressed by the rotation. Thisis the reason why the energy density in this range is relatively small for these mns as seen in Fig.3$n_{\beta}<\sim n\leq n_{f}$$n<\sim 6$(f).
$(x10^{-5})$$V)$$Oi\in$$\cross$$\supset$$\lrcorner$$\llcorner$$\succ$$\not\subset$$L\Delta$$\iota_{\lrcorner}\lrcorner$

TOT A L WAV E N UMB E
Fig.4: Energy flux function$\overline{\Pi}(n)$for six values of$\Omega/\Omega_{J}$$R$$\cap$. Ensemble averages of three $mns$ .

The $s\sigma eamfunction$ field at $t=1000$ J.days is shown in Fig.5 for six mns. The same randomsequance of the vorticity forcing is used for these $mns$ . For the case of no rotation (a), the streamfunction field has a very large pattem which is characterized by the lowest wavenumber $n=2$
in consistent with the fact that the transfemd energy is accumulating at the lowest wavenumber (Fig.3 (a), Fig.4). This flow pattem moves irregularly on the sphere withom changing thepattem largely. For the experiments with rotation (b-f), on the other hand, zonal band $smctu\infty s$
become dominant in the streamfunction field. As the rotation rate increases, the zonality of thestreamfunction field increases. Although details of the flow pattems change with time, the zonal
structures aoe unchanged panicularly for large rotation rates.

255

Fig.5: Streamfunction field at $t=1000$ J.days for six values of$2.5\cross 10^{8}m^{2}s^{-1}$and negative weas are drawn by dotted lines.

$\Omega/\Omega_{J}$. Contour interval is

256

T IM

$E$

T IM

$E$

T I ME $(J. doy)$$(J. doy)$200

al ari
$n$

’al

400

T IM

$(J. doy)$

$E$

600

800

1000

$(J. doy)$

gu ar
rval

$\cross$$a\infty$

257

Figure 6 shows the temporal variation of the zonal mean angular momenmm for the same six
mns as in Fig.5. For the case of no rotation (a), the intensity of the angular momentum is notvery large and the easterly or westerly flow dominates over a hemisphere. Temporal variationof the mean zonal flow is large in consistent with the $ii\tau egular$ movement of the flow pattem.
, on the other hand, the altemating easterly and westerly bands
For the cases with rotation
do not change their positions after the establishment of the band smcture. They become clear
and robust as the rotation rate increases. They are aloeady discemible in early stages by $t=1\alpha$), several mergers of westerly bands take place and the width of theJ.days. For large
bands increases. The number of the bands increases while their width decreases as the rotation
rate increases.
(1975):
Figure 7 (a) shows the time variation of an index of $aniso\sigma opyAi$ intrOduced by$(b- f\gamma$$\Omega/\Omega_{J}(e)f)$$He\pi\dot{m}g$$Ai= \frac{\langle u^{2}\}-\langle v^{2}\rangle}{\langle u^{2}\rangle+\langle v^{2}\rangle}$

,

(10)

where
and
are eastward and northward velocities, respectively, anddenotes the spherical average. These are ensemble averages of the three mns. For the case of norotation, the flow field is nearly isotropic $(Ai\sim 0)$ throughout the integration time. For the rotating cases, on the other hand, the anisotropy kcomes large with time;the flow field increases
its zonality. The anisotropy increases as the rota
rate increases in consistent
with the fact that the streamfunction field has soeonger zonality for larger (Figs.5 and 6). The
dependence of $Ai$ on
at $t=1\alpha$) J.days is shown in Fig.7 (b). For small rotation rate$(\Omega/\Omega_{J}<1.00)$ , the index of anisotropy is largely different between the three mns but it has atendency to increase with . For large , on the other hand, $Ai$ is nearly the same for the three$u(\lambda, \mu, t)$$v(\lambda, \mu, t)$$\langle\cdots\}$$(\langle u^{2}\}>\langle v^{2}\rangle)$$\dot{\mathfrak{a}}on$$\Omega$$0$$\Omega/\Omega_{J}$$\Omega$$\Omega$runs $(0.8\sim 0.9)$ .

Figure 7 (c) shows the time variation of the energy-weighted total wavenumber for the zonal

component$n_{0}$:$n_{0}= \sum_{n=2}^{N}nE_{0}(n, t)/\sum_{n=2}^{N}E_{0}(n, t)$

,

(11)

is a gross measure of the number of the altemating easterly and westerly bands. These are ensemble averages) J.days, and it $\Psi^{adually}$ decreasesof three $mns$ . For all the mns, decreases sharply by)}. $7he$is nearly constant except for the case ofJ.days,after that. For the last
increases in consistent with the fact that the number ofwavenumber becomes large as
as seen in Fig.6. The de ndence of on
the altemating bands increases withat$t=10\alpha)$ J.days is shown in Fig.7 (d), where
for the three mns are plotted for each. The
is nearly proportional toand has little dependence on the choice of thewavenumber
random numbers.
The relative vorticity field at $t=1000$ J.days is shown in Fig.8 for the same six mns as in Figs.5
nearlyand 6. The center of each figure is the same as that in Fig.5. Typical size of the vortices(a),the vorticity field isequal to that of the forcing as seen in Fig.1. For the case of no rotation
for the small and $m(rerate$ rotation rate (b-d). For
isotropic. It also seems to be nearly iso, on the other hand, the vorticity field is elongated zonally and the altemating positive
large
increases asincreases
and negative vonicity bands emerge. The intensity of thein consistent with tha fact that the enstrophy is larger for larger
as seen in Fig.2 (b).
where $E_{0}(n, t)$ is the energy density of the zonal component. The wavenumber$t\sim 1\alpha$$n_{0}$$2\alpha$$n_{0}$$\Omega=0.\alpha$$n_{0}$$\Omega/\Omega_{J}$$\Omega/\Omega_{J}$$\Psi$$\Omega/\Omega_{J}$$n_{0}$$\Omega/\Omega_{J}$$n_{0}$$n_{0}$$n_{0}$$\Omega/\Omega_{J}$$a\infty$$\sigma opic$$\Omega(e.f\gamma$$vor\dot{o}city$$\Omega/\Omega_{J}$$\Omega/\Omega_{J}$

258

$($

b

T I M $E=1000$ $(J. doy)$

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$1$

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$\bullet\bullet$$\phi$$0$$0$

T I ME$\{$(J. doy)2

34$\Omega/\Omega_{J}$

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T I M $E=1000$ $(J. doy)$$t5$$$\bullet$

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$\subset$1

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T I ME$\{$(J. doy)2

34$\Omega/\Omega_{J}$-dependence of $Ai(b)$ ,Fig.7: Temporal variation of an index of $aniso\sigma opyAi(a),$temporal variation of energy-weighted total wavenumber for zonal component $n_{0}(c)$ , and-dependence of $n_{0}(d)$ . Ensemble averages of three mns in (a,c). $Aniso\sigma opyAi$ andwavenumber obtained with the same random numbers is plotted with the same symbOl in (b,d).
$\Omega/\Omega_{J}$$\Omega/\Omega_{J}$$n_{0}$

259$\cross 10^{-5}$$(s^{-t})$

30

20
I

$0t\Delta$$0$$\supset$$0\vdash$$\vdash$$\triangleleft$$-10\lrcorner$

$-20$

$-30$
$-30$

$-20$

$-f0$

$0$

10

20

30

L ONC I TUD$t=10(n$Fig.8: Relative vorticity field at
J.days for the same mns as in Figs.5 and 6. The centerof each figure is the same as in Fig.5.
$E$

260

4 Discussion
For the case of no rotation, the energy supplied around is transferred upward and the rate of
the energy cascade is nearly constant in the range of $2\leq n<20$ as shown in Fig.4, which fact isconsistent with the $2D$ mrbulence theory. For the cases
rotation Malmd and Vallis (1991)
has already studied in their numerical experiment on a -plane. They showed that theenergy accumulates near because the upward energy cascade is suppressed by the rotation, and. Althoughthat the power law in the energy-cascading range remains nearly proponional tospecmm,energythe wavenumber we used is not identical toinfluenced
these feamres of theby the rotation was obtained in our experiments for small and moderate (Fig.3 (b-d), Fig.4).
For the cases with larger rotation rate $(\Omega/\Omega_{J}\geq 2.00)$ , on the other hand, the process of theenergy interchange is largely influenced by the rotation and the energy is accumulating even in
the energy-cascading range. Thus the energy specmm in this range becomes steeper thanas shown in Fig.3.
Williams (1978) and Nozawa and Yoden (1994) studied the forced two-dimensional turbulence
on a rotating sphere with a cyclic boundary condition. Although they showed some impressive
clear band smctures, the flow field in this study using the isotropic vonicity forcing function
does not show such a clear band smctures as in our previous study with the same experimental
parameters. Thus their results may be influenced by the anisotropic forcing functions and by thecyclic boundary condition.
Yoden and Yamada (1993) studied the decaying $2D$ turbulence on a rotating sphere, and showed
the emergence of strong easterly circumpolar-jets. In this study on the forced $2D$ mrbulence, clear
band $smc\iota ures$ which consist of the altemating easterly and westerly jets appear in all the latitudes,
and almost all the circumpolar-jets are easterly (Fig.6). The emergence of easterly jets in highlatitudes seems to be a general featuoe of $2D$ turbulence on a rotating sphere. Interactions between
the mean zonal flow and Rossby waves, of which energy is convened from the turbulent motions
by means of the upward energy cascade, seem to be a fundamental dynamics of the fomiationof the band smcture, namely, these Rossby waves might redisnibute the zonal mean angularmomentum to maintain the band smcmre.
Nozawa and Yoden (1994) have already studied the organization of the band smcture for different random numbers and showed that intensity and width of the jets are not very different,
although position of the jets depends on the choice of the random numbers. This is consistent
with the fact that the energy-weighted total wavenumber for the zonal component
doe not(d).
largely depend on the choioe of the random numbers as shown in Fig.7
The band smcture
is formed in early stages and then the smcmre becomes robust and persistent for the cases with(see also Appendix).
rotation as shown in Fig.6$n_{f}$$wit\tilde{h}$$\mathfrak{a}\cdot ansferred$$\beta$$k_{\beta}$$k^{-5/3}$$k_{\beta}$$n_{\beta}$$\Omega$$n^{-5/3}$$(e_{2}f)$$n_{0}$$s$$(b- f\gamma$

5 Conclusions
A series of numerical experiments on the forced $2D$ turbulence on a rotating sphere were donewith a high-resolution barotropic. The formulation of the homogeneous and isotropic vorticity forcing was adopted. Sensitivity of the energy specmm and the flow field to the rotation
rate and that to the choice of the random numbers for the forcing are investigated
The energy supplied at the forcing wavenumber, , cascades upwaM to low wavenumbers andthe enstrophy supplied at cascades downward to high wavenumbers for all the experiments inthis study. The upward energy cascade ceases around a wavenumber at which the -term” due$m\alpha lel$$n_{f}$$n_{f}$$\beta$$n_{\beta}$

261

to planetaiy rotation is comparable to the nonlinear Jacobian term. The energy-cascading range
becomes narrow as the rotation rate increases, because the upward enrgy cascade ceases aroundand the $\sigma ansferred$ energy accumulates in the range of $2\leq n<\sim n\rho$ . The slope in the energycascading range becomes a little steep while it in the enstrophy-cascading range becomes a little$resN^{C\dot{0}Vely}$.
gentle as the rotation rate increases, although they remain close toand
The energy specmm in these ranges does not depends on the choice of the random numbers.
For the case of no rotation, the streamfunction field shows a very large flow pattem and theflow field such as the relative vorocity field is isotropic. As the rotation rate increases, zonal bandsmctures become dominant in the sffeamfunction field;the flow field increases its anisotropy andthe number of altemating easterly and westerly bands increases. The band smcmres are alreadydiscemible in early stages and they are robust and persistent for the integration period.
$n_{\beta}$$n^{-5/3}$$n^{-4},$

GFD-DENNOU Library was used for drawing the figures. This work was supponed in part by
the Grant-in-Aid for the Cooperative Research with Center for Climate System Research, University of Tokyo, and was done in pan on the KDK system at Radio Atmospheric Science Center,
Kyoto University.

Appendix
In order to investigate the sensitivity of the zonal bandto the random sequanses ofthe vonicity forcing, we perform a series of supplementary experiments. For a certain experimentusing a random sequence of I, the random sequence is replaced to that of after the time J.days,
and integrated till $t=1000$ J.days. Figure A shows the dependence of the zonal mean angular) on the replaced time . The mean zonal angularmomentum at $t=1\alpha$ ) J.days for
momentum at $t_{c}=1000$ J.days is obtained using full random sequence of I, and that at $t_{c}=0$ J.days
is using full random sequence of II. The mean zonal angular momenmm is almost unchanged for$t_{c}\geq 180$ J.days;the zonal band smcture is already formed by thatand it becomes insensitive
to the choice of the random numbers after that time. In high latitudes this timing is earlier thanin low and middle latitudes;for example, the band smcture is unchanged for $t_{c}\geq 30$ J.days in$smc\iota u\infty$

$\Pi$

$0$

$\Omega/\Omega_{J}=4.\alpha$$t_{c}$$t_{c}$$\dot{\mathfrak{a}}me$southem high latitudes. When the replaced time is very early (is very sensitive to .

$t_{c}\leq 10$

J.days), the band smcmre$t_{c}$R EP L ACE D T I ME $(J. doy)$
Fig. : Dependence of zonal mean angular momentum at $t=1\alpha$) J.days on replaced time
$0$

$A$

$\Omega/\Omega_{J}=4.\alpha)$.

$t_{c}$.

262

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