7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe NagoyaBand structure formation in rotating systems

Shin-ichi Takehiro1
1
Research Institute for Mathematical Sciences, Kyoto University
e-mail (speaker): takepiro@gfd-dennou.org
The problem of 2D turbulence in rotating systems has
been intensively studied for a long time in relation to theformation and maintenance of zonal band structures andthe predominant east-west jets observed on various
planetary surfaces, such as Earth, Jupiter and Saturn.
The most fundamental model that have been consideredis the 𝛽-plane model, which describes two-dimensional
fluid motion on the tangent plane at a certain latitude on
a rotating sphere. In this model, randomly forced manysmall-scale vortices evolve to larger vortices through theisotropic inverse cascade process. However, when theeffect of rotation starts to dominate, the inverse cascade
would stop and a zonal structure has been considered toemerge[1]. From the balance between the advection and𝛽 terms in the vorticity equation, the characteristic√ 𝑈/𝛽 , so called Rhines scale, canlength scale 𝐿
be obtained, where 𝑈 and 𝛽 are the characteristic
amplitude of the flow and the background latitudinalpotential vorticity gradient, respectively. Rhines scale isoften used as an estimate of the order of magnitude of
the width of the band structure appearing in
two-dimensional turbulence of rotating systems.
In contrast, Vallis and Maltrud[2] performed numericalexperiments on two-dimensional forced turbulence in a
𝛽 plane model, and observed formation of large-scale
zonal banded structures through the energy spectrum on
a two-dimensional wavenumber space 𝑘 , 𝑘 . Here𝑘 and 𝑘 are the wavenumbers in the 𝑥 (longitude)
and 𝑦 (latitude) directions, respectively. Figure 2 shows
the 2D energy spectrum corresponding to Figure 1.
Immediately after the start of the integration, the energy
spectrum of the small-scale vortices produced by theforcing is distributed in a large circle (Fig. 2 left). After
that, an isotropic inversion cascade to the origin isobserved for large 𝑘 , 𝑘 region, whereas
dumbbell-like blank regions are seen near the origin,which is produced by the 𝛽 effect (Fig. 2 center andright). This means that the energy concentrates in theregion along 𝑘 -axis, avoiding the dumbbell region.
Such a spectral distribution means that 𝑘 ispredominantly close to 0, and corresponding to theemergence of zonal banded structures.
They have shown that this dumbbell structure can becalculated with the oscillation frequency of Rossby
waves and that in isotropic turbulence (reciprocal of theadvection time of the turbulent eddies). For example,
when the frequency of turbulence is estimated as 𝑈|𝑘|,we obtain, 𝑘𝑘 cos 𝜃 / , 𝑘/𝑘 sin 𝜃 cos 𝜃, where 𝑘
1/𝐿 . The outer regionof this dumbbell is dominated by the advection term,
meaning that the inverse cascade is possible there. This
analysis shows that the inverse cascade of turbulence is
not stopped by the 𝛽 effect, but it proceeds in an

anisotropic manner. After the appearance of the zonal
structure due to the inverse cascade in the 𝑥 direction,
the inverse cascade can proceed slowly in the 𝑦direction. This anisotropic inverse cascade in the 𝑦direction, although time-consuming, is not inhibited until
the spatial minimum wavenumber is reached. Therefore,a zonal banded structure with a width of the spatial
maximum scale is considered to emerge eventually.
Merger and disappearance of the narrow jets thatemerge at intermediate stages are observed in the forced
turbulence problem on rotating spheres[3] and zonal flows
driven by thermal convection in rapidly rotating thin
spherical shells[4]. These phenomena would be caused by
the similar mechanism to that operating in the 𝛽-planemodel. On the other hand, such the anisotropic inverse
cascade can be inhibited by other factors such as Ekmanfriction, which may cause narrow banded structures[5].

Figure 1. An example calculation for two-dimensional
forced turbulence in the 𝛽 plane. Zonal mean flows andvelocity component in the 𝑥 direction are shown. Lightand dark areas represent positive and negative velocity
components, respectively. Randomly forced many smallscale vortices (left) evolve to the zonal banded structure
after time integration (right).

Figure 2. Two-dimensional energy spectrum in an
example calculation of two-dimensional forced
turbulence in the 𝛽 plane. The left panel shows thespectrum for the state immediately after the start of time
integration. The center panel is the spectrum for the statewith the zonal banded structure. The right panel is theclose up around the origin of the center panel.
References[1] R.B. Rhines, J. Fluid Mech., 69, 3, 417 (1975)[2] G.K. Vallis et al., J. Phys. Oceanogr., 23, 1346 (1993)[3] K. Obuse et al., Phys. Fluids, 22, 056601 (2010)[4] S. Takehiro et al., 6nd Asia-Pacific Conference on
Plasma Physics (AAPPS-DPP), CD-I11 (2022)[5] S. Danilov et al., Phys. Rev. E, 65, 067301 (2002)