回転系での縞状構造の形成と消失について
竹広真一
京都大学数理解析研究所

2024 年 3 月 17 日

「GFD セミナー 2024 特別編」＠休暇村支笏湖
(RIMS, Kyoto Univ.)

Band structure in rotating systems

2024 年 3 月 17 日

1 / 30Outline

1Background2

β-plane model

3

Rotating sphere model4

Rapidly rotating thin spherical shell model (Boussinesq)

5Summary(RIMS, Kyoto Univ.)

Band structure in rotating systems

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2 / 30Outline

1Background2

β-plane model

3

Rotating sphere model4

Rapidly rotating thin spherical shell model (Boussinesq)

5Summary(RIMS, Kyoto Univ.)

Band structure in rotating systems

2024 年 3 月 17 日

3 / 30Surface flows of gas giant planets

Kaspi et al. (2020)

Surface flows of Jupiter and Saturn
Broad prograde jets around the equator (equatorial superrotation)
Nnarrow alternating jets in mid-/high-latitudes (band structure)

Origin of the surface jets
Convective motions in the “deep” region
Fluid motions in the “shallow” weather layer

Formation of band structure
Explanation by forced turbulence models of several rotating systems(RIMS, Kyoto Univ.)

Band structure in rotating systems

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4 / 30Zonal flow formation in rotating systems : outline
Forced turbulence models of several rotating systems
Initially, band structure emerges
But jets merge after long time
2D spectrum view : dumbbell feature

Important concepts
Potential vorticity (PV), β effect (spatial gradient of PV)
Anisotropic inverse cascade

Examples of rotating systems
2D β plane
2D rotating sphere
2D columnar vortices inside TC excited by thermal convection in a
rapidly rotating (thin) spherical shell(RIMS, Kyoto Univ.)

Band structure in rotating systems

2024 年 3 月 17 日

5 / 30Potential Vorticity (PV) conservation
Conservation of PV ∼ local angular momentum conservation
f +ζ∂+ u · ∇ q = 0, q =.
∂tH(f : vorticity of planetary rotation, ζ: vorticity of flows, H: height of fluid layer)

β effect : Variation of ambient PV f /H ⇒ vorticity of flow ζResult of conservation of PV : Rossby waves(RIMS, Kyoto Univ.)

Band structure in rotating systems

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6 / 30Outline

1

Background2

β-plane model

3Rotating sphere model4

Rapidly rotating thin spherical shell model (Boussinesq)

5Summary(RIMS, Kyoto Univ.)

Band structure in rotating systems

2024 年 3 月 17 日

7 / 30β plane model: Governing equations
2D barotopic vorticity equation∂ 2∂ψ∇ ψ + J(ψ, ∇2 ψ) + β= F − D,
∂t
∂x

u=−

∂ψ
∂ψ
, v=.
∂y∂x

ψ : stream function, F : small scale forcing, D : dissipation
Without forcing and dissipation ⇒ Hasegawa-Mima eq.

Conservation of PV∂q+ J(ψ, q) = F − D, q = ∇2 ψ + βy∂tOnly vertical component of vorticity affects the dynamics
f = 2Ω sin ϕ varies with latitude ϕ, β = df /dy = 2Ω cos ϕ/a(RIMS, Kyoto Univ.)

Band structure in rotating systems

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8 / 30β plane model : pioneering studies
Rhines (1975)
observation of band structure formation
order estimation of band width (Rhines scale)

Vallis and Maltrud (1993)
2D kinetic energy spectrum
Explanation of dumbbell shape spectrum

Stream function
Rhines (1975)(RIMS, Kyoto Univ.)

Time development of 2D kinetic spectrum(Vallis and Maltrud 1993)

Band structure in rotating systems

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9 / 30Rhines scale
Variation of f (and/or H) : β effect ⇒ emergence of zonal bandstructure
Conservation of PV : difficult to move in latitudinal direction⇒ Dominance of zonal flow, band structure
Rhines (1975) :advection of PV of flow (Jacobian) ∼ advection of environment PV (βterm)
U2∂ψ⇒ 2 ∼ βU
∂xLsU
Lβ =β|J(ψ, ∇2 ψ)| ∼ βRhines scale :U : Flow amplitude

Rhines scale is frequently used for estimation of band width
It can be derived by the homogeneity of PV.
ζ ∼ δf = βL ⇒(RIMS, Kyoto Univ.)U∼ βL
LBand structure in rotating systems

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β plane model : Numerical example
Snapshots of x-velocityt = 101t = 103t = 105

t = 5 × 106

Traditional knowledge and understanding
Small scale forcing, generation of small vortices⇒ larger vortices by N.L.T : inverse cascade⇒ inverse cascade stops at Rhines scale (dominance of β term)⇒ band structure is formed
Rhines scale ∼ expected band widthLonger time integration : jets merge, broader band structure appearsRhines scale is not good indicator for band width(RIMS, Kyoto Univ.)

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β plane model : example of 2D-spectrum

2D kinetic energy spectrum

t = 101t = 103t = 105
red : kE , blue : kR , green : kH

t = 5 × 106

Inhibit of fluid motion in “dumbbell shape region”
Anisotropic inverse cascade occurs around ky axis:⇒ transition to larger scale structures in y direction(RIMS, Kyoto Univ.)

Band structure in rotating systems

2024 年 3 月 17 日

12 / 30β plane model : 2D kinetic energy spectrum
Derivation of dumbbell like structure (Vallis and Maltrud, 1993)
βkx
Turbulent frequency ωt ∼ frequency of Rossby waves, ωR = − 2kx + ky2
T.F.= energy dissipation rate ε( 3 1/5kE ,x = kE cos8/5 θβkE =,εkE ,y = kE sin θ cos3/5 θT.F. = velocity scale U(rkR,xβ1kR ==,ULRkR,yT.F = vorticity scale Z(kH,xβ1=,kH =ZLZkH,y(RIMS, Kyoto Univ.)= kR cos3/2 θ= kR sin θ cos1/2 θ= kH cos2 θ= kH sin θ cos θBand structure in rotating systems(kE ,x , kE ,y )(kR,x , kR,y )(kH,x , kH,y )2024 年 3 月 17 日

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β plane model : summary

Homogeneous inverse cascade of small scale eddies
Inhibit of fluid motion in “dumbbell shape region”:⇒ homogeneous inverse cascade stops at Rhines scale
Anisotropic inverse cascade towards ky axis:⇒ band structure with multiple zonal jets appears
Anisotropic inverse cascade along ky axis:⇒ merger of thin multiple jets
⇒ eventually, a few jets with maximum width appear(RIMS, Kyoto Univ.)

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β plane model : Effects of friction
General dissipation term : D = −νn (−∇2 )n ζn = 1 : normal viscosity, n > 1 : hyper-viscosityn = 0 : Linear (Rayleigh,Ekman) friction, n < 0 : hypo-viscosity

Balance between energy input ε and dissipation:
εEk ∼.
2νn k 2nn ≥ 1 : Ek increases as k decreases ⇒ inverse cascade may not stopn ≤ 0 : Ek is constant or decreases as k decreases ⇒ inverse cascade
may stop

Spatial scale by dissipation:
1/2(2−n)ε,Lν =
νn β 2kν =νn β 2ε1/2(2−n),

For Ekman friction (n = 0) (Danilov and Guraeie, 2002, Vallis 2006),

1/4

1/4ε
ν0 β 2
LE =, kE =.
ν0 β 2ε(RIMS, Kyoto Univ.)

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Outline

1Background2β-plane model

3Rotating sphere model4

Rapidly rotating thin spherical shell model (Boussinesq)

5Summary(RIMS, Kyoto Univ.)

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Rotating sphere model : Governing equations
2D barotropic vorticity equation∂ζ12Ω ∂ ψ+ 2 J(ψ, ζ) += F − D,∂taa ∂λuλ = −∂ψ1 ∂ψ
, uϕ =.
∂ϕcos ϕ ∂λλ : longitude, ϕ : latitude, µ = sin ϕ.
ψ(λ, µ, t) : stream function, ζ(λ, µ, t) = ∇2 ψ : vorticity
J(ψ, ζ) = (∂λ ψ)(∂µ ζ) − (∂µ ψ)(∂λ ζ)
F : small scale forcing, D : dissipation

Conservation of PV∂q+ J(ψ, q) = F − D,∂tq = ∇2 ψ + 2Ω sin ϕ(RIMS, Kyoto Univ.)

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Rotating sphere model : pioneering studies
Williams (1978)
8-fold and equatorial symmetry
Random forcing is not isotropic

Nozawa and Yoden (1997)
Whole sphere, isotropic random
forcing
Various scale of random forcing androtation
Band widths consistent with Rhinesscale

NY1997 : stream function

Obuse et al. (2019)
Long time integration of Nozawa andYoden (1997)
Merging of zonal jets ⇒ Broader bandstructure appears(RIMS, Kyoto Univ.)

Band structure in rotating systems

Obuse et al. (2000)
final mean zonal flows
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Rotating sphere model : Numerical example
Time evolution of mean flow

Snapshots of azimuthal velocityt = 5 × 102

t = 5 × 103

t = 4 × 105

Jets merge, broader band structure appears
Seems to be a similar story with β plane(RIMS, Kyoto Univ.)

Band structure in rotating systems

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Rotating sphere model : 2D kinetic energy spectrum
Derivation of dumbbell like structure (Huang et al. 2001)
Turbulent frequency ωt ∼ frequency of Rossby waves,2Ωm
ωR = −n(n + 1)
T.F.= energy dissipation rate εmE (n) =ε1/3 5/3n (n + 1).
2Ω

T.F. = velocity scale UmR (n) =U 2n (n + 1)
2ΩaT.F = vorticity scale Z

mE , mR , mH

Z
mH (n) =n(n + 1).
2Ω(RIMS, Kyoto Univ.)

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Rotating sphere model : example of 2D-spectrum
2D kinetic energy spectrum

t = 103t = 105
red :mE , blue :mR , green:mHt = 5 × 106

Inhibit of fluid motion in “dumbbell shape region”
Anisotropic inverse cascade occurs around n axis:⇒ transition to larger scale structures in latitudinal direction(RIMS, Kyoto Univ.)

Band structure in rotating systems

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Outline

1

Background2

β-plane model

3

Rotating sphere model4

Rapidly rotating thin spherical shell model (Boussinesq)

5Summary(RIMS, Kyoto Univ.)

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Spherical shell model : outline
Rapidly rotating thin spherical shell model= 2D turbulence forced by thermal convection in a spherical shell
Outside TC (around equator, not focused on)Single equatorial prograde jet emerges outside TC

Inside TC (mid-/high latitudes)
Small scale convective motion is excited due to fast rotation
Convective motion induces 2D axial vortices
Multiple zonal alternating jets appears initially
After long time integration, jets merge, broader band structure appears(RIMS, Kyoto Univ.)

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Spherical shell model : Governing equations
Boussinesq fluid in a rotating spherical shell.
Scaling: the shell thickness, rotation period, temperature difference.
∇·u = 0,
RaE2 r∂u+ (u · ∇)u + 2k × u + ∇p − Ed (u ) =T,∂tPr ro∂TE+ (u · ∇)T =d(T ).
∂t
Prd (u ), d(T ) : Viscous/diffusion terms including hyper viscosity
depending on the total horizontal wavenumber
Parameters:Prandtl number: Pr = ν/καgo ∆TD 3
Rayleigh number: Ra =κννEkman number: E =ΩD 2
radius ratio: χ = ri /ro(RIMS, Kyoto Univ.)

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Rotating shell model : pioneering studies
Heimpel and Aurnou (2007)
Equatorial prograde jet, band structure in middle and high latitudes

longitudinal velocity

Mean zonal flows

1/8 sector domain : not whole shell! (resolution 128 × 512 × 64)
Short integration time : about 1600 rotation (0.02 viscous diffusiontime)(RIMS, Kyoto Univ.)

Band structure in rotating systems

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25 / 30Spherical shell model : Numerical example
Time evolution of mean flow

Snapshots of azimuthal velocityt = 103t = 104

t = 105

Time development of azimuthal velocity forχ = 0.9, Pr = 0.1, E = 3 × 10−6 , Ra∗ = 0.05.
Broader band structure appears in mid-/high-atitudes(RIMS, Kyoto Univ.)

Band structure in rotating systems

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Spherical shell model : Numerical example
surface lon.velocity

radial velocity

surface mean zonal flowt = 104

t = 105

Convective structure for χ = 0.9, Pr = 0.1, E = 3 × 10−6 ,Ra∗ = 0.05.
Broader band structure appears in mid-/high- latitudes(RIMS, Kyoto Univ.)

Band structure in rotating systems

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Outline

1Background2

β-plane model

3

Rotating sphere model4

Rapidly rotating thin spherical shell model (Boussinesq)

5

Summary(RIMS, Kyoto Univ.)

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Summary

Forced turbulence models of several rotating systems(β-plane, rotating sphere, rapidly rotating spherical shell)
Initially, band structure emerges
But jets merge after a long time

Rhines scale : not valid for band width estimation
Scale at homogeneous inverse cascade stops

2D spectral view : dumbbell structure
Inhibit of fluid motion in “dumbbell shape region”
Anisotropic inverse cascade progress slowly
Single band with largest width will eventually appear

Hypo-viscosity may prevent jets from merging(RIMS, Kyoto Univ.)

Band structure in rotating systems

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References I
Danilov, S., Gurarie, D., 2002 : Rhines scale and spectra of the beta-planeturbulence with bottom drag. Phys Rev. E, 65, 067301.
Huang, H.-P., Galperin, B., Sukoriansky, S., 2001 : Anisotropic spectra in
two-dimensional turbulence on the surface of a rotating sphere. Phys. Fluids, 13,
225–240.
Kaspi, Y., Galanti, E.,Showman, A. P., Stevenson, D. J., Guillot, T., Iess, L., ScottJ. B., 2020 : Comparison of the Deep Atmospheric Dynamics of Jupiter and Saturnin Light of the Juno and Cassini Gravity Measurements, Space Science Reviews,216, 84.
Rhines, R. B,. 1975 : Waves and turbulence on a beta-plane. J. Fluid Mech., 69,417–443.
Vallis, G. K., Maltrud, M. E., 1993 : Generation of mean flows and jets on a beta
plane and over topography. J. Phys. Oceanogr., 23, 1346–1362.
Vallis, G. P. : 2006, Atmospheric and Oceanic Fluid Dynamics. Cambridge
University Press, Cambridge, pp.745.
(RIMS, Kyoto Univ.)

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