Continuous spin-up and dynamo in a precessing sphere
Yuki Koike,1 Masaki Shimizu,1 Shigeo Kida,2 Genta Kawahara,1 Susumu Goto 1
1Osaka University, Graduate School of Engineering Science, Toyonaka 560-8531 Japan
2Doshisha University, Kyotanabe, 610-0394 Japan

1 INTRODUCTION
Flows in a precessing cavity such as a sphere, spheroid
and cylinder have been studied extensively since theseminal laboratory experiment by Malkus (1968) andthe analytical theory by Busse (1968). However, despite of the simple motion of cavity, ﬂows in a precessing cavity are too complex to be fully understood on
the basis of analytical theories or experiments. Hence,numerical simulations have been playing an importantrole to investigate characteristics of ﬂows in a precessing
cavity.
Recently, the numerical simulation by Kida andShimizu [1] discovered a turbulent ring in a precessing
sphere, along which strong vorticity as well as magnetic ﬂux are generated. Recall that quite a few geophysicists are interested in the dynamo action in theprecessing cavity, since the spin axis of the Earth isprecessing slowly. In the present study, we investigate
ﬂow structures at lower Reynolds numbers and reveal
the mechanism to create the ring.

2 GOVERNING EQUATIONS
We consider the MHD dynamo driven by incompressive ﬂows in a precessing sphere with the magnitudeof the spin and precession angular velocities being constant in time and two axes being orthogonal. The evolution equations in the sphere for the ﬂuid velocity u(r, t)
and the magnetic ﬂux density b(r, t) may be written in
the precession frame (x, y, z) which is rotating with theprecession angular velocity Ωp = Ωp ẑ as∇ · u = 0,

∇ · b = 0,∂u= u × (∇ × u) − 2Γ ẑ × u∂t
1 2− ∇P − b × (∇ × b) +
∇ u,
Re∂b
1= ∇ × (u × b) +
∇2 b,
∂t
Rem
where P is the modiﬁed pressure including the centrifugal force potential, Γ = Ωp /Ωs the Poincare number,
Re = a2 Ωs /ν the Reynolds number, Rem = a2 Ωs µσ
the Magnetic Reynolds number, Ωs the spin angularvelocity taken in the x direction, a the sphere radius,ν the kinematic viscosity, µ the magnetic permeabilityand σ the electrical conductivity of ﬂuid. The length
has been normalized by a, the time by 1/Ωs , and the√magnetic ﬂux density by ρµaΩs . The outside of thesphere is assumed to be vacuum, where the magnetic
ﬂux density b(o) obeys ∇ · b(o) = 0 and ∇ × b(o) = 0.
These equations are supplemented by (on r = 1),u = x̂ × r,b = b(o)(on r = 1)

which are the boundary conditions derived from the assumptions that the ﬂow is non-slip on the boundary
and that the magnetic permeability of the ﬂuid is equal
to that of vacuum. We also assume that b(o) is zeroat inﬁnity. Here note that the control parameters of
this system are the Poincare number Γ , the Reynolds
number Re and the Magnetic Reynolds number Rem .

3 NUMERICAL METHOD
We solve the above set of equations numerically by
spectral method. The velocity and magnetic ﬁelds areexpressed by the toroidal and poroidal scalar functions
U , W , B and J as u = ∇ × (∇ × rU ) + ∇ × rW andb = ∇ × (∇ × rB) + ∇ × rJ. These scalar functions
are expanded by the Zernike spherical polynomials andspherical harmonics; for example,U (r, t) =M∑L∑

m=−M l=|m|N∑Unlm (t)Φln (r)Ylm (ϕ, θ)n=|l|n+l=even

where Φln (r) is the Zernike polynomial and Ylm (ϕ, θ)is the spherical harmonics, (M, L, N ) is the number oftruncation modes, and (r, θ, ϕ) is the spherical polar coordinate with θ being the polar angle from the z-axis
and ϕ the azimuthal angle from the x-axis. The evolution equations for the expansion coeﬃcients Unlm andthe corresponding ones for W , B and J are derived
and integrated numerically by the Adams-Bashforthand Crank-Nicolson schemes. We use (M, L, N ) =(63, 63, 126)

4 RESULTS
Flow states are shown by symbols in ﬁgure 1 for thesimulated combinations of Re and Rem . Here we ﬁx
Γ = 0.1, since it has been experimentally shown thatthe ﬂow becomes unsteady at relatively low Re whenthe Poincare number Γ is around 0.1.[2] It is interesting to observe that in all the cases of these simulation
parameters there exists a high speed stream runningapproximately along the equator, and that the boundary layer near the wall is swelled into the interior along
the ring. This high speed ring is shown in ﬁgure 2by plotting high vorticity regions for a laminar case at
Re=1500 and a turbulent case at Re = 10000. Theserings are inclined slightly from the spin axis, and theangle of inclination seems independent of the Reynolds
number. The axially averaged velocity, ⟨u⟩ϕ , is shownin the right panel of ﬁgure 3 for Re = 1500. Note that
this velocity ﬁeld seems quite similar to that of spin-up.
Although it is diﬃcult, due to the nonlinearity, to describe analytically this velocity ﬁeld, the formation of
the ring structure may be understood qualitatively as
follows. Because the spin axis rotates about the precessing axis, ﬂuid inside the sphere continuously tries
to approach the solid-body rotation which is suitable
to the boundary condition at the instance, but which
is never established. It takes the duration of the orderof 1/Ωs to form the boundary layer and to start thespin-up. During this period, the ring with high angularvelocity inclines at an angle of the order Ωp /Ωs = Γ . It
becomes unstable as Re number increases and the turbulent regions with high-activity of vorticity and magnetic ﬂux are created along it.

Fig. 1: Flow states at Γ = 0.1.

Fig. 2: Ring structure. The region of large vorticity
is shown by iso-surface. Left: Re = 1500. Right: Re =10000.

xrrFig. 3: Spin-up. Left: Velocity ﬁeld during spin-up
from rest at Re ∼ 5000. Right: Axial averaged velocity
at Re = 1500 and
Rem = 8000. In-plane components are represented by allows and the perpendicularcomponent by gray scale.

5 CONCLUSION
Solving the MHD equations by the spectral method,
we have conducted highly precise numerical simulationsof ﬂuid motions and MHD dynamos in a precessing
sphere. In our simulations for the Poincare number
ﬁxed at 0.1, a ring-like high-speed stream is always observed irrespective of the Reynolds numbers. At higher
Reynolds numbers, turbulent vortical structures andmagnetic ﬂux are produced along the ring (see the rightpanel of ﬁgure 2), whereas this ring is also observed inlaminar ﬂows at lower Reynolds numbers (see the left
panel of ﬁgure 2). These observations suggest that sucha laminar solution is likely to be embedded in the turbulence, and may well be key ingredients to understandthe mechanism to sustain the strong turbulence and theMHD dynamo. In addition, it has been shown, in thepresent study, that the ring-like structure is created bya continuous spin-up process.

[References][1] Kida, S. and Shimizu, S., 13th European Turbulence
Conference, 268, (2011)[2] Goto, S. et al., Phys. Fluids, 19,061705 (2007)