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Vortexes in three dimensional systems are an emerging topic of study in the realm of quantum physics, particle physics and magnetism. In this study we describe a new Hamiltonian, and solve it numerically for the description of vortices in a spherical model subjected to an electromagnetic field. The numerical analysis shows also that supersymmetric Hamiltonian describes quantized orbitals for atomic systems with vorticity included, which is to our knowledge novel and also describes electricity as composed of quasi-symmetric vortex bundles. Further work on analytical solutions is under development. The 3D solutions of the Hamiltonian are relevant for nuclear physics, in the physics of elementary particles, in geophysics, and in the physics of stellar bodies.
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Formation of quantum vortices in a rotating
sphere in an electromagnetic field.
Sergio Manzetti 1,2and Alexander Trounev 1,3
1. Fjordforsk A/S, Bygdavegen 155, 6894 Vangsnes, Norway.
2. Uppsala University, BMC, Dept Mol. Cell Biol, Box 596,
SE-75124 Uppsala, Sweden.
3. A & E Trounev IT Consulting, Toronto, Canada.
December 17, 2018
1
1 Abstract
Vortexes in three dimensional systems are an emerging topic of study in the
realm of quantum physics, particle physics and magnetism. In this study we de-
scribe a new Hamiltonian, and solve it numerically for the description of vortices
in a spherical model subjected to an electromagnetic field. The numerical anal-
ysis shows also that supersymmetric Hamiltonian describes quantized orbitals
for atomic systems with vorticity included, which is to our knowledge novel and
also describes electricity as composed of quasi-symmetric vortex bundles. Fur-
ther work on analytical solutions is under development. The 3D solutions of
1Please cite as: Sergio Manzetti and Alexander Trounev. (2019) ”Formation of quan-
tum vortices in a rotating sphere in an electromagnetic field.” In: Modeling of quantum
vorticity and turbulence with applications to quantum computations and the quantum Hall
effect. Report no. 142019. Copyright Fjordforsk A/S Publications. Vangsnes, Norway.
www.fjordforsk.no
1
the Hamiltonian are relevant for nuclear physics, in the physics of elementary
particles, in geophysics, and in the physics of stellar bodies.
2 Introduction
Three-dimensional vortex systems are used to model several phenomena in
physics, ranging from modeling sunspot-dynamics [1, 2, 3], cyclones and anti-
cyclones in planetary systems [2, 4, 5, 6], vortex formation in fluids and vortex
formation in quantum fluids and superfluids [7, 8]. Additionally, vortex mod-
els were the first to be proposed to describe the forces which hold the nuclear
subparticles together [9] and were later published and accepted as a ”center
vortices” [10], nevertheless not empirically demonstrated. The vortices oc-
curring at a subatomic level are defined as magnetic vortices or vortex fluxes
which are modelled by a Wilson loop, also known as the Wilson loop average
in nonperturbative quantum-chromodynamics [11]. In exonucelar systems, the
Gross-Pitaevskii equation is widely used to model the magnetic vortex behaviour
in Fermi holes and in superfluids [7, 8, 12, 13], where some studies provide also
exact solutions to the superfluid vortices [14]. In one study in particular [8]
the authors model bose-eisntein condensates (BECs) with induced vorticity, by
applying a moving object with an elliptical paddle shape to create the vortices.
The authors use the Gross-Pitaevskii equation and generate a spherical sys-
tem with vortices formed on the surface boundary. The authors generate hence
long-lived vortices which are randomly distributed with turbulence [8]. In an-
other study, Bradley and Anderson [15] model the Gross-Pitaevskii equation
numerically and introduce the concept of a clustered fraction for vortices, which
describes that vortices of the same sign cluster as nearest neighbours, which
has significant implications for vortex condensation and annihilation. We also
found of particular relevance the results of Chesler et al [16], who modelled
vortex dynamics in a 2D system by using holographic duality as a gravitational
description for the vortex-system where vortex condensation and annihilation
was described numerically. Chesler et al reported that the intervortex spacing
2
remains in the inertial range and vortices avoided condensation. In view of
these important results, we present a new model Hamiltonian which is solved
for numerical solutions in a spherical system, yielding a 3D description of the
formation of vortices on the surface of a sphere. This model differs from pre-
vious models in that it allows for the formation of vortices without symmetry
breaking, and includes both vortex-condensation as well as fixed intervortex
spacing in respect to specific energy levels [17, 18]. It is also, to our knowl-
edge, the first model of vortex formation on a spherical system, which describes
the dynamics of vortex formation in the same fashion as for two-dimensional
systems, as previously published [17, 18], regulated by an internal dipole field
located at the center of the rotating sphere. This model can be applied to any
level of physics, where vortices occur in a spherical regime, with a modulated
center-dipole field.
2.1 Supersymmetric Hamiltonian for vortex formation
We present here the derivation of the supersymmetric operator for vortex sys-
tems in a sphere or in a 2D system. We develop the operator from the model
published by Laughlin [19], who defined the Hamiltonian operator for anyons
as
H=|~/i∇ − (e/c)~
A|2,(1)
where ~
A=1
2H0[xˆyyˆx], is the symmetric gauge vector potential, where
H0is the magnetic field intensity. However, presenting this treatise in a first
quantized formalism, we use the regular form:
H= [~/i∇ − (e/c)~
A]2,(2)
where the Hamiltonian is simply the normal square of the covariant deriva-
3
tive. At this point, looking at the factorized form of (2):
H= [~/i∇ − (e/c)~
A]×[~/i∇ − (e/c)~
A].(3)
we apply supersymmetry rules [20] on the two factorized components in eqn.
(3), and get the superpair of the Laughlin Hamiltonian:
HSU SY = [~/i∇ − (e/c)~
A]×[~/i∇ − (e/c)~
A].(4)
The SUSY counter-part of the Hamiltonian from (2) becomes therefore:
HSU SY =T T ,(5)
with
T= [~/i∇ − (e/c)~
A],(6)
and
T= [~/i∇ − (e/c)~
A],(7)
where Tand Tare the superpair in the SUSY Hamiltonian in (4) and one
anothers complex conjugate.
We consider then Tand T, with γ= (e/c)~
A, in one dimension:
T=h
i
d
dx γ·˜
I
T=h
i
d
dx γ·˜
I.
which commute by the relation:
[T T ][TT] = 0 (8)
4
making Tand Ttwo commutating complex operators in Hilbert space H.
Both Tand Tare unbounded operators given the condition:
||T x|| c||x||,(9)
Tand Tare also non-linear given the γconstant, being a constant trans-
lation from the origin. Tand Tare non-self-adjoint in Has the following
condition is not satisfied:
hT φ, ψi=hφ, Tψi,(10)
From this, it follows that:
HSU SY :D(HS U SY )H,
and
D(HSU SY )H,
where HSU SY is an unbounded linear and non-self-adjoint operator complex
on Hilbert space H=L2[−∞,+]. However, HSUS Y is self-adjoint in its
domain, D(HSU SY ) on L2[a, b].
From supersymmetry theory in quantum mechanics [20] it follows that
HΨ = HΨ=EΨ = EΨ, therefore we can assume that:
HSU SY Ψ = EΨ,(11)
where E is the energy of the system, which generates the boundaries of
D(HSU SY ) on L2[0, L], where the interval of quantization (L) is contained in
the zero-point energy term E= n2~2π2
2mL2.
5
2.2 Numerical study of vorticity dynamics in a nonlinear
quantum system in an electromagnetic field
In Ref. [17, 18], we investigated the formation of vortices in a homogeneous
and alternating magnetic field within the framework of the modified Gross-
Pitaevskii model. To derive the basic equation, we used the supersymmetric
Hamiltonian in (4), which is a generalization of the well-known Hamiltonian
proposed by Laughlin [19] to describe the quantum fractional Hall effect. In
the present paper, we investigate the case of the 3D nonlinear quantum system in
an electromagnetic field. The corresponding equation in dimensionless variables
has the form:
∂ψ
∂t =1
22ψ+iΩ(x∂ψ
∂y y ψ
∂x )β|ψ|2ψ+ ( ~
A)2ψ(12)
Here ~
Ais a dimensionless vector potential, β, Ω - parameters of the model de-
scribing the number of particles and the angular momentum, respectively. Eq
(12) describes evolution of the wave function from some initial state ψ(x, y, z, 0) =
ψ0(x, y, z) and up to the state describing a certain number of vortices, which
depends mainly on the angular velocity Ω. In the case of a homogeneous mag-
netic field with ~
B=B0(sin θ, 0, cosθ), we set ~
A= [ ~
B~r]/2. In the case of a
magnetic dipole ~m =m0(sin θ, 0, cosθ) and the electric charge qcombined with
it, we assume scalar potential ϕ= 0, and the vector potential
~
A(x, y, z, t) = [~m~r]
r3qt~r
r3(13)
For the equation (12) we consider the problem of the decay of the initial
state, which we set in the form
ψ(x, y, z, 0) = exp[(x2+y2+z2)/2] (14)
As boundary conditions, we will use the function of the initial state (14) given
on the boundaries of the computational domain. We note that for sufficiently
large dimensions of the domain this is equivalent to zero boundary conditions.
The amplitude of the wave function is shown in Fig 1-10, at different states
depending on parameters β, , B0, m0, q, θ, and at a fixed linear size of the region
6
x2+y2+z216. Fig. 1 shows the simulation data of the amplitude of the wave
function in cross section z= 0 at different times in a homogeneous magnetic
field with B0= 22, θ = 0, and for β= 100,Ω = 2. In this case, vorticity is
formed, and in Fig. 1.1, one can see the vorticity distribution in different cross
sections z= 0,0.5,1,1.5,2,2.5,3,3.5, and in Fig. 1.2 shows the distribution
of vorticity in the volume of the sphere. We note that the final state is not
symmetric, which is due to the presence of a system of vortices interacting in
the volume of the sphere. Vortices begin and end on the surface of the sphere,
some of them have a horseshoe shape, others are stretched from one hemisphere
to another.
.
Figure 1.The amplitude of the wave function at different instants of time cal-
culated for z= 0 with Ω = 2, β = 100, B0= 22, θ = 0.
7
.
Figure 1.1The amplitude of the wave function at different cross sections z=
0,0.5,1,1.5,2,2.5,3,3.5 calculated for t= 1 with Ω = 2, β = 100, B0= 22, θ =
0.
8
.
Figure 1.2The vorticity distribution in the volume of the sphere computed for
t= 1 with Ω = 2, β = 100, B0= 22, θ = 0.
It should be noted that vorticity is formed even at A= 0 - Fig. 2. However,
even if A2>0, then the main parameter influencing the vorticity is Ω. Fig.
2 shows the vorticity distributions in the volume of the sphere computed for
various Ω = 1.8,2,3 at t= 1 with β= 100, B0= 0,22, θ = 0.In the case of
Ω=1.8,2, the vorticity is distributed throughout the volume, whereas at Ω = 3
the vortices are concentrated near the axis of rotation.
9
.
Figure 2 The vorticity distribution in the volume of the sphere computed for
t= 1 with Ω = 1.8,2,3; β= 100; B0= 0,22; θ= 0.
Part of the calculations were performed for the modified model (12), in which
the sign of the term A2ψchanges to the opposite. In this case we have the Gross-
Pitaevsky model, in which vorticity under these conditions is not formed - Fig.
3-4. In Fig. 3 shows the distribution of the amplitude of the wave function
in the modified model (above) and in the model (12) - below, calculated for
t= 1 with parameters Ω = 1.8, β = 100, B0= 22; θ= 0. Fig. 4 shows the
10
distribution of the amplitude of the wave function calculated for t= 1 for the
values of the parameters Ω = 1.8, β = 100, B0= 22; θ=π/2 (in this case the
magnetic field is oriented perpendicular to the axis of rotation). From these
data it follows that in the modified model the wave function has the usual form
for quantum mechanics, while in the model (12) vortices are formed, distributed
over the volume of the sphere.
.
Figure 3.The amplitude of the wave function computed for t= 1 with Ω =
1.8, β = 100, B0= 22, θ = 0 in the modified model (above) and in the model
(12) - below.
11
.
Figure 4.The amplitude of the wave function computed for t= 1 with Ω =
1.8, β = 100, B0= 22, θ =π/2 in the modified model (above) and in the model
(12) - below.
In the case when the magnetic field perpendicular to the axis of rotation,
vortices appear that have the form of a spindle - Fig. 4. Fig. 4.1 shows the
simulation data of the amplitude of the wave function in cross section z= 0
12
at different times in a homogeneous magnetic field with B0=22, θ =π/2,
and for β= 100,Ω = 2. In this case, vorticity is formed (while in the modified
model is not formed), and in Fig. 4.2, one can see the vorticity distribution in
different cross sections.
.
Figure 4.1The amplitude of the wave function at different instants of time cal-
culated for z= 0 with Ω = 1.8, β = 100, B0= 22, θ =π/2.
.
Figure 4.2The amplitude of the wave function at different cross sections z=
0,0.5,1,1.5,2,2.5,3,3.5 calculated for t= 1 with Ω = 1.8, β = 100, B0=
22, θ =π/2.
13
The change in the orientation of the magnetic field relative to the axis of
rotation affects significantly the vorticity distribution. In Fig. 5 shows the
amplitude of the wave function in the cross section z= 0 at different times
computed for Ω = 2, β = 100, B0= 22, θ =π/4. In Fig. 5.1 and 5.2 are the
vorticity distributions at the time t= 1 in different sections and in the volume
of the sphere.
.
Figure 5.The amplitude of the wave function at different instants of time cal-
culated for z= 0 with Ω = 2, β = 100, B0= 22, θ =π/4.
.
14
Figure 5.1The amplitude of the wave function at different cross sections z=
0,0.5,1,1.5,2,2.5,3,3.5 calculated for t= 1 with Ω = 2, β = 100, B0= 22, θ =
π/4.
.
Figure 5.2The vorticity distribution in the volume of the sphere computed for
t= 1 with Ω = 2, β = 100, B0= 22, θ =π/4.
2.3 Vorticity in a charged field
The process of vorticity formation in the dipole and charge field (Fig. 6-10)
differs substantially from the analogous process in a homogeneous magnetic
15
field (Fig. 1-5). In the magnetic dipole field a dense core is formed around
which a finite number of vortices are distributed - Fig 6-7. In Figs. 6 and 6.1
show the distribution of vorticity around the central nucleus in different cross
sections and in the volume of the sphere, calculated for t= 1 with parameters
Ω=2, β = 100, m0= 1, θ = 0, q = 0. In Fig. 6.2 shows the process of formation
of the central core and vortices in the cross section z= 0.
.
Figure 6 The amplitude of the wave function at different cross sections z=
0,0.5,1,1.5,2,2.5,3,3.5 calculated for t= 1 with Ω = 2, β = 100, m0= 1, θ =
0, q = 0.
16
.
Figure 6.1The vorticity distribution in the volume of the sphere computed for
t= 1 with Ω = 2, β = 100, m0= 1, θ = 0, q = 0.
17
.
Figure 6.1The amplitude of the wave function at different tin cross sections
z= 0 calculated with Ω = 2, β = 100, m0= 1, θ = 0, q = 0.
As the intensity of the magnetic dipole increases, the density of the central
nucleus increases (Fig. 7). Changing the orientation of the axis of the magnetic
dipole relative to the rotation axis also affects the vorticity distribution. In Fig.
7 and 7.1 the vorticity distribution in different sections and in the volume of
the sphere is shown, and in Fig. 7.2 shows the formation of a dense core and
vortices in the cross section z= 0 at Ω = 2, β = 100, m0= 12, θ =π/6, q = 0.
18
.
Figure 7 The amplitude of the wave function at different cross sections z=
0,0.5,1,1.5,2,2.5,3,3.5 calculated for t= 1 with Ω = 2, β = 100, m0= 12, θ =
π/6, q = 0.
19
.
Figure 7.1The vorticity distribution in the volume of the sphere computed for
t= 1 with Ω = 2, β = 100, m0= 12, θ =π/6, q = 0.
20
.
Figure 7.2The amplitude of the wave function at different tin cross sections
z= 0 calculated with Ω = 2, β = 100, m0= 12, θ =π/6, q = 0.
In the field of the magnetic dipole and electric charge, a dense core is formed
on which vortices are closed (Fig. 8-9). In Fig. 8-9 shows the distribution of
vortices around a dense nucleus calculated at t= 1 with parameters Ω = 2, β =
100, m0= 7, θ = 0, q =±5. Vortices in this case have a pronounced funnel-
shaped form. In Fig. 8.1 and 9.1 shows the distribution of vortices in the cross
section z= 0, calculated at t= 1 with parameters Ω = 2, β = 100, m0= 7, θ =
0, q =±5. From the data in Fig. 8.2 and 9.2 it can be seen that the process of
forming a dense core and vortices occurs cyclically with a certain period.
21
.
Figure 8 The vorticity distribution around a dense nucleus computed for t= 1
with Ω = 2, β = 100, m0= 7, θ = 0, q =5.
.
Figure 8.1The amplitude of the wave function at different cross sections cal-
culated for t= 1 with Ω = 2, β = 100, m0= 7, θ = 0, q =5.
22
.
Figure 8.2The amplitude of the wave function at different tin cross sections
z= 0 calculated with Ω = 2, β = 100, m0= 7, θ = 0, q =5.
23
.
Figure 9 The vorticity distribution around a dense nucleus computed for t= 1
with Ω = 2, β = 100, m0= 7, θ = 0, q = 5.
.
Figure 9.1The amplitude of the wave function at different cross sections cal-
culated for t= 1 with Ω = 2, β = 100, m0= 7, θ = 0, q = 5.
24
.
Figure 9.2The amplitude of the wave function at different tin cross sections
z= 0 calculated with Ω = 2, β = 100, m0= 7, θ = 0, q = 5.
The electric charge also forms a dense core (Figure 10), around which vortices
are distributed (Figure 10.1), and the process of formation of the nucleus and
vortices occurs cyclically (Figure 10.2). In Fig. 10 shows the distribution of
vortices around a dense nucleus in the field of a point charge computed at t= 1
with parameters Ω = 2, β = 100, m0= 0, θ = 0, q = 12. In Fig. 10.1 and 10.2
show the amplitude of the wave function in different sections and at various
times calculated with the parameters as in Fig. 10.
25
.
Figure 9 The vorticity distribution around a dense nucleus computed for t= 1
with Ω = 2, β = 100, m0= 0, θ = 0, q = 12.
.
Figure 10.1The amplitude of the wave function at different cross sections cal-
culated for t= 1 with Ω = 2, β = 100, m0= 0, θ = 0, q = 12.
26
.
Figure 10.2The amplitude of the wave function at different tin cross sections
z= 0 calculated with Ω = 2, β = 100, m0= 0, θ = 0, q = 12.
3 Discussion
By fig 4 we see preserved structural similarities between the p-orbitals in quan-
tum mechanics for atomic systems, and the same orbitals with vorticity formed
around their nodular shapes. Our model describes hence the magnetic field
formed around the orbitals of quantized systems, also accounting for the s-orbital
displayed in fig 3. Here also, the ground state is reproduced with and without
vorticity, showing how the vortex forms in the center of the spherical structure
of the s-orbital. This shows a validity of our model in describing quantized
electronic systems, and also potential subatomic magnetic behaviours. From fig
10.1 we observe another intriguing point of the SUSY Hamiltonian, where an
electric field also generates vortices, as magnetic fields do. This is to our knowl-
27
edge new, and normally vorticity is attributed only to magnetic fields. The
similarities between the vortic behaviour of electric field (Fig 10) and magnetic
field (Fig 6, 7) are such that the electric field vortexes are more structurally
order in quasi-symmetric ensembles than are magnetic field vortices. The impli-
cations of the model are also significant for describing subatomic forces, where
the evolution of the wavefunction displays a pulsating character, expanding and
contracting periodically.
4 Conclusions
From the results we see that the spherical solutions to the SUSY Hamiltonian
are good models for atomic and nuclear magnetic field structures. In particular,
it is the evolution of the wavefunction and the vorticity patterns which arouse
interest. The evolution of the wavefunction of vortices suggests that magnetic
fields in spherical systems pulsate from the center to the boundaries of the
sphere in harmonical fashion. The results are also showing that the model
Hamiltonian can be used to describe the nature of the magnetic field vortexes
around electronic orbitals, which is a significant addition to orbital physics and
quantum physics.
28
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... Another possible mechanism is the excitation of an electron plasma in the form of vortices [52][53][54][55][56][57]. Such excitation exists both in the earth [55][56] and in interplanetary space [57]. To excite vortices, a combination of three factors is required: the presence of a magnetic field, the rotation of the system, and a sufficiently low temperature not exceeding the Fermi temperature of the degenerate electron gas. ...
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We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross-Pitaevskii (GP) equation in dimension N greater than or equal to 3. We also extend the asymptotic analysis of the free field Ginzburg-Landau equation to a larger class of equations, including the Ginzburg-Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if N = 3).
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Vortices and quasiparticles in 2D systems are studied for the last decades in relation to the development of technologies in quantum electromagnetics, optics, and quantum computation. Hamiltonians are a fundamental part for studying quasiparticles and vortices, and provide models to calculate the eigenvalues and eigenfunctions that describe a real physical state. By devising supersymmetry, a Hamiltonian for the description of vortices is developed forming in an 2D electron gas and study the numerical solutions under the effects of an alternating electromagnetic field. The numerical analysis shows that vorticity is formed spontaneously without symmetry‐breaking and vortices arise from the boundaries and converge toward the center of the system in a similar fashion to natural hydrodynamic phenomena in water or plasma. Additionally, the equation under damping conditions is studied, a homogenous magnetic field and under the absence of an electromagnetic field. The results and the study of the parameters indicate that the supersymmetic wave equation (SWE) may be a good model equation to describe vorticity for quantum electromagnetics, hydrodynamics, and other physical phenomena in the realm of physical and quantum physical sciences. A supersymmetric wave‐equation (SWE) is developed for the analysis of vortex dynamics in 2D quantum systems. The supersymmetry‐derived wave‐equation describes vortex formation without symmetry breaking. Numerical analysis of the vortex formation is carried out under damped conditions and under a variating electromagnetic field and an electric field. Results show that the SWE describes quantized vortex systems, and can be applied to large‐ and small‐ scale vorticity phenomena.
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A strong cyclonic vortex has been observed on each of Saturn's poles, coincident with a local maximum in observed tropospheric temperature(1-3). Neptune also exhibits a relatively warm, although much more transient(4), region on its south pol. Whether similar features exist on Jupiter will be resolved by the 2016 Juno mission. Energetic, small-scale storm-like features that originate from the water-cloud level or lower have been observed on each of the giant planets and attributed to moist convection, suggesting that these storms play a significant role in global heat transfer from the hot interior to space. Nevertheless, the creation and maintenance of Saturn's polar vortices, and their presence or absence on the other giant planets, are not understood. Here we use simulations with a shallow-water model to show that storm generation, driven by moist convection, can create a strong polar cyclone throughout the depth of a planet's troposphere. We find that the type of shallow polar flow that occurs on a giant planet can be described by the size ratio of small eddies to the planetary radius and the energy density of its atmosphere due to latent heating from moist convection. We suggest that the observed difference in these parameters between Saturn and Jupiter may preclude a jovian polar cyclone.
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We construct exact solutions of the Gross-Pitaevskii equation for solitary vortices, and approximate ones for fundamental solitons, in two-dimensional models of Bose-Einstein condensates with a spatially modulated nonlinearity of either sign and a harmonic trapping potential. The number of vortex-soliton (VS) modes is determined by the discrete energy spectrum of a related linear Schrödinger equation. The VS families in the system with the attractive and repulsive nonlinearity are mutually complementary. Stable VSs with vorticity S⩾2 and those corresponding to higher-order radial states are reported, in the case of the attraction and repulsion, respectively.
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Jupiter's Great Red Spot is viewed as a vortex that arises naturally from the equations of motion of the jovian atmosphere. Here I solve numerically the equations governing fluid motion in a model of the jovian atmosphere for a variety of initial conditions. Large spots of vorticity form spontaneously in chaotic azimuthal flows and are stable if the vorticity of the spots has the same sign as the shear of the surrounding azimuthal flow. The Great Red Spot is compared with these solutions and a new prediction of its vertical structure is made.
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