Modelling the formation of atmospheric vortices
on planet earth using a supersymmetric operator.
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.
March 14, 2019
Main point 1: A supersymmetric Hamiltonian for electromagnetic vortices is
used to describe vorticity on planet earth.
Main point 2: A Master equation is generated which describes the phenomenon
of vortex formation in the atmosphere in a 3D system.
Main point 3 : Interplay between electromagnetic ﬁeld of the earth and the
stratiﬁed gravity yields vorticity.
Hamiltonian; magnetism; planetary; gravity; supersymmetry; numerical; anal-
Macrophysical phenomena, such as turbulence, vorticity are normally mod-
elled using ﬂuid-mechanical diﬀerential equations and geodesics systems. Other
methods may also be of importance to the meteorology community, such as
quantum physical operators. In this study we use a novel Hamiltonian and
study the vortex formation in the atmosphere of planet Earth under the eﬀects
of the gravity density stratiﬁcation and the electromagnetic ﬁeld of the planet.
The results propose that vorticity in the atmosphere (high and low pressure sys-
tems) is driven in major part by the interplay between the earths magnetic ﬁeld
and gravity density. The results show that the quantized behaviour of atmo-
spheric vortices lies in their dominant occurrence on the northern and southern
hemispheres. The use of quantum mechanical operators in modelling planetary
vorticity reveals also that these vortices arise from the core of the planet and
manifest in a most pronounced manner on the surface of the earth where gravity
density is experiencing an abrupt phase change. Further research is made on
combining this model with earths atmospheric parameters, such as ocean tem-
peratures and circulation, terrestrial oscillation and the sun’s magnetic ﬁeld.
The results are important for future developments of climate and weather pre-
1Please cite as: Sergio Manzetti and Alexander Trounev. (2019) ”Modelling the formation
of atmospheric vortices on planet earth using a supersymmetric operator.” In: Modeling of
quantum vorticity and turbulence in two- and three-dimensional systems. Report no. 142021.
Copyright Fjordforsk A/S Publications. Vangsnes, Norway. www.fjordforsk.no
Hamiltonians are mathematical operators used for modelling various phenomena
in physics, ranging from quantum physics and quantum ﬁeld theory to relativis-
tic physics [1, 2, 3, 4]. The advantage of Hamiltonian models over classical
physics models is the derivation of eigenvalues and eigenfunctions that describe
the wave-behaviour of energy, such as in wind waves, ocean waves, acoustic
waves as well as electromagnetic waves. Gravity is also a phenomenon that is
known to occur as waves [5, 6, 7, 8, 9, 10], and describing new models for its
eﬀects on its domain of objects (planets, stars, galaxies etc) can increase the
knowledge of how it shapes and aﬀects planets and its components, solar systems
and larger ensembles of objects. In this study, we are interested in modelling
the eﬀects that gravity has on the atmosphere, by using a model Hamiltonian
derived by applying supersymmetry rules on a Hamiltonian for free particles
rotating around a center [11, 12, 13].
2.1 Gravity and the atmosphere
Recent studies suggest that gravity aﬀects the atmosphere and its dynamics
[14, 15, 16, 17]. The ﬂuctuating character of gravity  and its continuously
oscillating nature  suggest that the intricate behaviour of atmosphere, be-
having as a ﬂuid, can be modelled using model Hamiltonians which consider also
the gravity ﬂuctuations. The conventional models to model the atmosphere ori-
gin from the ﬁrst equations published by Edward Lorenz  in the early 1960’s.
Since then and with the development of supercomputers, several state of the art
approaches have emerged; such as application of grid-deﬁned symmetry and ge-
ometry for simulating an atmospheric environment in silico [21, 22] simulation
of atmospheric components by Cartesian meshes or other geometric forms ,
application of Double Fourier Series and other models for simulating non-linear
dynamics [24, 25] use of semi-implicit semi Lagrangian discrete models for simu-
lating the ﬂuid-dynamic behaviour of weather systems , and implementation
of other mathematical models to simulate atmospheric ﬂuctuations for weather
prediction and atmosphere study [27, 28, 22]. These models and the derived al-
gorithms apply physical qualities to the simulated atmosphere and its dynamical
behaviour is represented by classical vector calculus based on scalar ﬁelds .
Fluctuations by angular momenta are simulated using continuous equations,
also known as mimetics . Also, large scale atmospheric ﬂows are mod-
elled by ﬂuid mechanics, where large-scale bodies (i.e. a low pressure system)
are adjusted with a dynamical core to represent the non-linear vortical motions
through a mass-conservation scheme [21, 30]. State of the art methods in atmo-
spheric modelling do however not consider the spatio-temporal variations in the
gravitational ﬁeld and its inﬂuence on atmospheric processes. These missing re-
lationships, although quite small at the regional scale, may nonetheless inﬂuence
atmospheric processes via highly non-linear phenomena such as instability and
turbulence and promote large scale dynamics on the atmosphere which have not
been mapped before. Both turbulence and instability have been modelled ac-
curately , however their linear or non-linear dependency on the gravity ﬂux
has not been proposed before, and thus introduces a novel idea. Furthermore,
gravity’s dynamical ﬂuctuations on the uppermost parts of the atmosphere, also
known as gravity waves [10, 9] and gravity-induced wave propagation across
troposphere and ionosphere [6, 5] have been modelled and studied previously.
Still however, a derivation of linear/non-linear models for gravity-atmosphere
interaction has not been derived, also considering the quantized behaviour of
gravity . Several approaches to describe mesospheric systems including nu-
merical studies [32, 6, 5, 7, 8, 33] do therefore support the eﬀect from gravity
on the atmosphere, including the temperature-inversion patterns from gravity
, momentum-alteration of atmospheric bodies and diﬀusion , and also
the triggering of mechanical impulses across the atmosphere . By this, we
are interested in modeling the vorticity that is induced in the atmosphere by
the gravity ﬁeld, by solving our model Hamiltonian numerically. In this en-
deavour we ﬁrst focus on describing the Hamiltonian for the physical problem
and then apply mathematical software to solve numerically the corresponding
master equation. The results of the analysis are presented throughout a series
of ﬁgures and illustrations, and discussion is presented afterwards.
2.2 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 , who deﬁned the Hamiltonian operator for anyons
H=|~/i∇ − (e/c)~
2H0[xˆy−yˆx], 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)~
where the Hamiltonian is simply the normal square of the covariant deriva-
tive. Supersymmetry rules are applied on eqn. (2), which is composed of
the two ﬁrst-order diﬀerential operators:
H= [~/i∇ − (e/c)~
A]×[~/i∇ − (e/c)~
and we get the superpair of the Laughlin Hamiltonian:
HSU SY = [~/i∇ − (e/c)~
A]×[−~/i∇ − (e/c)~
Hence, the SUSY counter-part of the Hamiltonian from (2) becomes:
HSU SY =T T ∗,(5)
T= [~/i∇ − (e/c)~
T∗= [−~/i∇ − (e/c)~
where Tand T∗are the superpair in the SUSY Hamiltonian in (4) and one
anothers Hilbert-adjoint operator.
We consider then Tand T∗, with γ= (e/c)~
A, in one dimension:
which commute by the relation:
[T T ∗]−[T∗T] = 0 (8)
making Tand T∗two commutating complex operators in Hilbert space H.
Both Tand T∗are unbounded operators given the condition:
||T x|| c||x||,(9)
Tand T∗are also non-linear given the γconstant, being a constant trans-
lation from the origin. Tand T∗are non-self-adjoint in Has the following
condition is not satisﬁed:
hT φ, ψi=hφ, T∗ψi,(10)
From this, it follows that:
HSU SY :D(HS U SY )−→ H,
D(HSU SY )⊂H,
where HSU SY is an unbounded linear and non-self-adjoint complex operator
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  it follows that
H†Ψ = 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
3 Materials and methods
In Ref. [11, 12], 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 supersymmetric generalization of the well-known Hamiltonian
proposed by Laughlin  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 and in the gravitational ﬁeld, taking into account
the stratiﬁcation of the density of matter. The corresponding master equation
in dimensionless variables has the form:
∂y −y∂ ψ
∂x )−β|ψ|2ψ−V ψ + ( ~
Ais a dimensionless vector potential, V=V(x, y, z), β =β(x, y, z),Ω -
gravity potential and parameters of the model describing 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 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]
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. We
used the ﬁnite element method with a division of the sphere into tetrahedrons
and an explicit Euler in time. Typical calculations used 14286 elementary cells,
in test problems this number increased to 142860 and 1428600, respectively.
The time step varied from 0.002 to 0.02. As the main solver and for visualizing
the data, we used Wolfram Mathematica 11.3.
Our model describes vorticity on a sphere and we apply this to study the in-
ﬂuence of gravity on the process of formation of quantum vortices. Gravity is
modelled by the parameter βin the master equation (see materials and meth-
ods), which yields a stratiﬁcation of the gravitational potential with higher
gravity density at the core of the sphere. The stratiﬁcation of the parameters β
varies inversely with the density of matter, and hence the stratiﬁcation leads to
an increase in the quantum vorticity in the spherical model which arises from
the nucleus. The results show that ﬁrst phenomenon of importance is that the
gravity of the planet suppresses vorticity in the spherical system, which can be
seen over several ﬁgures of the solutions to the master equation (Figure 1-4, 9-
11). The results show furthermore that vorticity is fueled by the electromagnetic
ﬁeld of the planet, and hence counterbalanced by the gravitational ﬁeld. We
modelled the electromagnetic ﬁeld by (13), which leads to that vorticity arises
entirely from the nucleus of the planet and is projected towards the atmosphere
(Figure 5-8). In order to conceive in detail these results we look deeper at the
analysis behind this model. Let us ﬁrst consider the eﬀect of stratiﬁcation on
quantum vorticity. Suppose that the beta parameter changes as the earth’s
density, the gravitational potential is zero, and the remaining parameters of the
model are Ω = 2, m0= 6, θ = 0, q =−6 (Fig. 1-4). By following this we see
in Fig. 1 the distribution of the βparameter (gravity - on the left) and the
quantum vorticity near the core of the planet (right). Following, ﬁg. 2 shows
the quantum vorticity near the core of the planet.
Figure 1 The beta parameter (on the left) and the vorticity distribution around
a dense nucleus (right) computed for t= 1 with Ω = 2, m0= 6, θ = 0, q =−6.
Figure 2 The quantum vorticity near the core of the planet (on the left) and
the vortex output to the planet’s surface (right).
Here we see the emergence of the vortices that we propose are the driv-
ing force behind the cyclonic and anti-cyclonic vortices behind low and high
pressure systems manifested in the troposphere. The arise of the vortexes is
interestingly projected towards the poles of the planet, as shown in ﬁgure 3,
which respects the patterns of the distribution of low and high pressure systems
frequently circulating on the northern and southern hemispheres . Fig 3
shows furthermore in the cross-sections that vorticity is strongest closer to the
nucleus, and thus the troposphere and mesosphere manifest the most marked
atmospheric phenomena of vorticity (hurricanes, cyclones, low pressures) com-
pared to the thermosphere, ionosphere and stratosphere. The stratiﬁcation of
the gravity gives therefore a good model for modelling vorticity on planet earth
using quantum mechanical operators. The regions on ﬁgure 3 correspond to
planetary latitudes of 50 to 60 degrees. We wish further to study how the wave-
function solution behaves over time, and by modelling the amplitude from the
initial Gaussian distribution one can follow how the wavefunction is at the ini-
tial stage a dense core, which then decays to form a quantum vorticity. This is
of particular interest, because the density of the core is critical for generating a
dynamo eﬀect on the planet, which is a driving force of the vortex formation.
Planetary dynamos require a particular density of the core , which we model
by studying the evolution of the wavefunction (Fig. 4).
Figure 3 The amplitude of the wave function at diﬀerent cross sections calcu-
lated for t= 1 with Ω = 2, m0= 6, θ = 0, q =−6. z= 0,0.5,1,1.5,2,2.5,3,3.5
Figure 4 The amplitude of the wave function at diﬀerent tin cross sections
z= 0 calculated with Ω = 2, m0= 6, θ = 0, q =−6.
We furthermore move to the second problem, where we consider a system
with an inverse stratiﬁcation of the beta parameter, when beta increases from
the center to the periphery with a jump from 10 to 1000 at r= 2. This is
done in order to model the surface of the planet, where the density of matter is
drastically reduced. At this level of the planet, the quantum vortices distribute
in an entirely arbitrary fashion, without any particular symmetry (Fig 5). The
model shows a nucleus of the origin of the vortices at the rupture region to
which the vortices adjoin. The vortices gain here a more pronounced tubular
structure (Fig 5). The amplitude of the wavefunction for this problem shows
further a more delocalized pattern of vortices forming across the globe, without
a predominance of higher latitudes (Fig 6). When a comparison is made of this
ﬁgure with the ﬁrst four ﬁgures, one can see the structure of the nucleus, whose
density reaches 100, is well seen at the top, whereas in the region of the existence
of vortices the typical density (the square of the amplitude) is of the order of
0.01. This suggests that vortices manifest in less dense areas of the planet, as
in the atmosphere and vanish in the higher strata of the planet.
Figure 5 The distribution of quantum vortices in the half sphere (left) and the
vorticity distribution around a dense nucleus (right) in a system with an inverse
stratiﬁcation of the beta parameter computed for t= 1 with Ω = 2, m0= 6, θ =
0, q =−6.
Figure 6 The amplitude of the wave function at diﬀerent cross sections calcu-
lated for t= 1 with Ω = 2, m0= 6, θ = 0, q =−6.
The wavefunction which describes these vortices evolves over time from a
dense gravity nucleus to a more pronounced state of chaos and turbulence to
higher levels of the planet, with a higher density of vortices at time 0.5-0.8 (Fig.
7).This suggests that the interplay between the compressive force of gravity in
a sphere and the eﬀect of magnetism forms a basis for vorticity on the planet,
which is ﬁnely tuned by the density of the strata. This can be seen more
clearly in ﬁgure 8, where the vorticity distributes spherically from the nucleus.
When stratiﬁcation is applied on gravity (β) it is proportional to the density of
the planet and hence in the presence of a gravitational potential, the quantum
vorticity is less pronounced.
Figure 7 The amplitude of the wave function at diﬀerent tin cross sections
z= 0 calculated with Ω = 2, m0= 6, θ = 0, q =−6.
Figure 8 The quantum vorticity near the core of the planet (on the right) and
the vortex output to the planet’s surface (left).
The distribution of the density across the planetary globe and the gravi-
tational potential play therefore a major role on the vortex formation in the
vicinity of the nucleus (Fig 9). Comparing these data with the data in Fig. 1,
2, 5, and 8, we ﬁnd the diﬀerence in the form of vortices, due to gravitational
potential. We can therefore assume that by this model other phenomena such
as the Jupiter’s spot and other planetary atmospheres can be modelled by using
the combination of quantum mechanical operators and stratiﬁcation of gravity.
We furthermore calculated the the distribution of the vortices in diﬀerent cross
sections, z= 0,0.5,1,1.5,2,2.5,3,3.5 for this case, where the vortices also reach
the surface at a latitude between 50 and 60 degrees (Fig 10).
Figure 9 The density of the planet ρand β=ρ(top left), gravitational poten-
tial (top right), and the vorticity distribution around a dense nucleus (at the
bottom) computed for t= 1 with Ω = 2, m0= 6, θ = 0, q =−1.
Figure 10 The amplitude of the wave function at diﬀerent cross sections cal-
culated for t= 1 with Ω = 2, m0= 6, θ = 0, q =−1.
By increasing the beta parameter by two orders of magnitude with the grav-
itational potential unchanged, we ﬁnd a new state in which the vortex tubes are
thinned out (Fig. 11) and ﬁnally, as the gravitational potential is increased by
two orders of magnitude, vorticity completely disappears (data not shown). In
conclusion, we note that a numerical solution to the supersymmetric Hamilto-
nian by using the ﬁnite element method makes it possible to visualize vorticity
with a good resolution (Fig. 12).
Figure 11 The quantum vorticity near the core of the planet in the state with
β= 100ρin the gravitational potential as in Fig. 9 computed for t= 1 with
Ω=2, m0= 6, θ = 0, q =−6.
Figure 12 Spherical mesh and visualization of quantum vorticity in a homoge-
neous magnetic ﬁeld parallel to the axis of rotation.
However, in the three-dimensional problem the solution has no symmetry,
although the initial data are symmetric. This is explained by the fact that
the vortex system is unstable in 3D, since here we have the problem of many
bodies.The results obtained earlier for planar systems show the presence of
symmetry in the distribution of vortices. This symmetry, obviously, disappears
as we go over to 3D, since the vortices interact in a volume in a diﬀerent way
than in the plane.This problem is closely related to the problems of quantum
vortex turbulence and hydrodynamic turbulence. In this sense, model (12) is a
convenient tool for investigating these problems.
Our model shows a representation of the vorticity in the atmosphere on the
earths surface, with a pronounced vorticity arising from the nucleus of the
planet, localizing over the northern and southern hemispheres according to a
quantized form without following a particular symmetry. The model accounts
for a stratiﬁcation of the gravity ﬁeld with the variable β. Compared to other
models, such as the models for simulating non-linear dynamics [24, 25] and
the Lagrangian discrete models for simulating the ﬂuid-dynamic behaviour of
weather systems  and other models [27, 28, 22] we diﬀerentiate by a simpli-
ﬁcation of the ensemble of vortices, which from our model solely arise from one
source (the interplay of magnetism and gravity). This simpliﬁcation can give
advantages when modelling atmospheric behaviour on other planets ab initio,
by accounting for only basic parameters as the strength of the gravity ﬁeld, the
properties of the core and the intensity of the magnetic ﬁeld, which can also give
an eﬃcient computational time for atmospheric predictions. When modelling
weather on planet earth however, the implementation of parameters such as
ocean temperatures, sun light reﬂection, circulation of the great conveyor belt
as well as the circulation of the trade winds are required to provide exact fore-
cast data and diﬀerentiate between the vortices accurately. The Hamiltonian
used in this study origins from particles oscillating in an electromagnetic ﬁeld
in a quantized system [12, 11, 13] and by describing quantized phenomena can
suggest that the results presented by  on the quantized behaviour of gravity
can have impact on the vorticity in the atmosphere on planet earth and other
planets as well. The quantization manifests at macroscale by a dominant popu-
lation of vortices forming closer to the poles, which is observed on planet earth
indeed . It grants however further scrutiny to study atmospheric phenomena
using quantum models, as many of the macroscopic phenomena are triggered
by ensembles of classical mechanic eﬀects (i.e. thermal diﬀusion), which play
a marked role on the dynamics of atmospheres. A combination of quantum
representation of major eﬀects on planetary systems with classical models may
therefore open for a new model to study atmospheric dynamics.
This study shows that a supersymmetric quantum mechanic operator (4) solved
by using the master equation in (12) can be used as a foundation to model
vorticity on planet earth, considering only gravity as a stratiﬁed potential and
the electromagnetic ﬁeld. The results require further development to include
parameters such as ocean temperatures and circulation, reﬂection, planetary
oscillation around the sun, as well as magnetic eﬀects from the moon and the
sun’s magnetic ﬁelds. The evolution of the wavefunction requires furthermore
localizing the exit of the vortices on the surface in detail, so a coordinate sys-
tem can be attributed to the supersymmetric Hamiltonian model for planetary
vorticity. The authors are working further to include these properties in future
All data related to this project can be accessed at http://www.fjordforsk.no
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