Modelling the formation of atmospheric vortices

on planet earth using a supersymmetric operator.

Sergio Manzetti 1,2 and 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.

February 26, 2019

Key points

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.

Keywords

Hamiltonian; magnetism; planetary; gravity; supersymmetry; numerical; anal-

ysis

1

Citing information1

1 Abstract

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-

diction models.

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

2

2 Introduction

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 [18] and its continuously

oscillating nature [19] 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 [20] 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 [23],

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 [26], and implementation

of other mathematical models to simulate atmospheric fluctuations for weather

3

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 [21].

Fluctuations by angular momenta are simulated using continuous equations,

also known as mimetics [29]. 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 [31], 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 [19]. 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

[34], momentum-alteration of atmospheric bodies and diffusion [35], and also

the triggering of mechanical impulses across the atmosphere [36]. By this, we

are interested in modeling the vorticity that is induced in the atmosphere by

the gravity field gravity, by solving our model Hamiltonian numerically. In this

endeavour 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

4

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 [37], who defined the Hamiltonian operator for anyons

as

H = |~/i∇− (e/c) ~A|2, (1)

where ~A = 12H0[xy − yx], is the symmetric gauge vector potential, where

H0 is 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-

tive. Supersymmetry rules [38]are applied on eqn. (2), which is composed of

the two first-order differential operators:

H = [~/i∇− (e/c) ~A]× [~/i∇− (e/c) ~A], (3)

and we get the superpair of the Laughlin Hamiltonian:

HSUSY = [~/i∇− (e/c) ~A]× [−~/i∇− (e/c) ~A]. (4)

Hence, the SUSY counter-part of the Hamiltonian from (2) becomes:

HSUSY = T T ∗, (5)

with

T = [~/i∇− (e/c) ~A], (6)

5

and

T ∗ = [−~/i∇− (e/c) ~A], (7)

where T and T ∗ are the superpair in the SUSY Hamiltonian in (4) and one

anothers Hilbert-adjoint operator.

We consider then T and 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 ∗]− [T ∗T ] = 0 (8)

making T and T ∗ two commutating complex operators in Hilbert space H .

Both T and T ∗ are unbounded operators given the condition:

||T x|| c||x||, (9)

T and T ∗ are also non-linear given the γ constant, being a constant trans-

lation from the origin. T and T ∗ are non-self-adjoint in H as the following

condition is not satisfied:

〈T φ, ψ〉 = 〈φ, T ∗ψ〉, (10)

From this, it follows that:

HSUSY : D(HSUSY ) −→H ,

and

D(HSUSY ) ⊂H ,

6

where HSUSY is an unbounded linear and non-self-adjoint complex operator

on Hilbert space H = L2[−∞,+∞]. However, HSUSY is self-adjoint in its

domain, D(HSUSY ) on L2[a, b].

From supersymmetry theory in quantum mechanics [38] it follows that

H†Ψ = HΨ† = EΨ = EΨ†, therefore we can assume that:

HSUSY Ψ = EΨ, (11)

where E is the energy of the system, which generates the boundaries of

D(HSUSY ) on L2[0, L], where the interval of quantization (L) is contained in

the zero-point energy term E= n2~2π2

2mL2 .

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 [37] 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:

∂ψ∂t = 1

2∇2ψ + iΩ(x∂ψ∂y − y

∂ψ∂x )− β|ψ|2ψ − V ψ + ( ~A)2ψ (12)

Here ~A is 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

7

the electric charge q combined with it, we assume scalar potential ϕ = 0, and

the vector potential

~A(x, y, z, t) = − [~m~r]r3 −

qt~rr3

(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. 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.

4 Results

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 β in 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

8

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 β parameter (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.

.

9

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 [39]. 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 [40], which we model

by studying the evolution of the wavefunction (Fig. 4).

10

.

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 t in 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

11

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.

12

.

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.

13

.

Figure 7 The amplitude of the wave function at different t in 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-

14

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 ρ and β = ρ (top left), gravitational poten-

tial (top right), and the vorticity distribution around a dense nucleus (at the

15

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).

.

16

Figure 11 The quantum vorticity near the core of the planet in the state with

β = 100ρ in 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.

5 Discussion

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

17

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 [26] 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 [19] 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 [40]. 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.

18

6 Conclusions

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

models.

Acknowledgements

All data related to this project can be accessed at http://www.fjordforsk.no

19

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