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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

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, 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

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=1

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)~

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:

HSU SY = [~/i∇ − (e/c)~

A]×[−~/i∇ − (e/c)~

A].(4)

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

HSU SY =T T ∗,(5)

with

T= [~/i∇ − (e/c)~

A],(6)

5

and

T∗= [−~/i∇ − (e/c)~

A],(7)

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:

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 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,

and

D(HSU SY )⊂H,

6

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 [38] 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

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 ~

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

7

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

the vector potential

~

A(x, y, z, t) = −[~m~r]

r3−qt~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. 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 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

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 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.

.

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 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

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 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-

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 ρand 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= 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.

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

References

[1] Izrail Moiseevich Gel’fand and I Ya Dorfman. Hamiltonian operators and

algebraic structures related to them. Functional Analysis and Its Applica-

tions, 13(4):248–262, 1979.

[2] Richard J Szabo. Quantum ﬁeld theory on noncommutative spaces. Physics

Reports, 378(4):207–299, 2003.

[3] Barry Simon. Schr¨odinger operators in the twenty-ﬁrst century. In Math-

ematical physics 2000, pages 283–288. World Scientiﬁc, 2000.

[4] Alexander A Balinsky and William Desmond Evans. Spectral analysis of

relativistic operators. World Scientiﬁc, 2011.

[5] Thomas S Lund and David C Fritts. Numerical simulation of gravity wave

breaking in the lower thermosphere. Journal of Geophysical Research: At-

mospheres, 117(D21), 2012.

[6] David C Fritts and Thomas S Lund. Gravity wave inﬂuences in the ther-

mosphere and ionosphere: Observations and recent modeling. In Aeronomy

of the Earth’s Atmosphere and Ionosphere, pages 109–130. Springer, 2011.

[7] Rolando R Garcia and Susan Solomon. The eﬀect of breaking grav-

ity waves on the dynamics and chemical composition of the mesosphere

and lower thermosphere. Journal of Geophysical Research: Atmospheres,

90(D2):3850–3868, 1985.

[8] Saburo Miyahara and Jeﬀrey M Forbes. Interactions between gravity waves

and the diurnal tide in the mesosphere and lower thermosphere. Journal

of the Meteorological Society of Japan. Ser. II, 69(5):523–531, 1991.

[9] Colin O Hines. Internal atmospheric gravity waves at ionospheric heights.

Canadian Journal of Physics, 38(11):1441–1481, 1960.

20

[10] RR Hodges Jr. Generation of turbulence in the upper atmosphere by in-

ternal gravity waves. Journal of Geophysical Research, 72(13):3455–3458,

1967.

[11] Sergio Manzetti and Alexander Trounev. Formation of quantum vortices in

a rotating sphere in an electromagnetic ﬁeld. Technical Reports, 14(2019):1–

31, 2019.

[12] Sergio Manzetti and Alexander Trounev. Derivation of a hamiltonian for

formation of particles in a rotating system subjected to a homogeneous

magnetic ﬁeld. Technical Reports, 14(2020):1–19, 2019.

[13] Sergio Manzetti and Alexander Trounev. Quantum vorticity in a rotating

magnetic ﬁeld. Technical Reports, 15(2020):1–20, 2019.

[14] Igor Chunchuzov, Sergey Kulichkov, Vitaly Perepelkin, Astrid Ziemann,

Klaus Arnold, and Anke Kniﬀka. Mesoscale variations in acoustic signals

induced by atmospheric gravity waves. The Journal of the Acoustical So-

ciety of America, 125(2):651–663, 2009.

[15] Ruocan Zhao, Xiankang Dou, Dongsong Sun, Xianghui Xue, Jun Zheng,

Yuli Han, Tingdi Chen, Guocheng Wang, and Yingjie Zhou. Gravity waves

observation of wind ﬁeld in stratosphere based on a rayleigh doppler lidar.

Optics express, 24(6):A581–A591, 2016.

[16] Colin C Triplett, Richard L Collins, Kim Nielsen, V Lynn Harvey, and

Kohei Mizutani. Role of wind ﬁltering and unbalanced ﬂow generation in

middle atmosphere gravity wave activity at chatanika alaska. Atmosphere,

8(2):27, 2017.

[17] J¨orn Callies, Raﬀaele Ferrari, and Oliver B¨uhler. Transition from

geostrophic turbulence to inertia–gravity waves in the atmospheric energy

spectrum. Proceedings of the National Academy of Sciences, 111(48):17033–

17038, 2014.

21

[18] David J Stevenson. Fluctuating gravity of earths core. Proceedings of the

National Academy of Sciences, 109(47):19039–19040, 2012.

[19] Valery V Nesvizhevsky, Hans G B¨orner, Alexander K Petukhov, Hartmut

Abele, Stefan Baeßler, Frank J Rueß, Thilo St¨oferle, Alexander Westphal,

Alexei M Gagarski, Guennady A Petrov, et al. Quantum states of neutrons

in the earth’s gravitational ﬁeld. Nature, 415(6869):297, 2002.

[20] Edward N Lorenz. Deterministic nonperiodic ﬂow. Journal of the atmo-

spheric sciences, 20(2):130–141, 1963.

[21] Andrew Staniforth and John Thuburn. Horizontal grids for global weather

and climate prediction models: a review. Quarterly Journal of the Royal

Meteorological Society, 138(662):1–26, 2012.

[22] John Thuburn. Numerical wave propagation on the hexagonal c-grid. Jour-

nal of Computational Physics, 227(11):5836–5858, 2008.

[23] Luca Bonaventura, Ren´e Redler, and Reinhard Budich. Earth System

Modelling-Volume 2: Algorithms, Code Infrastructure and Optimisation.

Springer Science & Business Media, 2011.

[24] Myung-Seo Koo and Song-You Hong. Double fourier series dynamical core

with hybrid sigma-pressure vertical coordinate. Tellus A: Dynamic Mete-

orology and Oceanography, 65(1):19851, 2013.

[25] Hoon Park, Song-You Hong, Hyeong-Bin Cheong, and Myung-Seo Koo. A

double fourier series (dfs) dynamical core in a global atmospheric model

with full physics. Monthly Weather Review, 141(9):3052–3061, 2013.

[26] Nigel Wood, Andrew Staniforth, Andy White, Thomas Allen, Michail

Diamantakis, Markus Gross, Thomas Melvin, Chris Smith, Simon

Vosper, Mohamed Zerroukat, et al. An inherently mass-conserving semi-

implicit semi-lagrangian discretization of the deep-atmosphere global non-

hydrostatic equations. Quarterly Journal of the Royal Meteorological Soci-

ety, 140(682):1505–1520, 2014.

22

[27] Peter H Lauritzen, Christiane Jablonowski, Mark A Taylor, and Ramachan-

dran D Nair. Rotated versions of the jablonowski steady-state and baro-

clinic wave test cases: A dynamical core intercomparison. Journal of Ad-

vances in Modeling Earth Systems, 2(4), 2010.

[28] Paul A Ullrich. A global ﬁnite-element shallow-water model supporting

continuous and discontinuous elements. Geoscientiﬁc Model Development,

7(6):3017–3035, 2014.

[29] Mark A Taylor. Conservation of mass and energy for the moist atmospheric

primitive equations on unstructured grids. In Numerical Techniques for

Global Atmospheric Models, pages 357–380. Springer, 2011.

[30] Mohamed Zerroukat, Nigel Wood, and Andrew Staniforth. Slice-s: A semi-

lagrangian inherently conserving and eﬃcient scheme for transport prob-

lems on the sphere. Quarterly Journal of the Royal Meteorological Society:

A journal of the atmospheric sciences, applied meteorology and physical

oceanography, 130(602):2649–2664, 2004.

[31] David C Fritts, Ling Wang, Joe Werne, Tom Lund, and Kam Wan. Gravity

wave instability dynamics at high reynolds numbers. part ii: Turbulence

evolution, structure, and anisotropy. Journal of the Atmospheric Sciences,

66(5):1149–1171, 2009.

[32] Sergio Manzetti. A quantistic interpretation of the relationship between

the earth-core and the atmosphere. Atmospheric and Climate Sciences,

4(04):508, 2014.

[33] Marie-Lise Chanin and Alain Hauchecorne. Lidar observation of gravity

and tidal waves in the stratosphere and mesosphere. Journal of Geophysical

Research: Oceans, 86(C10):9715–9721, 1981.

[34] A Hauchecorne, Mo L Chanin, and R Wilson. Mesospheric tempera-

ture inversion and gravity wave breaking. Geophysical Research Letters,

14(9):933–936, 1987.

23

[35] James R Holton. The role of gravity wave induced drag and diﬀusion in the

momentum budget of the mesosphere. Journal of the Atmospheric Sciences,

39(4):791–799, 1982.

[36] Mark R Schoeberl, Darrell F Strobel, and John P Apruzese. A numerical

model of gravity wave breaking and stress in the mesosphere. Journal of

Geophysical Research: Oceans, 88(C9):5249–5259, 1983.

[37] Robert B Laughlin. Anomalous quantum hall eﬀect: an incompressible

quantum ﬂuid with fractionally charged excitations. Physical Review Let-

ters, 50(18):1395, 1983.

[38] Fred Cooper, Avinash Khare, and Uday Sukhatme. Supersymmetry and

quantum mechanics. Physics Reports, 251(5-6):267–385, 1995.

[39] Brian J Hoskins and Kevin I Hodges. New perspectives on the north-

ern hemisphere winter storm tracks. Journal of the Atmospheric Sciences,

59(6):1041–1061, 2002.

[40] David J. Stevenson. Planetary magnetic ﬁelds. Earth and Planetary Science

Letters, 208(1):1 – 11, 2003.

24