ESEIAAT STUDY: Numerical simulations of
OBSERVED PHENOMENA IN THE ATMOSPHERES OF THE GIANT PLANETS
This document is issued by Kevin Ahrens Vel´asquezfor the Bachelor Science Degree in Aerospace Vehicle Engineering
30th of June, 2020
THESIS DOCUMENTATION
Directed by: Enrique Garc´ıaMelendo
Co-director: Manel Soria Guerrero
Abstract
The sky has been always in the mind of humans, as a target, a challenge and a source of knowledge. In the late 20th century, the Human Race has managed to significantly improve technology aimed at the exploration of space and the acquisition of knowledge about the Solar System and beyond it. However, knowledge can be also acquired from Earth in a easier and cheaper way. With the intention of providing and completing scientific information about Jupiter and Saturn, particularly its atmospheric nature and dynamics, this thesis is carried out. This file is the result of a thesis based on the study of numerical simulations about phenomena observed in the Gas Giants. Firstly, it establishes a reference frame about Jupiter and Saturn which includes the most relevant information about the planets such their characteristics, nature, phenomena and performed exploration. The introduction is focused on the atmosphere of the planets and the phenomena which has been observed to take place there. In addition, some characteristics about the dynamics that govern them are also presented. Once the scientific reference frame is established a general explanation about the physical model used in the software is explained. This part of the report provides the main equations and few details regarding model and the computational aspect, which have to be considered. In third place, the different cases studied in the simulations are explained. Each case presents the scientific reference followed and then presents the methodology used and results obtain as well as some comments about them. At the end, the conclusions and recommendations for the follow up of the study are given, together with an economic and environmental study.
3 I declare that,
the work in this Degree Thesis is completely my own work.
no part of this Degree Thesis is taken from other people’s work without giving them credit,
all references have been clearly cited,
I’m authorised to make use of the research group related information I’m providing in this document.
I understand that and infringement of this declaration leaves me subject to the foreseen disciplinary actions by The Universitat Polit`ecnica de Catalunya - BarcelonaTECH.
Kevin Ahrens Vel´asquez July 30 of 2020
Numerical simulations of observed phenomena in the atmospheres of the giant planets Acknowledgement
The acknowledgements are for everyone which has helped for the performance of this thesis. Firstly, I want to thank Enrique, Manel and Marc, which have been main re- sponsible for the development of this work. Enrique has been a reference of passion and enthusiasm for the science and he has been always kind no matter the conditions. Manel has been a reference of hard work, passion and critical thinking during this four-month period. Marc has been a very good mate always helpful and available when it has been necessary, the reference of good student. All three have been there helping me and have given to me this opportunity. Secondly, I want to thank my family. To my mother and father for the unconditional support as well as for the love always provided. They both are the main reference of hard work and personal growth in any circumstance. To my brother for playing the role of big brother and teaching me when it has been necessary. To my sister for being there no matter what and making me grow as brother and person. I want to thank my partner Marta, which has walked with me during this adventure and has filled hard moments with smiles and love. She is a reference of unconditional love. Finally I want to thank my friends, which have been there when it has been necessary during the four-year experience of the grade, in particular to my flatmates and those who almost are.
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6 Contents
Aim ix
Scope xi
Requirements xiii
Nomenclature xv
Acronyms xvii
1 The Giant Planets 1 1.1 Introduction to Jupiter and Saturn ...... 1 1.2 Jupiter ...... 2 1.3 Saturn ...... 9
2 The Shallow-Water Model 17 2.1 Introduction to the Shallow-Water Model ...... 17 2.2 Obtaining of the Shallow-Water Model equations ...... 19 2.2.1 Dimensional analysis ...... 21 2.2.2 Demonstration ...... 26 2.3 Physical bases of the Algorithm ...... 29 2.4 Potential Vorticity ...... 30
3 Reference frame of the numerical simulations 33 3.1 Barcelona Supercomputer Center ...... 33 3.2 Data input and running process ...... 34 3.2.1 Planet data ...... 34 3.2.2 Template ...... 36 3.2.3 Run script ...... 37 3.3 Clarifications of the simulation process ...... 37 3.3.1 Numerical parameters ...... 37 3.3.2 Results presentation ...... 39
4 The simulations 41 4.1 Introduction ...... 41 4.2 Study of storms in the northern hemisphere of Saturn ...... 41 4.2.1 Scientific reference ...... 42 4.2.2 Computational work ...... 44 4.3 Study of feasible origins of the Great Red Spot ...... 59 4.3.1 Scientific reference ...... 59 4.3.2 Computational Work ...... 62 4.4 Validation of software module ...... 78
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4.4.1 Scientific reference ...... 78 4.4.2 Computational Work ...... 79
5 Conclusions and follow-up of the study 85
Environmental awareness 89
ii List of Figures
1.1 Jupiter seen from Voyager 1. Instrument: ISS - Narrow Angle Camera. Extracted from [6] ...... 3 1.2 Vertical temperature structure of Jupiter’s atmosphere. Extracted from [8] 5 1.3 Zonal winds of Jupiter’s atmosphere. Data extracted from [9] ...... 6 1.4 Sketch of Jupiter drawn by Giovanni Cassini in December 1690. Extracted from [11]...... 7 1.5 Image of the Great Red Spot taken by Juno Spacecraft. Instrument: Jun- oCam. Extracted from [6] ...... 8 1.6 Saturn seen from Cassini spacecraft. Instrument: ISS - Wide Angle Cam- era. Extracted from [6] ...... 9 1.7 North Pole’s hexagon as seen from Cassini. Instrument: ISS - Wide Angle Camera. Extracted from [6] ...... 11 1.8 Images captured by Cassini orbiter. Instrument: ISS - Wide Angle Camera. Extracted from [6] ...... 12 1.9 Zonal winds of Saturn’s atmosphere. Data extracted from [15] ...... 13 1.10 Panoramic view of the rings as seen from Cassini. Instrument: ISS - Narrow Angle Camera. Extracted from [6] ...... 14 1.11 Wavy structures in the A ring as it crosses the gap. Image acquired by Cassini spacecraft. Instrument: ISS - Narrow Angle Camera. Extracted from [6] ...... 15
2.1 Fluid layer of the model. Extracted from [17] ...... 18 2.2 Diagram of a rotating sphere with a reference frame established over its surface at a random point ...... 20 2.3 Diagram of the motion of a particle ...... 22 2.4 Diagram of constant pressure straight lines and the gradient direction . . . 24 2.5 Diagram of constant pressure concentric curves and the gradient direction . 25 2.6 Fluid layer under hydrostatic conditions ...... 25 2.7 Diagram of a fluid layer whose surface is perturbed ...... 27
3.1 SaturnEGM.planet file ...... 34 3.2 Saturn Voyager.wind file ...... 35 3.3 Profile of zonal wind used for Jupiter simulations. Data extracted from [9] 35 3.4 Profile of zonal wind used for Saturn simulations. Data extracted from [15] 36
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4.1 The upper figure shows a four processed images mosaic of the region studied in [22]. It is a Mercator projection of Voyager 2 images 4.2 days before its encounter with Saturn in 1st of September of 1981. The article uses planetocentric. The figure below is a drawing of the distinctive traits of the atmospheric phenomena. In addition to the drawings, each feature is labeled with a particular name so that the reader can understand easily the explanation as information is provided. The figure has different axis such the latitude axis in the right side, the longitude axis in the bottom, a wind profile with the velocity value in the bottom axis too as well as determined latitudes they have selected. Extracted from [22] ...... 42 4.2 Arrangement of 6 figures which represent the evolution of the REGION 4 presented in the drawing of Figure 4.1. The image a is taken 20 days approximately before the encounter. An arrow indicates the position of bright spot c throughout the observation period...... 43 4.3 Sketch of t1 structure evolution during a period of 10 steps. A dual repres- entation is provided where the first one (a) explains the evolution regarding the time step. The numbered dots represent the position of the features for each moment (each time step is a 5 Saturn rotations period). The drawing (b) below provides a scheme of the mean flow as well as arrows which in- dicate the direction of the clouds from step 9 to step 10. The frame moves at 16 m/s westward. The left axis contains latitude coordinates while the horizontal one is intended for the longitude. Extracted from [22] ...... 45 4.4 Structure of the feature to be simulated which contains the sizes of the storm, each couple follows the same pattern. tinjection is the moment when the gaussian is injected ...... 46 4.5 Diagram of the first injection instant and the following one. It includes the imaginary circumference red coloured as well as the distance between the first double-storm and the second one...... 47 4.6 Frame of day 9 of the simulation. A cluster of 3 double-storms represented in Potential Vorticity. From latitude of 34◦ to 46◦. Resolution of 0.02◦/VC 49 4.7 Diagram of the cluster of 3 double-storms represented with Tracer. Latit- ude from 34◦ N to 46◦ N. Resolution of 0.02◦/VC. Day 9 of the simulation 49 4.8 Diagram of BS-1 represented in Potential Vorticity. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/VC. Day 150 of the simulation ...... 51 4.9 Numerical dissipation of energy during the 200 day simulation process . . . 52 4.10 Diagram of the perturbances intended to create turbulence represented in Potential Vorticity. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 35 of the first simulation ...... 53 4.11 Diagram of the perturbances intended to create turbulence represented in Potential Vorticity. Final scenario. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 60 of the first simulation ...... 54 4.12 Diagram of the perturbances intended to create turbulence represented in Potential Vorticity. Poor turbulent scenario. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 25 of the second simulation ...... 54
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4.13 Diagram of the cluster under turbulence effects represented in Potential Vorticity. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 71 of the simulation ...... 55 4.14 Diagram of the cluster under turbulence effects represented with Tracer. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 71 of the simulation ...... 55 4.15 Improved result of the cluster under turbulence effects represented with Tracer. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 71 of the simulation ...... 57 4.16 Improved result of the cluster without turbulence effects represented with Tracer. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 71 of the simulation ...... 58 4.17 Two massive storms captured by Juno 21st December 2018. Instrument: JunoCam. Exracted from [24] ...... 60 4.18 Two storms caught merging in Jupiter South Hemisphere near Oval BA captured by Juno on 26th December 2019. Instrument: JunoCam. Extrac- ted from [24] ...... 60 4.19 Diagram of Jupiter high pressure cells taken by the HST in 2007. Extracted from files provided by [25] ...... 61 4.20 Great Red Spot captured by Pioneer 10 in 1974. Extracted from [26] . . . 61 4.21 Comparison between the old Great Red Spot, of 1890 (left) and the current one from a image taken in 2014 (right). Extracted from [27] ...... 62 4.22 MS1 result. Diagram of the reference vortex A1 represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 100 of the simulation ...... 64 4.23 Diagram of two vortices, A1 and A2, with different tangential velocities represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 150 of the simulation ...... 64 4.24 Diagram of the moment when A1 and A2 merge represented in Potential Vorticity. A2 is injected at 21◦ S. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 200 of the simulation ...... 65 4.25 Diagram of two vortices, A1 and A2, with different tangential velocities represented in Potential Vorticity. A2 is injected at 23◦ S. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 150 of the simulation . . . . . 66 4.26 MS2 final result. Diagram of vortex A12, A1 and A2 represented in Po- tential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 200 of the simulation ...... 67 4.27 MS3 final result. Diagram of vortex A123, A1 and A2 represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 300 of the simulation ...... 68 4.28 Isolated test final result. Diagram of resultant vortex represented in Poten- tial Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 250 of the simulation ...... 69 4.29 Numerical dissipation of energy during a 250 day simulation process . . . . 69
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4.30 CS1 Result. Diagram of resultant structure represented in Potential Vor- ticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 75 of the simulation ...... 70 4.31 CS1 Result. Diagram of resultant structure represented in Potential Vor- ticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 75 of the simulation ...... 71 4.32 CS3 Result. Numerical dissipation of energy during a Diagram of resultant structure represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 75 of the simulation ...... 72 4.33 CS4a Result. Diagram of resultant structure represented in Potential Vor- ticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 140 of the simulation ...... 72 4.34 CS4c Result. Diagram of resultant structure represented in Potential Vor- ticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 140 of the simulation ...... 73 4.35 OS1 Result. Diagram of resultant structure represented in Potential Vor- ticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 147 of the simulation ...... 74 4.36 OS4 Result. Diagram of resultant structure represented in Potential Vor- ticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 147 of the simulation ...... 75 4.37 OS7 Result. Diagram of resultant structure represented in Potential Vor- ticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 131 of the simulation ...... 75 4.38 Caption ...... 76 4.39 Images captured by Juno orbiter. Instrument: JIRAM. Extracted from [6] 78 4.40 The northern ”eye” of Saturn seen from Cassini. Instrument: ISS - Narrow Angle Camera. Extracted from [6] ...... 79 4.41 Set 1. Diagram of Jupiter’s polar region represented with Potential-Vorticity...... 81 4.42 Set 1. Diagram of Saturn’s polar region represented with Potential Vorti- city...... 82 4.43 Set 2. Diagram of Jupiter’s polar region represented with Potential Vorti- city...... 83 4.44 Set 2. Diagram of Saturn’s polar region represented with Potential Vorti- city...... 84
vi List of Tables
1.1 Comparison between the Giant Planets and the Earth. Data extracted from [1] ...... 2
3.1 Domain limits for the first simulation case ...... 38 3.2 Domain limitis of the ”merger” and ”observed” simulations cases . . . . . 38 3.3 Domain limits of the ”columns” simulation case ...... 38 3.4 Domain limits of the third simulation case ...... 38
4.1 First stage: t1 final parameters ...... 48 4.2 Second stage: BS-1 final parameters ...... 51 4.3 Turbulence parameters ...... 53 4.4 Fourth stage: t1 final parameters under turbulence effects ...... 56 4.5 Merger Reference anticyclonic storm ...... 63 4.6 Final parameters for A1 and A2 ...... 66 4.7 Final parameters for A1, A2 and A3 ...... 67 4.8 Parameteres of Column simulation set ...... 70 4.9 Parameteres of Column simulation set ...... 73 4.10 Set 1. Parameters of perturbances in polar regions ...... 80 4.11 Set 2. Parameters of perturbances in polar regions ...... 80
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viii Aim
The aim of this Bachelor final project is to reproduce atmospheric phenomena observed by Voyager, Cassini and Juno space missions in Jupiter and Saturn. The intention is to represent features such as vortices in Jupiter’s Poles and storms in Saturn’s North Hemisphere obtaining successful and reliable results using high performance computation tooling. In order to obtain highly accurate results, the parameters introduced to run the sim- ulation have to be carefully established. A process aimed at assuring a high quality performance of simulations as well as the acquisition of necessary knowledge is followed. It includes few secondary objectives such as the familiarisation and understanding of the physical model on which the code is based, an introduction to Jupiter and Saturn’s nature and atmosphere as well as the analysis of scientific articles related to the corresponding features
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x Scope
As it is explained below, the scope covers a first familiarisation with the topic of the thesis, the study and learning from scientific information yet published and a path of continuous analysis of numerical simulations.
• The analysis and study of scientific documentation such as papers and articles about:
– General information of Jupiter and Saturn’s nature. The information includes their general characteristics, an atmospheric analysis, their composition and research as well as observations already performed. – Most of the different space missions accomplished or currently being carried out, aimed at providing new knowledge about the Giant Planets. Spacecrafts such as Ulysses, Cassini-Huygens or Juno are considered. – Specific features such as the Great Red Spot in Jupiter or the Dragon Storm in Saturn.
• An approach and familiarisation with high performance computation tools such as C language, ParaView software and Ubuntu operative system, which are the ones used to manage the computational environment with regard to the code.
• The implementation of a program aimed at generating multiple cases for different combinations of values, which sends these at once. It will be based on C language too, ensuring the compatibility with the main code, and it is intended to accelerate the process of the study.
• The accomplishment and analysis of manifold simulations regarding the Storm Alley and the pole of Saturn as well as the poles of Jupiter and the GRS. The intention is to improve the parameters introduced so as to obtain better results. The corres- ponding analysis of the result will allow to find which are the correct parameters and reestablish others.
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xii Requirements
As the project is developed, determined requirements have to be met so as to assure a good performance of the next stages. These requirements are strongly related to the scope which this thesis covers.
• To understand the physical model, Shallow Water, which is the base of the solver. Its understanding will lead to a better selection of the parameters since the model is derived from the Navier-Stockes Equations and it will condition the numerical simulation.
• To learn the functioning of the code regarding to simulation. The process to compile the code, that includes the way in which the entry files should be introduced and the steps to call the code and finally run the simulation are considered in this requirement. It has to be taken into account the fact that the software used to simulate has been implemented using parallel programming techniques and there are some parameters which depend on that.
• Simultaneously with the points above mentioned, to understand and acquire the necessary knowledge about Jupiter and Saturn’s nature as well as their general characteristics and atmosphere. Information about the space missions sent to these planets and particular phenomena which take place in their atmospheres is also considered.
• To use high computing power and tooling, such as the ones provided by the Bar- celona Supercomputing Center which will lead to a faster and complete performance of the tasks.
• To get access to scientific bibliography related to the scope of the thesis.
• The following two points will be determinant factors taking into account the high quantity of data produced throughout the project:
– To develop an organized and structured method aimed at helping to codify, manage and store all the data related with the simulations, including the input folder, the results and the images obtained.
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– To develop a record intended to be an easy access source of the relevant data of each simulation.
• To have the necessary knowledge about C language so as to implement the necessary algorithms during the project.
• To develop the capability to analyse and draw conclusions of simulations results after understand the dynamics of the phenomena studied and the physical laws which govern them.
xiv Nomenclature
δ Aspect ration ac Coriolis acceleration f Coriolis term
ρ Density
H Difference between fluid layer top and bottom surface
Ω~ Angular velocity of the planet
λ Fluid property
∇ Gradient of a vector g Gravity h Height of the fluid layer
L Horizontal dimension scale
U Horizontal velocity scale
A Integration constant
φ Latitude
ζ Local vorticity
D Dt Material derivative
δ δx Partial derivative in x-direction
δ δy Partial derivative in y-direction
δ δz Partial derivative in z-direction r Position vector
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Π Potential Vorticity ap Pressure acceleration P˜ Pressure excess
P Total pressure po Reference pressure Ro Rossby number
η Difference between height and horizontal dimension scale t Time variable
T Time scale
~i Unitary reference vector in x-direction
~j Unitary reference vector in y-direction
~k Unitary reference vector in z-direction
~u Velocity vector
D Vertical dimension scale
W Vertical velocity scale
ω Vorticity x Position in x-direction u Velocity in x-direction y Position in y-direction v Velocity in y-direction z Position in z-direction w Velocity in z-direction
xvi Acronyms
GRS Great Red Spot
GWS Great White Spot
JP Jupiter’s pole
SP Saturn’s pole
SED Saturn Electrostatic discharges
SS Solar System
NCAS National Center of Atmospheric Science
BSC-CNS Barcelona Supercomputing center - Centro nacional de supercomputaci´on
RES Red Espa˜nolade supercomputaci´on
NAC Narrow Angle Camera
WAC Wide Angle Camera
JIRAM Juno Infrared Auroral Mapper
EPIC Explicit Planetary Isentropic Coordinate
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xviii 1 The Giant Planets
This chapter is intended establish a reference frame about the Giants of the Solar System. After a comparison between the gaseous planets and the Earth, general charac- teristics such as size, orbit, dynamics and composition will be explained, as well as enough information so that the following chapters and the numerical simulations explanation can be well understood.
1.1 Introduction to Jupiter and Saturn
As it is known, our Solar System is formed by eight planets. These celestial bodies are classified in two categories. The Inner Planets and the Outer Planets. The latter category includes the planets beyond the Asteroid Belt. The outer region planets are characterized by presenting very large size, in comparison with the four first planets of the Solar System. Secondly, their composition is mainly gaseous, although they may present a solid core. In addition, they are surrounded by a great number of satellites, unlike the rocky planets. In particular, Jupiter and Saturn, which belong to the Outer Planets group, have been studied since hundreds of years ago, through earth-based observations. These ob- servations have improved since Galileo Galilei invented the first telescope, so as to gain insight about them. The amazing nature that govern their atmospheres provides amaz- ing planetary-scale phenomena depicted in their outermost atmospheric layers of gas. These two planets are presented below and will be the frame of reference of the different numerical simulations carried out throughout the thesis.
1 2 CHAPTER 1. THE GIANT PLANETS
Table 1.1: Comparison between the Giant Planets and the Earth. Data extracted from [1]
Data Jupiter Saturn Earth Avarage orbital distance (AU) 5.2 9.54 1 Mean orbit velocity (km/h) 47.002 34.701 107.218 Orbit eccentricity 0.04838 0.05386 0.01671 Orbit period (in Earth years) 11.86 29.45 1.0001 Orbit inclination (degrees) 1.304 2.49 0.00005 Equatoria inclination (degrees) 3.1 26.7 23.4 Equatorial radius (km) 69.911 58.232 6.371 Volume (km3) 1.43x1015 8.27x1014 1.08x1012 Density (g/cm3) 1.326 0.687 5.513 Mass (kg) 1.89x1027 5.68x1026 5.97x1024 Surface gravity (m2/s) 24.79 10.4 9.81 Rotation period (in Earth days) 0.413 0.444 0.997
The Table 1.1 provides the main characteristic of Jupiter and Saturn as well as the Earth. It is intended to present the data in a way that it is easy to understand, so that the human mind can establish a first idea of these planets taking as reference a well known planet as the Earth. The data and information acquired to present this chapter has been extracted from [2] and [3], as well as [4] and [5].
1.2 Jupiter
Jupiter is the fifth after Mars, from which is separated by the Asteroids Belt and the first one of the Outer Planets. Although it is mainly gaseous and its density is considerably lower than that of a Rocky Planet, its weight is the largest one and its mass is more than two times that of the rest of the planets all together. The origin of his name comes from the Roman God Jupiter, which was the most powerful god in the Roman mythology. This fact leads to a comparison between the God and the planet since Jupiter is the biggest too, among the eight planets. It is easy to see it from the Earth to the naked eye due to its brightness. Jupiter is the third brightest object in the night sky after the Moon and Venus. The light that Jupiter provides during the Earth nights has been traveling for 40 minutes approximately until reach the human eye. There are proves that this celestial body has been observed from Earth since the 7th century BC. by Babylonial astronomers. Later, Roman and Greek civilisation also knew him and introduced it both Roman and Greek mythology. The latter named the body for Phaethon. It is known that more cultures such the Germanic later, or the Asian 1.2. JUPITER 3
Vietnamese also knew about this planet. Asian people, particularly, named the body for the ”wood star”. In fact, the adjective form for Jupiter, Jovian, comes from that Middle Ages. It was until 1610, when Galileo Galilei, an Italian scientific, could see for the first time the planet through a telescope made by himself. In addition to that, he saw that the planet was surrounded by smaller bodies, the four Galilean Moons. Nowadays it is known that the Jovian System is composed by the planet itself and more than 75 natural satellites, including the four Galilean Moons. Some of these satellites are not confirmed yet. Currently, the exploration of space bodies is carried out by different Space organisa- tions. Since the 20th Century, crafts and probes have been sent to the Jovian System, letting a new age of robotic exploration begin. Probes such Pioneer, Ulysses, Voyager or New Horizons, among others, have visited the planet providing information and images like the one presented below.
Figure 1.1: Jupiter seen from Voyager 1. Instrument: ISS - Narrow Angle Camera. Extracted from [6]
As it is stated above, and thanks to all the exploration that has been carried out aimed at the Jovian System, it is known that its size is huge. The dimensions of its equatorial radius is of 69.911 km and could be able to contain more than 1300 Earth size planets. Its axis is tilted 3.1 degrees. In the same way as the Earth, Jupiter also rotates around its axis, however its rotation period is the shortest one in the Solar System, with a day length of 9 hours and 55 minutes, this means that a day in in the Jovian Planet lasts 0.41 Earth days. See Table 1.1. As it 4 CHAPTER 1. THE GIANT PLANETS rotates so fast its shape is not a perfect sphere but an oblate spheroid. In addition, Jupiter rotation can follow three different systems, and this is because of the different rotation velocities that its atmosphere presents. When it comes to its orbit, the planet takes almost 12 Earth years to complete one complete orbit around the Sun, at an average distance of 778.340.821 km, five times that between our planet and the Sun. Its trajectory around the Sun is elliptical, which eccentricity value is about 0.04838 and it has an inclination angle of 1.3 degrees, approximately, with respect to the Plane of the Ecliptic1. Its composition is very similar to the Sun’s composition. It consists of hydrogen and helium particles, with much more quantity of hydrogen than helium. Its large atmosphere give cause for a strong variation of pressure and temperature from the outermost layer to a liquid surface. This surface is the result of lots of bars of pressure forcing the hydrogen to a liquid state. Although it may have a rocky core formed by other elements there is not enough information so as to demonstrate the existence of this core.
Jupiter’s atmosphere The Giant Planet presents the widest atmosphere of the our System, which has an altitude of approximately 5000 km. Since the atmosphere does not have a clear transition from liquid state to gaseous state, the base of the atmosphere is taken as the height at which the pressure is of 1 bar. It is divided, similarly to the Earth, in troposphere, stratosphere, termosphere and exosphere, as it is shown in Figure 1.2. If the temperature gradient throughout the atmosphere of Jupiter is compared to the one of Earth it can be seen that they do follow a similar pattern. Starting from the troposphere up to the stratosphere the temperature decreases with the altitude. As the stratosphere is analysed a it can be noticed that the temperature remains in a constant range of values. Then, in the termosphere, as altitude increases temperature rises, contrary to the troposphere. Finally, in the exosphere, which is the transition between the atmosphere itself and the outer space, the temperature is about 900 k. While the atmospheric temperature structure of Jupiter is similar to that of the Earth, the Jovian planet provides astonishing phenomena and dynamics which can be object of a endless list of studies. The atmosphere, in particular, the troposphere contains different types of clouds, seen as bands located in determined latitudes. It has been determined that there are mainly three different levels of clouds. Those which are at the upper height are made of ammonia ice particles, the second level clouds main compound is ammonium sulfide and the lower ones are mostly made of water particles. The latter have a higher influence in the atmospheric phenomena. More clouds exist under the main layer, which compounds different from those which form the main cloud roof.
1The Plane of the Ecliptic is an imaginary plane which contains the orbit of the Earth around the sun, as it is established in [7] 1.2. JUPITER 5
Figure 1.2: Vertical temperature structure of Jupiter’s atmosphere. Extracted from [8]
As it can be seen in Figure 1.1, its atmosphere provides an astonishing scenario where any fluid interaction governed by planetary scale fluid dynamics can be perceived. The visible atmosphere is organized in bands 2. On the one hand, if the bands are brighter, they are known as zones, whose clouds are made of ammonia ice particles which remain at high altitudes. On the other hand, the darker bands are known as belts, which have a lower concentration of ammonia and whose clouds remain at lower altitudes and are hotter than those of the zones. Belts and Zones’ origin is still unknown. The interaction between these bands is the reason of the giant storms observed. This atmospheric movement can be attributed to strong winds which take place when a zones and bands boundaries meet. Jupiter presents one of the most strongest wind jets in the Solar System, which can reach up to 530 km/h. The strongest jets observed occur in the equator as the Figure 1.3 presents. In addition, the direction of these winds can be towards the east, prograde, if the transition is from zones to belts. On the contrary, at a transition from belts to zones, the jet is westward or retrograde.
2Band is the name given to a planet surface portion with kind of ring shape 6 CHAPTER 1. THE GIANT PLANETS
Figure 1.3: Zonal winds of Jupiter’s atmosphere. Data extracted from [9]
The diagram of Figure 1.3 provides the wind structure of the outermost atmospheric layer of Jupiter. The variation of the wind profile along the whole Gas giant can be observed. The strongest jets remain in latitudes from -20 up to 20 degrees. The Great Red Spot remains south of the westward strongest peak at -20 degrees of latitude, close to it.
The Great Red Spot Jupiter is host of the biggest known storm in the Solar System, which size currently is of 16.350 km. This feature is an icon of the planet. It is the oldest feature known and still active in the Jovian Planet and in the Solar System. It is characterized by its anticyclone dynamics and its high-pressure. Although a high amount of information and knowledge has been acquired, its origin is not determined yet. The first observations of the current storm date from the 19th century, in 1831. However, previous observations of a similar phenomena in the Jovian planet exist, but since those observations lack of well detailed information, a link between the current feature and that observed before cannot be established. 1.2. JUPITER 7
The first observation ever performed took place in the 17th century, when in 1664 Robert Hook first saw according to [10], however the explanation he gave was not enough consistent in comparison with the one provided by Giovanni Cassini in the same century, with observations that took place from 1665 to 1713. Figure 1.4) shows one of Cassini’s sketches of the planet. In addition to that, a time gap of a hundred of years separates these incidents from the first recorded observation of the current storm, which took place in 1831. Then it was until the 1879, that the scientist started to explore and study the feature.
Figure 1.4: Sketch of Jupiter drawn by Giovanni Cassini in December 1690. Extracted from [11].
When it was first saw it had a reddish color, and the storm was bigger than twice the size it has nowadays, of about 40.000 km long, three times the Earth’s diameter. Currently, its size is of 22 degrees, and it is expected to have a round shape by 2040, due to it is shrinking since it was discovered. The storm has been reducing its size, and changing its color. It is thought that its reddish color is related with the interaction between its chemical composition, mainly ammonium hydrosulfide, and the ultraviolet incidence from the Sun. However some observations reveal that the color is not constant and has changed, from red to white. 8 CHAPTER 1. THE GIANT PLANETS
Figure 1.5: Image of the Great Red Spot taken by Juno Spacecraft. Instrument: Jun- oCam. Extracted from [6]
The figure Figure 1.5 shows the Great Red Spot as seen from Juno Spacecraft in 2019. It is a recent image in which it can be noticed that the spot remains between a belt and a zone, following anticyclonic dynamics. The cloud color is a combination of orange and red result of the interaction of the gases it contains with the Sun’s radiation. A very violent region west to the storm can be seen as a result of its rotation and the interaction with the belt and the zone. One the one hand, the Great Red Spot region is known to be colder due to the high altitude of the storm. The clouds of the storm are above the rest of the surrounding clouds. On the other hand, the storm atmosphere is warmer than its surroundings, at a temperature of 1.600 K, due to the amount of energy in form of gravity and acoustic waves that the storm emanate and produce. Those waves also help to increase the turbulent behaviour of the feature. Another interesting fact about the storm is the wind profile velocity which increases in the periphery and slows down in the rotating edge. Studies reveal that its peak velocity has reached up to 432 km/h. Regarding its drift velocity, it is thought to change depending on the brightness of the belt where it remains. The Great Red Spot average rotation period was the main reference on which the System II was based on, whose length is of 9 hours 55 minutes and 42 seconds. From a research point of view, it has to be considered that this amazing phenomena has been a strong reason, among others, to encourage the scientific community so that missions towards Jupiter System can be funded and accomplished. Thus, spacecrafts missions such as Voyager II in 1979, which flew from a distance of 9.200.000 km or Juno Spacecraft, which visited the planet in 2017 and flew closer at a distance of 8.000 km, have been carried out aimed at providing new information so that the dynamics of the Great Red Spot can be better understood. 1.3. SATURN 9
1.3 Saturn
Moving on a distance twice that between Jupiter and the Sun, at 9.5 AU Saturn is reached. It is the sixth planet from the Sun, and after Jupiter, it is the biggest compared with the rest, its surface radius is nine times that of the Earth. Its color is kind of beige and its caused by ammonia ice particles. The planet presents an amazing and complex rings system. It makes Saturn to be considered as the most beautiful planet for a lot of people. More than a planet characteristic, the rings have become an icon of Saturn. In the same way of the other outer Solar System bodies, it is mostly made of gas and presents the lower density, which value is 0.687 g/cm3 as the Table 1.1 shows. However, this fact is counteracted by the planet’s size, which leads the Saturn to be the second heaviest body. The gas giant is named after the Roman god of agriculture and wealth, Saturn, which it the mythology was also the father of Jupiter. The system of Saturn is not only composed by the planet but a large quantity of moons, 53 which are already named and confirmed and 29 still pending to be investigated. The biggest of the natural satellites is Titan, although other very interesting moons are found such as Enceladus or Mimas.
Figure 1.6: Saturn seen from Cassini spacecraft. Instrument: ISS - Wide Angle Camera. Extracted from [6]
The human being knows about the existence of Saturn since prehistoric ages. As well as Jupiter, Saturn was first studied by ancient civilisations such the Babylonian, the Greek which named the planet for Phanion or the Roman which knew the planet as Star of Saturn. Other Asian people such Indian or Japanese also knew about its science. After this first age of naked-eye study of the planet, terrestrial observations using telescopes started. Galileo Galilei was one of the first humans to see its rings, in addition, he discovered a gap between two of the rings which is currently known as Cassini Division. However, the discoverer of Saturn’s Rings was Christiaan Huygens, when it 1659 using a telescope with better characteristics than the one used by Cassini, saw the rings. Later, 10 CHAPTER 1. THE GIANT PLANETS with the contribution of different astronomers the system got complex and complete, and moons number and data to the history of Saturn increased. The last and current stage of exploration of Saturn is carried out by the Space Agencies, all over the world. Different missions and spacecraft have already flown to Saturn system such Cassini-Huygens, Voyager’s or Pioneer and other such Dragonfly mission which have to visit it. The Figure 1.6 above presents a view of the gas giant. There the ring system can be seen as well as some other iconic features such the hexagon of the pole. As the image shows, the atmosphere of Saturn is not as band-limited as that of Jupiter, the transition between the different regions from pole to pole is smoother, a layer of clouds and haze covers the turbulence occurring at lower altitudes. In order to put numbers to its characteristics, the equatorial radius of Saturn measures 58.232 km, the volume is 765 times approximately larger than the Earth’s one and has a gravity of 10.4 m/s2, similar to that of the Earth. As it is stated above, Saturn orbits around the Sun at a distance of 9.54 times the distance between Earth and the star, and each orbital completion takes the planet 29.4 Earth years, at a velocity of 34.7 km/h. The light takes almost 84 minutes one way from Saturn. Moreover, it is tilted 26.7 degrees with respect to the Plane of the Ecliptic, thus, the planet hosts a climate behaviour similar to that of the Earth, though the effect of the Sun is weaker in comparison with the effect produced on the Earth. Unlike its years, the rotation of Saturn quite fast, it takes the planet 10.7 hours to spin around itself. This time is not constant along all the planet surface, and that is because the shape of the body is an oblate spheroid rather than a sphere due to the high rotation velocity. Thus, three different systems were established. System I and System II were based on the rotation rate of determined regions of the planet, while the third system was based on the internal rotation rate, determined by its radio emissions. Later it was demonstrated that the radio emissions do not match with the internal rotation rate. Saturn is mostly made of hydrogen and helium. Its core is formed by metals like iron and nickel inside a rocky layer. The materials reach a solid state due to temperature and high pressure. Outside this inner part, there is a layer of metallic hydrogen covered by liquid hydrogen. Saturn’s atmosphere In the same way of Jupiter, the surface of Saturn does not have a clear transition from liquid to gas state, so scientists have decided to establish a standard pressure of 1 bar. The altitude at that pressure is thought to be the reference altitude at which atmosphere starts. Its vertical structure also varies with height and pressure. The outermost layer of the atmosphere contains ammonia ice particles of yellow, grey or brown colors and it is observed to remains between 0.5 and 2 bars of pressure with a temperature between 100 and 160 k. As the pressure increases the composition of the clouds changes. Under the 1.3. SATURN 11 ammonia ice layer, a wide layer of water ice particles is found, bounded by the pressure range between 2.5 and 9 bars, which temperature range goes from 185 k to 270 k. Within this layer, it has been noticed that a cloud level composed by ammonium hydrosulfide remains between 3 and 6 bars of pressure. It can be noted how similar are the Giant Gas planets structures and how many characteristics the atmospheres of each planet share. Finally, between a pressure from 10 to 20 bars, it is thought that the atmosphere contains water droplets mixed with ammonia at a temperature above 270 k up to 330 k. Saturn’s inner atmosphere provides an unfavourable outlook regarding scientific ex- ploration since its conditions, and particularly its pressure are still a challenge for the technology today. All the data is recorded from the craft during its orbit, only a small piece of the information available was acquired during Cassini plunge inside the atmo- sphere of the planet. However, outer observations have shown different features and phenomena which study is hard and tough and leads to enough work and investigation. The planet hosts different types of atmospheric events located or appearing along all the surface, as manifold observations and studies have shown. However, it has been observed that particular zones are prone to be home to this features. These zones can be divided by the poles, high and equatorial latitudes and the Storm Alley, which as the name suggest, is a zone where storms regularly appear.
Figure 1.7: North Pole’s hexagon as seen from Cassini. Instrument: ISS - Wide Angle Camera. Extracted from [6]
The polar regions are zones of permanent appearing random disturbances and storms with a central main vortex. A beautiful enormous hexagonal pattern has been discovered in the north pole , first seen by Voyager mission, and seems to be stable and remain 12 CHAPTER 1. THE GIANT PLANETS constant. The poligon wave is presented in Figure 1.7 at a latitude of 78 degrees north. Its side measures almost 14000 km long and its rotation period is of 10 h 39 minutes and 24 seconds, apparently. No drift has been observed and its nature is not completely known. Many scientist say that it may be a Rossby wave. 3. In addition to phenomena in polar regions, different storms have been observed in Saturn’s surface. On the one hand, another famous and studied case is the Great Whit Spot. This feature is a giant storm which takes place in the Saturnian surface once every 30 Earth years, approximately, during the northern hemisphere’s summer solstice. Previous similar features have been observed since 19th century. The last one observed took place in 2011 and scientist thing that the next one will occur in 2020. This repetitive feature is characterized by its powerful nature, gathered during a short period of time as well as a high drift velocity. While the head of the feature, the main and strongest part of the storm, moves westward, the tail of the storm covers the planet towards the east, at the same latitude as the Figure 1.8a shows. When the storm is about to disappear, in some cases it can be observed that the head and the tail clouds have merged and no difference between them exists. Only a ring along the band is seen. Information about previous Great White Oval events is provided in [13]
(a) Great White Spot captured in the (b) Convective storm in the southern north in 2011 hemisphere
Figure 1.8: Images captured by Cassini orbiter. Instrument: ISS - Wide Angle Camera. Extracted from [6]
On the other hand, storms smaller than the Great White Spot have been observed in the planet’s surface. The Figure 1.8b provides an example. The phenomena seems to
3Rossby wave, also named planetary wave, belong to the group of inertial waves. It appears in planetary scale size phenomena and it is thought to be based on the rotation of the planet. Its movement can be understood as an oscillation based on the conservation of potential vorticity. The first to define it was Carl-Gustaf Rossby. See [12] 1.3. SATURN 13 begin its activity in lower levels of cloud, then erupt upwards until it can be noticed in the top atmospheric layer of Saturn. It is thought that these storms are accompanied by Saturn electrostatic discharges as it is explained in [14]. The storms are intermittent, thus, they can appear and be active during several weeks and then disappear by weeks or months. The region where the storms take place is known as the Storm Alley. Finally, as well as Jupiter atmosphere, Saturn also provides a wind structure which varies along the latitude of the planet. Thanks to image navigation studies developed using space probes data, the wind structure has been determined during the late 20th century, mainly with Voyager and Cassini probe data. Moreover the bands in the atmosphere of Saturn are fainter than those of the fifth planet, thus, interaction between zonal winds and the clouds is harder to be observed. A diagram of Saturn’s zonal wind variation is provided in Figure 1.9.
Figure 1.9: Zonal winds of Saturn’s atmosphere. Data extracted from [15]
The Figure 1.9 shows the wind function obtained from image navigation. It can be noted that two different diagrams are provided. The black curve corresponds to Voyager data while the red one to Cassini mission data. Unlike the Jovian wind structure, this one presents an evolution between the missions. The main difference observed is in the 14 CHAPTER 1. THE GIANT PLANETS equator jet, which has suffered a decrease in its velocity during the period between the two probes. This means that the atmosphere over the equatorial region presents a slower rotation rate. As the diagram shows, such high velocities take place in the Saturnian atmosphere, with values of 500 km/h approximately. Saturn’s wind velocity is the second higher velocity observed in the Solar System, after Neptune’s. The atmospheric dynamics of the planet are still being studied and there is not a de- termining model which solutions for the understanding of the phenomena can be extracted from. Rings system Although Saturn is not the only planet with a ring system, it is the most beautiful and visible disk-like structure. Its rings are a reference source of data which have helped for the study of other planetary rings. Saturn’s rings remain at a distance of 6 thousand kilometers approximately with regard to the surface of the planet, up to 127700 km. They are composed of ice particles mainly. Nevertheless, rocky materials as well as moon remains can be found in the rings. More accurate information is presented in [16] not only about its composition but the whole topic. The size of the particles goes from dust-like particles up to pieces as big as a house and their thickness value can oscillate between 10 m and 1 km.
Figure 1.10: Panoramic view of the rings as seen from Cassini. Instrument: ISS - Narrow Angle Camera. Extracted from [6]
They are named alphabetically, according to the discovery order. The first three rings discovered, A, B and C, are the most relevant ones, while the D, E, F and G are thinner and less visible. An important gap, of about 4700 km, separates the A and B rings, and it is named for the Cassini Division, as it has been stated above. The closest ring to Saturn’s surface is D ring, followed by the C ring, B ring, the Cassini Division, the A ring, then the F and G ring, and finally the E ring. Far from the E ring, a ring called Phoebe ring exists, which matches with the orbit of the moon Phoebe and it is thought to be originated by that satellite. In addition to Cassini Division or Encke Gap, data from Voyager mission showed that there are more gaps than those visible, many other smaller gaps have been discovered during the Voyager mission. The origin of the rings is not clear, some scientists thing they were created in the beginning of Solar System formation. However, data from Cassini suggests the opposite, 1.3. SATURN 15 it is an early structure. Since they are not visible to the naked eye, the first person to observe them was Galileo with its telescope in 1610, though he was not able to identify them. Currently, the rings are known thanks to Huygens which thanks to an improved telescope was able to identify and present the structure to the rest of the people. Within the rings, different moons orbit around the planet. It is thought that these moons work as a mechanism so that the ring particles do not escape from Saturn’s gravity. Enceladus has also an effect in the rings, since part of the ice which can be found in the disks is thought to derive from the moon. Moreover, during some missions such Voyager or Cassini, the scientists have studied these structures not only by imaging them but experimenting with radio occultations when the spacecraft flies behind the rings.
Figure 1.11: Wavy structures in the A ring as it crosses the gap. Image acquired by Cassini spacecraft. Instrument: ISS - Narrow Angle Camera. Extracted from [6]
Nowadays the nature of the rings is not well known, but the information available leads to thing that the physics which govern the structure are not as simple as they seems to be. Different factors can affect to the ring structure such as moons’ gravity effect, resonance of the orbital period of moons and that of gap particles or gravitational waves. The Figure 1.11 shows a type of perturbance captured by Cassini-Huygens orbiter in the A ring produced by Pan 4. Furthermore, the variability of brightness observed from the Earth in Saturn is the result of changes in the rings. Moreover, the disks present an almost negligible atmosphere which is composed of OH mainly. It is formed after the disintegration of water particles ejected by Enceladus when interacting with energetic ions. Although the disks seem to be thin, smooth and perfectly plane, disturbances in its shape have been observed, which are not easy visible due to the small size in comparison with the rest of the rings.
4Pan is the closest moon to Saturn. It has a raviolli-like shape and orbits within the Encke Gap. It maintains the gap stable, though it has gravitational effects on the rings. 16 CHAPTER 1. THE GIANT PLANETS
To sum up, both planets share a similar nature and characteristics, however, there are manifold details which establish a difference between the Gas Giants. On the one hand, Jupiter and Saturn are the Giants of the Solar System, they present a gaseous nature still pending to be understood by Scientist, due to the unfavourable scen- ario regarding pressure and temperature conditions. Nevertheless, scientist have managed to deal with the constraint for hundred of years as Galileo did at that time. Humans have extracted as much data as available from observations so that a reference frame such the one presented can be exposed. Thanks to the observations in any form, today it is known that the giants are host of unbelievable landscapes and atmospheric phenomena such the Great Red Spot, the Storm Alley’s long-lived storms or the similar pole dynamics occur- ring in both planets. And exploration is not finished yet, projects such Dragonfly probe or Europa Clipper orbiter are being prepared so that new information about the system of Jupiter and Saturn aimed at completing knowledge will be acquired. Both Jupiter and Saturn planets are surrounded by a complete system of moons, some confirmed and others still being studied. As well, the planets present a ring system, fact that is obvious for Saturn but not so much for the Jovian system. When it comes to its atmospheric composition, it is known that the leading role belongs to hydrogen followed by helium, moreover, the cloud levels in Jupiter and Saturn are similar, with a great amount of am- monia and water. On the other hand, many properties differ from a planet to the other, starting from the size. Jupiter is bigger than any other body, as well, it is denser than Saturn. Saturn’s density is such low that it could float in water. Saturn is twice farther than Jupiter, and its rotation and orbital period are slower than that of Jupiter, however winds in Saturn are much faster than Jupiter’s. In addition, when the planets are observed, Jupiter presents a mix of brown, white and reddish bands’ color while Saturn hue mix is smoother. Put- ting aside main characteristics, the landscape provided by each planet is quite different. Jupiter atmosphere is faint and the phenomena such the Great Red Spot which takes place in lower levels can be easily observed, as well as a great part of the interaction between well defined bands moved by winds along its surface. In turn, Saturn panorama is different, while the interaction between bands is not as astonishing as that provided by the jovian planet, the ring system together with the planets smooth varying hue are object of many amazing portraits in the Solar System which cannot be found anywhere else. 2 The Shallow-Water Model
The intention of this chapter is to present and explain the physical model used which is the base of the software run for the numerical simulations. In addition, the hypothesis taken as well as the bases of the model will be explained in this stage of the thesis. The software is the result of a beautiful project managed and carried out by members of UPC Tuareg, Professor Enrique Garcia Melendo and Professor Manel Soria Guerrero, in which some former and present students have participated. Since the great computational part of the project is out of the scope, the part regarding the code is not be explained. However, some details may be cited so as to benefit the understanding of the explanation. An explanation of the main equations is presented, followed by the development of them and the process of assumptions taken as well as the particularities of the model caused by the assumptions. Finally, the resulting equations which are used and com- puted are stated, and the procedure to solve them are described. The main part of the information presented has been extracted from the book [17] whose 3rd chapter explains the physical model.
2.1 Introduction to the Shallow-Water Model
Its function is to study and reproduce motion of geophysical fluids which are charac- terized by being shallow, a thin layer of fluid which is affected by gravity and Coriolis accelerations, thus, the fluid layer is considered to be rotating. The system is based on the Navier-Stokes equations. It has to be considered that the model uncouples the physical motions from the thermodynamics. Thus, the dependency of the stratification of density along the height in this type of fluids is not considered. However, the model is still capable of provide good results and reproduce with accurate details the motion of the fluid layer. In addition, the fluid is considered to be geostrophic which means that Coriolis effects are supposed to counteract the pressure gradient forces and the motion of the flow is parallel to the constant pressure lines. Thus, perpendicular to its gradient.
17 18 CHAPTER 2. THE SHALLOW-WATER MODEL
Figure 2.1: Fluid layer of the model. Extracted from [17]
Figure 2.1 shows the model layer which is used for the calculations. As it can be noted, different parameters are included. First of all, the reference frame is determined by x, y and z axis, where x and y form the horizontal plane and z corresponds to the vertical dimension. Density, ρ, is constant; µ, which represents friction is considered to be zero, in the types of motion studied it is negligible. The body force is represented with a vector named g, perpendicular to the z = 0 plane but in the opposite direction of z-axis. Regarding the rotation of the fluid, the angular velocity is represented with Ω symbol, and the rotation axis matches with the z-axis. Derived from angular velocity, Coriolis is 2Ω. When it comes to the fluid layer, two heights are presented, one for the top surface, h, and the other for the bottom, hb. Unlike the latter, the first one depends on time. With the frame axis, the velocities are also established as u, v and w. Finally and most important, the dimensions of the layer. The shallow-model is characterized by
D δ = << 1 (2.1) L
Equation 2.1 represents the aspect ratio, the relation between the vertical and hori- zontal scale of the layer. As the name of the model suggests, the layer of fluid is thin, the vertical dimension is very small compared with the horizontal one. The atmosphere and the ocean meet this condition, they are layers of fluid, much more larger in horizontal scale than in vertical. Think about a hurricane in the Earth, the height of the feature can reach up to 18 km approximately and it can reach up to 480 km, as [18] presents. Now, if the situation is brought to Jupiter and Saturn, the feature horizontal span can be much higher while the difference of height between the case in the Earth or the scenario in the Gas Giants is not so different. 2.2. OBTAINING OF THE SHALLOW-WATER MODEL EQUATIONS 19
2.2 Obtaining of the Shallow-Water Model equations
In first place, a reference frame of the equations is provided. The intention particular terms which are involved during the procedure and justify them. The model is based on Navier-Stokes equation as it is stated above. The first condition assumed is that the fluid is incompressible, and the equation used to establish this is the Continuity Equation:
δu δv δw + + = 0 (2.2) δx δy δz
The Momentum Equations determine the motion of the fluid, and the expression used is presented below. See that Coriolis term is included and considered, since the phenomena studied has planetary scale size and rotates.
Du δu δu δu 1 δP + u + v + w − fv = − (2.3) Dt δx δy δz ρ δx
Dv δv δv δv 1 δP + u + v + w + fu = − (2.4) Dt δx δy δz ρ δy
Dw δw δw δw 1 δP + u + v + w = − − g (2.5) Dt δx δy δz ρ δz
The figure below presents a diagram which includes the angular velocity of the planet, Ω, the reference frame established for any point P over the surface of the globe and the latitude angle, φ, at which the point is located. 20 CHAPTER 2. THE SHALLOW-WATER MODEL
Figure 2.2: Diagram of a rotating sphere with a reference frame established over its surface at a random point
So, in order to explain where does the Coriolis term come from, the following expression is presented where ac is the Coriolis acceleration:
~ ac = 2Ω × ~u (2.6)
The application of Equation 2.6 provides the breakdown of Coriolis acceleration in the vertical plane yz of the reference frame presented in Figure 2.1. Particularly, the Coriolis component of interest is the one perpendicular to the globe surface on the point P:
~i ~j ~k
ac = 0 2Ω cos φ 2Ω sin φ (2.7)
u v w
The Coriolis components are obtained applying the operation of Equation 2.7.
~ ac = 2Ωw cos φ~i − 2Ωv sin φ~i − 2Ωu cos φk + 2Ωu sin φ~j (2.8)
The result can be simplified if it is assumed that w has a value much lower than u and v. In addition, since only horizontal velocities are considered in the model, the third term of the result can be neglected too, obtaining the following value:
ac = −2Ωv sin φ~i + 2Ωu sin φ~j (2.9) 2.2. OBTAINING OF THE SHALLOW-WATER MODEL EQUATIONS 21
The terms obtained previously are included in x and y Momentum Equations, as it can be observed. Finally, notice that f = 2Ω sin φ. Putting aside Coriolis and focusing in pressure field, the term of the right in equations 2.3, 2.4 and 2.5 is analysed. The pressure field follows the next expression:
P (x, y, z, t) = −ρgz + P˜(x, y, z, t) (2.10)
The first term is only dependent on z direction, hence if derivative operation is applied in x and y direction, it can be noticed that the gravitational term disappears:
1 δP 1 δP˜ = (2.11) ρ δx ρ δx
1 δP 1 δP˜ = (2.12) ρ δy ρ δy
Following the same procedure for z-direction, the result involves the gravitational term, but when the result of the derivative is brought to the equation, the gravitational term of the result is cancelled with the gravitational term of regarding the Momentum Equation. The procedure is presented below: ! 1 δP 1 δP˜ = −ρg + − g (2.13) ρ δz ρ δz
Thus the resulting term has the same form of that for x and y direction.
1 δP 1 δP˜ = (2.14) ρ δz ρ δz
2.2.1 Dimensional analysis
Once the equations are presented and its terms’ obtention is justified, a dimensional analysis is carried out. Not only scales of distance are used, but also velocity, time and pressure scales. This procedure is intented to provide the value order of the variables involved. For instance, in the case of δ, it has a value order of a hundredth which is represented as the ratio between the vertical and horizontal scale:
δ = o (0.001)
With respect to the velocities, two different scales are used. One is for the horizontal velocities and the other for the vertical one since the latter is much smaller than u and v. 22 CHAPTER 2. THE SHALLOW-WATER MODEL
u = v = o (U)
While the horizontal velocity scale is stated as U, the vertical velocity scale is obtained using the aspect ratio. Think about the motion of a particle inside the fluid layer. Ac- cording to Figure 2.3, the particle which moves from A to B, has vertical and horizontal velocity and the time that it takes to move a distance L at a velocity u is the same time which takes to cross the from the bottom to the top of the layer at a velocity w.
Figure 2.3: Diagram of the motion of a particle
Thus, considering that the time is the same for the vertical motion and for the hori- zontal one, the next equation states the times as equivalent and provides a solution for the obtaining of the vertical velocity scale.
W U = D L
D W = U = δU L
Velocity analysis is left behind now that the scales for each one is known and the Momentum Equations are written in dimensional form:
U U 2 U 2 WU 1 P + + + − fU = − (2.15) T L L D ρ L
U U 2 U 2 WU 1 P + + + + fU = − (2.16) T L L D ρ L
W UW UW W 2 1 P + + + = − (2.17) T L L D ρ D 2.2. OBTAINING OF THE SHALLOW-WATER MODEL EQUATIONS 23
The x and y equations are similar, while the z equation is quite different. For the L term of the time derivative, a scale is also easy to be established. Since T = U , the first term of the Momentum Equation is written as:
U U 2 = (2.18) T L
WU With the objective of establishing a common scale for all the terms, the scale of D U is rewritten. See that the term is equivalent to L :
WU DU 2 U 2 = = (2.19) D LD L
Finally, two different scales are obtained for the left side of the equation. Now that the Momentum Equation is reduced so that only to different scale values are used, it is easy to work with and the scale of the pressure field can be established. Using Equation 2.15, for instance, the intention is to leave alone the P variable:
U 2 P , fU = o (2.20) L max ρL
See Equation 2.20, two scales are proposed which can correspond to the pressure field. The biggest of the terms is the one used and the other term is neglected. Furthermore, the equation states that the term to be used is of the order of the pressure field term. But the pressure field is not alone yet. Thus variable ρ and L are moved to the left side:
U 2 P = ρL , fU (2.21) L max
Observe that Equation 2.21 establishes a comparison between inertial and Coriolis terms. As it is mentioned above the smaller value of both is neglected. The ratio which provides the comparison between these forces is the Rossby number
DU/DT U 2/L U = = ≡ Ro (2.22) fU fU fL
According to [17], two cases of interest appear regarding the value of Ro. The value of the number can be higher or lower than 1. If the non-dimensional number is higher than one it means that the inertial acceleration is greater than the Coriolis acceleration, thus the latter can be negligible. Otherwise, if Rossby number is Ro << 1 the inertial term is neglected and a situation of hydrostatic equilibrium can be assumed. That means that pressure is always equal to the weight of the fluid column which is over the object point. Furthermore, only the first term of the pressure field depends on z and the excess 24 CHAPTER 2. THE SHALLOW-WATER MODEL of pressure which apppears as P˜ becomes P˜(x, y, t). Hydrostatic equilibrium leas to a velocity definition such:
1 δP˜ v = (2.23) ρf δx
1 δP˜ u = − (2.24) ρf δy
The result is a cross vector product obtained form the Momentum Equations whose inertial term is deleted. Moving on the effect of pressure in this situation, it is usually explained using the acceleration caused by the pressure gradient, instead of explaining it as a force effect. The definition of this ap states that it increases in the opposite direction of the pressure:
1 a = − ∇P (2.25) p ρ
Figure 2.4: Diagram of constant pressure straight lines and the gradient direction
Figure 2.4 presents a scheme which represents Equation 2.25. The diagram contains a pressure field which includes the constant pressure lines and its value as well as the direction of the gradient and that of the pressure acceleration. In hydrostatic equilibrium, the fluid tends to rotate around the center of the pressure perturbance, instead of filling the hole that pressure creates such normal fluid does. The flow velocity is perpendicular to the pressure gradient. 2.2. OBTAINING OF THE SHALLOW-WATER MODEL EQUATIONS 25
Figure 2.5: Diagram of constant pressure concentric curves and the gradient direction
Figure 2.5 is intended to give an idea of how does the fluid moves under Coriolis effects and assuming hydrostatic equilibrium. The concentric black curves are constant pressure lines, and the direction of the pressure variation is outwards the center. The red lines represent a time line of fluid particles in three different instants. The fluid moves in order to compensate the difference of pressure, and the motor is Coriolis effect. Other form to understand the horizontal motion which appears is to think about a fluid column inside a layer. See Figure 2.6, imagine that the lines which link 1 and 2 with the top surface are the walls of the fluid column and its length is D + η − z. The wall of point 1 will be put down under a pressure p1 = −ρg(D + η1 − z) > p2 = −ρg(D + η2 − z) which is the pressure of the second point. Thus, a difference of horizontal pressure along the walls appears and causes the horizontal movement of the whole column.
Figure 2.6: Fluid layer under hydrostatic conditions 26 CHAPTER 2. THE SHALLOW-WATER MODEL
To sum up, now it is known that the second term of pressure field of Equation 2.10 does not depend on z and the velocities u and v neither, under this conditions. The pressure field expression is rewritten as:
P (x, y, z, t) = −ρgz + P˜(x, y, t) (2.26)
Applying the conditions to Momentum Equations, their expression changes to:
Du δu δu 1 δP˜ + u + v − fv = − (2.27) Dt δx δy ρ δx
Dv δv δv 1 δP˜ + u + v + fu = − (2.28) Dt δx δy ρ δy
1 δP˜ = 0 (2.29) ρ δz
2.2.2 Demonstration
Now that the necessary knowledge is exposed, the demonstrations of the Shallow- Water Model equations can be presented. The equations have been transformed and a reference of the terms used is established. First of all, pressure field is mathematically manipulated. As it can be seen, Figure 2.7 shows an example of a fluid layer which has been perturbed through pressure variation. The reference frame is established by x, y and z. The function which represents the surface is η, which only depends on horizontal coordinates and time. The blue wave represents the difference between D and the height of the surface h. At a height of scale D the pressure has a value of p0. The layer meets the aspect ratio condition, δ << 1. Notice that the diagram is coherent with Figure 2.1. 2.2. OBTAINING OF THE SHALLOW-WATER MODEL EQUATIONS 27
Figure 2.7: Diagram of a fluid layer whose surface is perturbed
The height can be written as:
h = D + η(x, y, t) (2.30)
In order to determine P˜, the first boundary condition is applied. The pressure field is analysed at a height h: ˜ P (x, y, h, t) = − (ρg(D + η)) + P = po (2.31)
˜ P = po + (ρg(D + η)) (2.32)
Using P˜ expression in pressure field, the pressure at any point is given by:
P (x, y, z, t) = −ρgz + po + (ρg(D + η)) (2.33)
Since only P˜ is dependent of x and y, the derivative operation regarding these direc- tions is only applied to that term. See equations below:
δP˜ δη = ρg (2.34) δx δx
δP˜ δη = ρg (2.35) δy δy
Finally, if the results of 2.34 and 2.35 are introduced in equations 2.27 and 2.28, the final Momentum Equations of Shallow-Water Model are obtained: 28 CHAPTER 2. THE SHALLOW-WATER MODEL
Du δu δu δη + u + v − fv = −g (2.36) Dt δx δy δx
Dv δv δv δη + u + v + fu = −g (2.37) Dt δx δy δy
Two of the three equations which determine the physics of the model are stated. The third equation is derived from the incompressibility condition of Equation 2.2. Focusing on the Continuity equation, if horizontal direction terms are moved to the right side, the expression results in:
δw δu δv = − + (2.38) δz δx δy
Equation 2.38 is now integrated in z direction:
δu δv w = − + z + A(x, y, t) (2.39) δx δy
In order to determine A, boundary conditions at the surface and at the bottom of the Dz layer are established. Firstly, it is known that w is equivalent to Dt . If the equation is analysed at the top of the layer, the expression is written as:
D(D + η) δu δv = − + (D + η) + A(x, y, t) (2.40) Dt δx δy
Since D is a constant, it is not considered in the left term. Leaving alone A:
Dη δu δv A(x, y, t) = + + (D + η) (2.41) Dt δx δy
Replacing A in Equation 2.39, the expression obtained is:
δu δv Dη δu δv w = − + z + + + (D + η) (2.42) δx δy Dt δx δy
The second boundary condition is applied now. Equation 2.42 can be analysed at the bottom of the fluid layer, where w = z = 0:
Dη δu δv 0 = + + (D + η) (2.43) Dt δx δy 2.3. PHYSICAL BASES OF THE ALGORITHM 29
If Material Derivative’s terms are written:
δη δη δη δη δu δv + u + y + w + + (D + η) = 0 (2.44) δt δx δy δz δx δy
Take into account that since eta does not depend on z there are terms which can be deleted:
δη δη δη δu δv + u + y + + (D + η) = 0 (2.45) δt δx δy δx δy
Now it can be seen that the derivative terms can be rewritten as:
δh δhu δhv + + = 0 (2.46) δt δx δy
Equation 2.46 becomes the third final equation which determines the physics of the Shallow-Water Model. Particularly, this equation provides the information about the perturbance in the layer surface while equations 2.36 and 2.37 are used to calculate the velocity field.
2.3 Physical bases of the Algorithm
Now that the Shallow-Water Model equations are established, a scheme of the al- gorithm process is provided.
Du δu δu δη + u + v − fv = −g (2.47) Dt δx δy δx
Dv δv δv δη + u + v + fu = −g (2.48) Dt δx δy δy
δh δhu δhv + + = 0 (2.49) δt δx δy
In order to reproduce and then study the phenomena, the initial case of the fluid layer needs a perturbance input. The fluid layer remains stable until this instability is introduced. There are two ways to do it. The perturbance can be an instability located in the surface of the layer or it can be created by modifying the velocity field so that the stability is lost. In any case, the algorithm calculates the evolution of the system. 30 CHAPTER 2. THE SHALLOW-WATER MODEL
Firstly adn in general terms, the code first calculates solves Equation 2.49 obtaining the function η, which as it is mentioned above, represents the perturbance of the surface, it is to say the difference between D and the height of the layer’s surface. Once η function is known, the software uses the result to solve Equation 2.47 and Equation 2.48. The result obtained from the latter equations is the velocity field.
2.4 Potential Vorticity
First of all, vorticity is explained in general terms. Vorticity, which definition is ω, is a very important variable in the field of geophysics fluid motion and dynamics. For a fluid under an uniform rotation rate, whose angular velocity is Ω, the vorticity can be understood as the curl drawn by the fluid as seen from a non-rotating inertial frame of reference. The expression of vorticity is:
ω = ∇ × u (2.50)
Notice that since the vorticity is defined as the rotational of the velocity, it is non- divergent:
∇ · ω = ∇ · (∇ × u) = 0 (2.51)
The variable can be local or can be absolute. The absolute vorticity for a rotating fluid is defined as:
ωa = ∇ × {u + (Ω × r)} = ωl + 2Ω (2.52)
It includes the local vorticity, ωl and the planetary vorticiy, 2Ω. The first term is related to the relative velocity, the linear velocity of the fluid while the second is caused due to Coriolis effect. In Chapter 2 Fundamentals of [17], a complete and very clear explanation is provided. The explanation includes the demonstration of the conservation of vorticity along a vortex tube which is bounded by vortex lines.1 Leaving behind voriticy and focusing on potential vorticity, as it is stated above, it is very useful for the analyses of meteorologic phenomena, geoplanetary rotating fluid. Although it is difficult to understand, a general and easy explanation is given in this section. Potential voriticity is a variable used for the analysis of data. If atmospheric phenom- ena studied had to be analysed using variables such temperature, the process would be
1A vortex line is that line whose points are everywhere tangent to the vorticity of the fluid. See reference [17]. 2.4. POTENTIAL VORTICITY 31 difficult because temperature changes during the time and the tracking of fluid particles during the phenomena period becomes difficult. The main point of potential vorticity takes place when a scalar fluid property λ which meets:
dλ = Ψ = 0 (2.53) dt In addition, the friction force is neglected, and either the fluid has to be barotropic or λ is thought to considered a function only fo p or ρ. If the requirements are met then the potential vorticity is conserved for each fluid element. Since potential vorticity is conserved, it can be used as a tracker, in addition it gives information of the meteorologic feature and the future behaviour that the phenomena can present as it is mentioned by scientists of NCAS in [19].
ζ + f Π = (2.54) H
where ζ is the relative or local vorticity, f is the Coriolis term and H is defined as h−hb in accord with Figure 2.1. From Equation 2.54 it can be seen that in order to conserve the value of Π, if H increases, ζ value becomes smaller and vice versa. The change in H can be understood, thinking about a fluid column, as if it stretches if H grows or as it the column shrinks and gets thicker if the value is reduced. If the fluid element is elongated it tends to rotate faster, zeta increases, otherwise the rotation rate slows down. The reference [20] good explanation of the conservation of potential vorticity. A special emphasis is put in this property because most of the results presented in following chapters are done using potential vorticity. As it is stated above, it gives a lot of information and is properties benefit the analysis. In summary, Shallow-Water Model provides results which still involve important phys- ical characteristics and behaviour despite of the fact that it does not consider density stratification of the atmosphere and consequently presents some limitations regarding thermodynamic effects. It is based on Navier-Stokes equations and considers a very small vertical velocity in comparison with the horizontal one, as well as the non-dependency on the z-coordinate. It is important to remark that the fluid layer studied has to meet the requirement of δ << 1, since the model is aimed at providing results of fluid systems such the atmosphere or the ocean. The scheme which follows is based on the resolution of the continuity equation, which is written in terms of layer height and then the calculation of the velocity field by the resolution of Momentum Equations. In order to reproduce phenomena, the system needs to be perturbed and become unstable. That is performed with g δη which in a way can δxi be assumed as the motor of the system. Finally, notice the importance of potential vorticity in meteorological science. The fact of being conserved for the fluid element ease the analysis and understanding of atmospheric phenomena dynamics. 32 CHAPTER 2. THE SHALLOW-WATER MODEL 3 Reference frame of the numerical simulations
The previous chapters have been moulding the necessary knowledge so that everything which has to do with the simulation work of the thesis can be well understood. The purpose of this one is to establish a reference frame for the reader regarding the simulation part of the thesis and explain the final results. The chapter includes a brief explanation of the resource used during the task accomplishment and also an explanation about how does the input process of the software works, it is to say the preparation of different files such the template, or the planet’s data, among others.
3.1 Barcelona Supercomputer Center
The part regarding to the numerical simulation is different to most of project types in many aspects. Since the objective of the thesis is to be able to reproduce phenomena or reach a determined result, a long path of continuous simulations running and data processing is carried out. To do so, a high computational effort is required. So that the reader can have an idea of the data volume produced, consider that at least 150 simulations have been carried out, and the computational space occupied is of almost 2000 GB of memory. The main tool used to complete the work is provided by Barcelona Supercomputing Center. Barcelona Supercomputing Center is one of the main centers of High Performance Computation in Spain located in Torre Girona chapel. It manages Red Espa˜nolade Su- percomputaci´on, RES, and provides computational tools and resources so that scientific and engineering projects can be carried out within the national territory. As well, the or- ganization has participated and actively participates in European computational projects as well as to work with well-known companies in the computational field, such NVIDIA, and others leading enterprises such Iberdrola. The simulation results presented in this thesis are part of a computational project founded by BSC, leaded by Professor Manel Soria as Principal Investigator and Professor Enrique Garcia Melendo. BSC has given the resources in order to accomplish the work in
33 34 CHAPTER 3. REFERENCE FRAME OF THE NUMERICAL SIMULATIONS a fast and practical way. The development of the simulations has been performed using Marenostrum. Marenostrum is the name given to the different versions and updates of its main supercomputer. Currently, four versions have been installed. Marenostrum 4 installation began in 2017. More detailed information about the center, its activities and the technical characteristics of the super computer are provided in its page [21].
3.2 Data input and running process
The software used is designed to read determined files so that the simulation can take place. This files are data input from numerical parameters up to planet’s data. Then, when the input data for the simulation is organised, determined commands are typed in order to start the simulation. Four different files are necessary to run a single simulation. There are two files which contain data about the planet, a template in order to set the parameters of the simulation and a script that includes the commands to run the case.
3.2.1 Planet data
The first file necessary is a .planet file. It contains the equatorial and polar radius value as well as the gravity value. For instance, an example of Saturn’s planet file:
Figure 3.1: SaturnEGM.planet file
The second file regarding the planet is the .wind file. It is composed by two columns, the first one provides the latitude angle and the second column includes the zonal wind velocity at that latitude. The values of the latitude go from southern latitudes towards the north. And the range of the latitudes depends on the case studied. If a non-polar region is being studied, the latitude data does not cover the whole globe. For instance, Figure 3.2 belongs to Saturn planet zonal wind obtained from Voyager probe, as the figure shows, the latitude range begins at -70 degrees. The first two numbers of the file represent the number of rows and the number of columns. 3.2. DATA INPUT AND RUNNING PROCESS 35
Figure 3.2: Saturn Voyager.wind file
It has to be mentioned that the data selected is thought to be as coherent as possible with the case studied. If a feature observed by Cassini is the object of the study, accurate necessary data belonging to the Cassini spacecraft age is used. A great part of the data regarding wind profiles is taken from [9] if the wind profile is Jupiter’s or from [15] regarding Saturn. It is important to remark that two different zonal wind profiles have been used for the simulation work.
Figure 3.3: Profile of zonal wind used for Jupiter simulations. Data extracted from [9] 36 CHAPTER 3. REFERENCE FRAME OF THE NUMERICAL SIMULATIONS
3.2.2 Template
Figure 3.4: Profile of zonal wind used for Saturn simulations. Data extracted from [15]
Figure 3.3 is the diagram of the zonal wind used for Jupiter simulations. The data used is extracted from image navigation of Voyager age. Besides, the diagram of Figure 3.4 is used for Saturn simulations. Since the case regarding Saturn is based on references from Voyager era, the wind profile selected corresponds to data acquired from Voyager mission. The template file contains all the parameters which will determine the result and it is the .sw file. It starts with the definition of specific variables such π number, the seconds that a day has or the time step used, DT. After that, the selection of the numerical scheme used for the software to solve the case. Throughout the accomplishment of this thesis, the selected scheme is muscl. The domain is also determined in the template file. To do so the latitudinal and longitudinal limits are stated. Furthermore, the file needs the resolution of the mesh, it is to say the number of degrees intended for a contol volume. The file also includes the number of days to be simulated, which start at t0 and end at t1, as well as a numerical dissipation constant, the number of time steps between each information saving, or the option of continuing the simulation starting from a determined point or result. Finally, the depth of fluid layer and a part intended for the parameters of geostrophic or gaussian perturbances is included. The parameters of the perturbance are initial day and final day of perturbance injection, the latitude and longitude of the first 3.3. CLARIFICATIONS OF THE SIMULATION PROCESS 37 injection day, its velocity, an energy dissipation constant, the volume of mass injected and the size of the feature. In addition to all the parameters which are considered, the file also includes the .planet and .wind file. The template file has all the physical and computational information which is needed to run the software.
3.2.3 Run script
At the moment of executing the software some commands are needed. These com- mands are included in the script and then the script is executed. The script’s name follows runjob.sh.
3.3 Clarifications of the simulation process
Before explaining the cases it is important to mention how the results are presented as well as particular details which have to be considered.
3.3.1 Numerical parameters
The final results have been simulated with standard parameters. The intention of this is to formalize and present in a uniform way the different data obtained, as well as to take advantage of that standardisation providing the best result. When a simulation is performed, in addition to the parameters of the perturbances inserted in the system, the system itself has to be considered. The domain limits are intended to establish a system which does not interact with the feature, otherwise, the result would lose veracity. The dynamics in Jupiter and Saturn’s atmosphere are not bounded nor enclosed in a domain. So although a domain for the sim- ulation is necessary, it is intended to be as real as possible. The domain is implemented in a periodic way, which means that phenomena which crosses the longitude boundary in westward direction will reappear in the east boundary of the domain without interaction with the longitude limits, keeping its dynamics and path unaltered. It functions as a con- nection. However, this does not happen with latitude, the phenomena in the boundaries remains there, there is no connection between northern boundary and southern one. The limit values for the first simulation case are presented in Table 3.1. The longitude limits are not relevant, unlike the latitude. The latter determines the behaviour of the zonal wind and its value. 38 CHAPTER 3. REFERENCE FRAME OF THE NUMERICAL SIMULATIONS
Table 3.1: Domain limits for the first simulation case
Northern, southern, eastern and western limits Latitude 0 34◦ N Latitude 1 46◦ N Longitude 0 0◦ Longitude 1 90◦
In the same way, the limit boundaries corresponding to the Great Red Spot case are presented below. Two tables are provided since two main domains are used in the Great Red Spot simulations. Table 3.2 provides the domain of merger and observed cases, while Table 3.3 provides the domain boundaries used for the columns set. The reason of a bigger domain in the last case is due to the size of the features simulated. The intention is to avoid interaction between the latitudinal and longitudinal limits of the domain and the feature, as well as possible interaction of the feature with turbulence produced by itself.
Table 3.2: Domain limitis of the ”merger” and ”observed” simulations cases
Northern, southern, eastern and western limits Latitude 0 40◦ S Latitude 1 5◦ S Longitude 0 0◦ Longitude 1 90◦
Table 3.3: Domain limits of the ”columns” simulation case
Northern, southern, eastern and western limits Latitude 0 40◦ S Latitude 1 5◦ S Longitude 0 0◦ Longitude 1 180◦
The pole’s simulation has a different domain. Its boundaries are lines tangent to the latitude selected of the polar cap. For example, if the latitude selected is 30◦ it means that the polar cap will go from latitude of 90◦, the pole, to a latitude of 60◦. The domain is the square whose sides are tangent to the cap circumference.
Table 3.4: Domain limits of the third simulation case
Center of the domain, cap radius and side’s length Side length 60◦ S Center 0◦, 0◦ Cap radius 30◦ 3.3. CLARIFICATIONS OF THE SIMULATION PROCESS 39
Notice that the reference frame of coordinates for the determination of polar domain has its origin in the pole itself. As well, the resolution is an important fact when simulating. It can be cause of numeric dissipation, that is, the energy calculated by the software may be affected by the resolution if it is very high, and decrease as consequence. The resolution used for the Saturn’s case has a value of 0.02◦/cv, that of the Great Red Spot case is of 0.04◦/cv while for the pole case a bigger resolution of 0.1◦/cv is selected. Moreover, the time step used for the iteration process in the first case is DT = 1 s, the time step for the GRS case is of DT = 1 s too. The DT used for the pole is also bigger, of 1 s, since the simulations are longer and simulation time would increase a lot with a small time step. The latter parameter, DT is also a cause of numerical dissipation. If the value of time step is high, energy loses can appear. It is important to take numerical dissipation factor into account particularly in long simulations such those of the pole. This energy dissipation is not physical but numerical. Depending on the conditions and the model implementation, if a long simulation takes place, the algorithm can suffer loses of energy during the iterations. So in order to check that this aspect is controlled, the following graph is provided.
3.3.2 Results presentation
The results are presented in two different ways. The first option is to provide a Po- tential Vorticity result. Potential Voriticity is a very interesting variable since it provides a lot of information due to its conservation in fluid elements. However, it is considered for all the domain and problems may occur. If the feature simulated has a strong nucleus and leaves behind a plume, the nucleus is easily visible but the cloud is mixed with the system background because its Potential Vorticity value may be similar to that of the system and its visibility is poor. The second option for results presentation is quite unnatural, however it solves the problem above presented. The variable used in this way is a tracer. As the name suggests it reveals the path followed by the feature. It is based on the Potential Vorticity mechanism but only applied in the perturbance. In that way the background has a null value of the tracer but the storm is well identified. It can be understood as a ink drop that is thrown in a pool swirl. The vortex of the pool can be easily identified. The Great Red Spot case as well as the Pole case are presented only in Potential Vorticity. 40 CHAPTER 3. REFERENCE FRAME OF THE NUMERICAL SIMULATIONS 4 The simulations
This chapter includes the explanation of each case and the corresponding results ob- tained. It present the whole path as well as the different procedures followed to accomplish the numerical simulations. The steps followed during the simulation performance are men- tioned. As well, the decisions taken as long as the results were obtained are exposed. The three cases studied are exposed separately one from the other. Moreover, important top- ics are also mentioned such as the objective or reason of the cases study as well as the references taken in order to check, if it were needed, the result.
4.1 Introduction
Three different cases have been carried out. The intention of the cases is to learn more about the nature dynamics of the Gas Giants by numerical simulations of atmospheric phenomena. The cases studied can be listed:
• The principal case of study is intended to reproduce features observed during Voy- ager 2 visit to Saturn in its northern hemisphere, in the known Storm Alley.
• The second case is aimed at studying the possible origin methods of the Great Red Spot, so that an approach of different feasible origin process is provided.
• The third study is intended to the validation of a module of the software by repro- ducing polar dynamics of Jupiter and Saturn.
4.2 Study of storms in the northern hemisphere of Saturn
This section provides the study process and reproduction path of features observed by Voyager 2 in the northern hemisphere of Saturn. The study is based on a special study leaded by L. A. Sromovsky, H. E. Revercomb, R. J Krauss and V. E. Suomi presented in
41 42 CHAPTER 4. THE SIMULATIONS the article [22]. The intention is to study the feasibility of reproducing one of the features presented in the article and compare the results with what the study of [22] states.
4.2.1 Scientific reference
These scientists present a study of storms observed in the Voyager 2 visit to the sixth planet of the Solar System, which starts takes place in late August of 1981. The study consists in the analysis of motion, morphology and evolution of the storms. The are focused on a range of latitudes between 30◦ N and 38◦ N where three main structures are identified. The one feature object of study in this thesis is that named for t1 in the article.
Figure 4.1: The upper figure shows a four processed images mosaic of the region studied in [22]. It is a Mercator projection of Voyager 2 images 4.2 days before its encounter with Saturn in 1st of September of 1981. The article uses planetocentric. The figure below is a drawing of the distinctive traits of the atmospheric phenomena. In addition to the drawings, each feature is labeled with a particular name so that the reader can understand easily the explanation as information is provided. The figure has different axis such the latitude axis in the right side, the longitude axis in the bottom, a wind profile with the velocity value in the bottom axis too as well as determined latitudes they have selected. Extracted from [22] 4.2. STUDY OF STORMS IN THE NORTHERN HEMISPHERE OF SATURN 43
As Figure 4.1 shows, t1 is the most eastern structure. It is also called a ”v-shaped feature” or ”convective storm” due to its nature and shape. The feature is thought to have a convective nature, which makes the cloud blow into the upper layer of the atmosphere, which dynamics are similar to the long-lived storms mentioned in chapter 1. After is upwards shifting, it seems that the clouds disappear due to temperatures changing or convection and wind effects or advection. The strongest westward jet occurs at L4. The article proposes that the storm has already interacted with the Brown Spot BS- 1, presented in the diagram at -100◦ of longitude and at latitude L5 and attributes its turbulent behaviour to this fact. As it can be seen in Figure 4.1, t1 and t3 structures seem to have a similar nature and dynamics, though a big difference in clouds sizes can is observed.
Figure 4.2: Arrangement of 6 figures which represent the evolution of the REGION 4 presented in the drawing of Figure 4.1. The image a is taken 20 days approximately before the encounter. An arrow indicates the position of bright spot c throughout the observation period. 44 CHAPTER 4. THE SIMULATIONS
The scientist also mention that its turbulent motion in outermost levels of the atmo- sphere can be produced by interaction in lower levels with a turbulent background. This scenario can be produced by the bright c oval presented in the drawing map as it shifts eastwards in lower latitudes, as it is represented in Figure 4.2. This suggestion is based on previous observations which show similar events. Moreover, Figure 4.2 shows the shape evolution during a period of 16 Earth days approximately, from image a to f. Image b is 8 Saturn’s rotations after a, frame c is acquired after 15 rotations, 23 for d, 28 rotations from a to e and finally after 38 rotations f is taken. The composition shows the evolution of the bright clouds as well as the interaction with bright (c spot) and dark spots (d1 spot). It is thought that the origin of the structure are the bright spots shown in a frame labeled as b1 and b2. Notice that as the bright c spot gets closer to b1 big shaped clouds appear in b and c frames. The same happens when c spot approaches to b2, the region is perturbed. Finally, observe frame f and notice that the clouds near b1 and the spot itself rotate around the dark spot in clockwise circulation, it is to say anticyclonic, which suggests the rotation behaviour of d1.
4.2.2 Computational work
The numerical simulation study is focused on reproducing bright spots b1 and b2 evolution. In a first moment the objective was to reproduce f frame of Figure 4.2, but as the project is carried out, it is realized that to reach such result has a high level of complexity due to software limitations, or the conditions which have to be stated for the simulation in order to acquire that result. However, Figure 4.2 is taken as reference since a clear evolution not only of t1 structure but all the region and smaller features is provided. It is important to mention that the reference frame adopted in [22] for background velocity in Figure 4.2 is of 16.6 m/s, according to Saturn’s System III coordinates reference. The main objective is to study the evolution of a t1 -like structure and check the feasibility of the Shallow-Water regarding the reproduction of such features. The path followed for the simulations begins with the determination of the charac- teristics of perturbances intended to represent b1 and b2. Then the same procedure is followed with the d1 spot. After that, a scenario of turbulence is created in order to ana- lyse the response of the perturbances interacting under turbulence effects together with the effect of d1 vortex which is determined in the previous step. Finally, the best results are selected with the objective of improving them. Stage 1: Determination of perturbances’ characteristics The perturbances are introduced in as gaussians, which contain material to be injec- ted so that the model becomes unstable. The first thing done is to determine where to inject the perturbances. Unlike the article, the latitudes used in the software are plan- etographic latitudes because these latter are easy to implement. Thus, a conversion from 4.2. STUDY OF STORMS IN THE NORTHERN HEMISPHERE OF SATURN 45
planetocentric, Φc, to planetographic latitude, Φc, is performed.
Figure 4.3: Sketch of t1 structure evolution during a period of 10 steps. A dual represent- ation is provided where the first one (a) explains the evolution regarding the time step. The numbered dots represent the position of the features for each moment (each time step is a 5 Saturn rotations period). The drawing (b) below provides a scheme of the mean flow as well as arrows which indicate the direction of the clouds from step 9 to step 10. The frame moves at 16 m/s westward. The left axis contains latitude coordinates while the horizontal one is intended for the longitude. Extracted from [22]
In Figure 4.3 it can be noticed that the latitude range where the relevant b1 and b2 dots develop goes from 33.5◦ N to 36◦ N with a planetocentric reference. This range is converted following:
2 Re tan Φg = tan Φg (4.1) Rp
Where Re and Rp represent the equatorial radius and the polar radius. Using Equa- tion 4.1, the range in planetographic latitudes goes from 39.1◦ N to 41.8◦ approximately. Now that a reference latitudinal position is obtained, the storm structure is determined. It is to say, the layout of the structure to be simulated is designed. Its design does not follow any particular criteria, the intention is to set a similar scenario regarding that observed in [22]. 46 CHAPTER 4. THE SIMULATIONS
Figure 4.4: Structure of the feature to be simulated which contains the sizes of the storm, each couple follows the same pattern. tinjection is the moment when the gaussian is injected
The selected arrangement is presented in Figure 4.4. It is composed by 6 gaussian perturbances that form 3 couples, two of them at the same latitude and the other in a lower latitude. Moreover, each couple is formed by a smaller storm concentric to other more powerful and bigger in size. As the figure shows, the little one has a radius r and that of the bigger is represented by R. Notice that moment when the storms are injected in the channel1 is different for each couple, a difference of a day and a half between each injection is established. In addition to this time gap, a difference of 5 degrees between centers is stated. It is to say, when the second double-storm is injected, there has to be a distance of 5 degrees between the position of the previous double-storm, after a day and a half, and the new one. This double-storm pattern is intended to represent a nucleus of the storm as well as the clouds that appear later, as Figure 4.2 shows. The nucleus role is developed by the small perturbance which is more intense and compact than the bigger storm, which is aimed at representing the plumes which appear as the feature develops. Notice that a distance of separation of 1◦ in latitude is established. Regarding the longitude value, there is one for each storm couple and the calculation of the parameter is not trivial. Since the insertion of the double-storm is determined by a time value, only the latitude of the first couple can be established, while the other two are determined by the time between each injection. The procedure followed to determine the longitude where to insert the second and third couple starts with the conversion of the latitude where the previous couple pattern is put,
1Channel is the name given to a non-polar region of the planet limited by to latitudes from a longitude to other. 4.2. STUDY OF STORMS IN THE NORTHERN HEMISPHERE OF SATURN 47 from planetographic to planetocentric, using Equation 4.1. Once the conversion is done, the radius from the center of the planet to the latitude is calculated with Equation 4.2. R · R R = e p (4.2) Φc q 2 2 2 2 Re · sin φc + Rp · cos φc
Now this distance is projected using Equation 4.3 in a horizontal plane at that latitude. This projection provides the radius of the circumference formed at that latitude.
rΦc = RΦc · cos Φc (4.3)
Then, the perimeter of the imaginary circumference formed at that latitude, whose plane is parallel to the equatorial plane, can be calculated with the expression provided below:
pc1 = 2πrΦc (4.4)
Figure 4.5: Diagram of the first injection instant and the following one. It includes the imaginary circumference red coloured as well as the distance between the first double- storm and the second one.
The intention is to get the number of degrees which separate the previous storm from the currently injected just at the moment when the new one is injected. So, once the perimeter of the imaginary circumference is calculated, the space traveled by the first storm during 1.5 days determined. The ratio between the whole perimeter and the distance travelled by the first storm multiplied by 360◦ provides the degrees travelled. 48 CHAPTER 4. THE SIMULATIONS
For instance, if the longitude of c2, according to Figure 4.5, wants to be obtained, the procedure followed is the calculation of the traveled distance of the first couple and the addition of 5 degrees so that the position where the new storm has to be injected is obtained. The following expression is used:
(−20) · 1.5 · 86400 ∆Dist = lon0c1 + + 5 (4.5) pc1
Where lon0c1 is the initial longitude, -20 is the velocity, 1.5 days is the interval of time between the two insertions, 86400 is the number of seconds that a day has and pc1 is the perimeter of the circumference where the first double-storm drifts. Then, if 5 degrees are summed, the position of the second double-storm is obtained. A graphic example is provided in 4.5. The white dots represent each storm couple. After different test performed following the decisions previously exposed, the para- meters chosen are presented in Table 4.1. Firstly, the injection time is stated to be of one day which means that material is inserted in the system during that period of time. No dissipation is selected for this case nor the others. The latitude coordinates finally selected in this first stage of the process are 39.5◦ for the big perturbance, labeled 1 in Figure 4.4, and 38.5◦ for gaussian 2. Other relevant parameters are the caudal which value for the bigger storm is 5·109 m3/s and that of the smaller is 2·109 m3/s. The gaussian 1 is defined by a σ value of 40◦ and a radius of 2◦ while the smaller has a σ value of 0.5◦ and a radius of the same value. Notice that the shape of the gaussian 1 is much less compacted than the gaussian 2. As it is mentioned above, the little storm plays the role of the storm nucleus.
Table 4.1: First stage: t1 final parameters
Perturbance parameters Injection time 1 day Injection interval 1.5 days Dissipation 0 Latitude coordinates 39◦ N ± 0.5 Velocity -20 m/s 9 3 Caudal1 5·10 m /s 9 3 Caudal2 2·10 m /s ◦ σ1 40 ◦ σ2 0.5 ◦ Radius1 2 ◦ Radius2 0.5 Tracer1 1 Tracer2 2
Notice that only perturbances’ values are provided since the main objective of this first stage is to determine a base for the parameters of the structure intended to reproduce 4.2. STUDY OF STORMS IN THE NORTHERN HEMISPHERE OF SATURN 49 t1. Another important characteristic is the sign of the caudal, that indicates the rotation behaviour of the perturbance. Throughout this case, the double-storms are treated as anticyclonic features, with clockwise rotation direction. If caudal had negative sign the character would be the opposite, cyclonic.
Figure 4.6: Frame of day 9 of the simulation. A cluster of 3 double-storms represented in Potential Vorticity. From latitude of 34◦ to 46◦. Resolution of 0.02◦/VC
Figure 4.6 shows the result the frame 9 corresponding to simulation 9th day of the simulation. The nucleus of the double-storms is well distinguished while the plumes are harder to be observed. In order to get a result easier to be analysed, the frame is post-processed using the tracer variable, which acts like the potential vorticity with the difference that it is only applied in the perturbance injected so that the hue scale is clearer. Figure 4.7 shows that the structure obtained acquires a shape similar to that presented in Figure 4.2. As it is mentioned in [22], t1 is also known as ”v-shape feature” and the result obtained acquires a similar shape. The cause is mainly attributed to the zonal winds which deform the feature modeling it in this v-like shape.
Figure 4.7: Diagram of the cluster of 3 double-storms represented with Tracer. Latitude from 34◦ N to 46◦ N. Resolution of 0.02◦/VC. Day 9 of the simulation
The basic structure and parameters are established, however, as the tests advanced changes in some of the characteristics are tested with the intention of improving the result. 50 CHAPTER 4. THE SIMULATIONS
Stage 2: Determination of BS-1 parameters It is thought that in order to reproduce REGION 4 as better as possible, the main features have to be simulated. Those features which are thought to highly influence the dynamics of that region. After the determination of the t1 basic parameters, the big brown spot BS-1 is reproduced. The main characteristics of the spot are extracted from Figure 4.2 and Figure 4.3. It can be observed that the vortex size is of 2◦ approximately. In addition, it drifts towards east and mainly at planetocentric latitude 36◦, however, when it encounters t1 its movement is distorted and the direction changes heading the south. See the numbered dots which correspond to the steps analysed. As [22] mentions, it is thought that the spot has anticyclonic character. Finally, it is interesting to obtain velocity of the spot, however it is not specified in the figures nor the paper. A simple procedure is carried out so that an approximation of the drifting value can be extracted. The article establishes a reference frame velocity of 16 m/s. In addition, if the drawing is analysed, an approximation of the distance advanced during the time steps of interest can be obtained. To do so, two calculations of the spot velocity are carried out. The first one is intended to provide the drifting velocity from step 1 to step 6, while the second approximation only analyses from third step to the sixth. Performing this double calculation, a mean velocity can be extracted, moreover, it can be seen the effect that the direction change in the third step has on the drifting velocity of the spot. Since the reference frame has an own velocity when the drifting velocity of the spot from Figure 4.3 is obtained, this velocity is a relative one. In order to obtain the absolute velocity, a reference frame change is performed.
distance(◦) 1 v = · p · (4.6) r 360 lat time See that Equation 4.6 is simple. A ration of the circumference advanced, in degrees, is multiplied by the perimeter of the imaginary circumference at that latitude and then divided by the time spent during the travelled distance. Using the equation, vr1−6 = 6.59 m/s is obtained. Applying the same procedure for the second calculation, vr3−6 = 7.9 m/s. The the absolute velocity is calculated:
va = vframe + vr (4.7)
The absolute velocity for the travel during the first step and the sixth is va1−6 = −10 m/s and the one for that during the third and the sixth is va1−6 = −8.7 m/s. From the previous values, a mean velocity is obtained,v ¯ = −9.35 m/s, which is used for the simulations of the vortex. In order to reach an appropriate result, different tests are performed. The tests are simulations of the vortex where different caudal values, coordinates and sizes are stra- 4.2. STUDY OF STORMS IN THE NORTHERN HEMISPHERE OF SATURN 51 tegically combined. Firstly, a set of simulations are run where the changing parameter is the latitude of the vortex. The results obtained show where the vortex has to be placed. Within a range from 38◦ up to 44◦, the results suggest that if the vortex is simulated under 39.5◦ the wind shear breaks the feature and a big amount of material is pulled towards wind direction, eastward. In opposite, if the dot is placed farther than 41◦ in north direction, it is also destroyed due to strong wind jet eastwards. Moreover, after a detailed analysis of the results focused on the latitudes of interest, it is observed that the vortex stability together with an accurate intensity 2 in the in latitudes between 40◦ and 41◦ results in an appropriate behaviour.
Table 4.2: Second stage: BS-1 final parameters
Perturbance parameters Injection time 1 day Injection interval 0 Dissipation 0 Latitude coordinates 40.75◦ N Velocity -9.35 m/s Caudal 1.5·1010 m3/s σ 1◦ Radius 0.7◦ Tracer 1
Table 4.2 provides the final parameters used for the brown spot reproduction. The resulting vortex presented in Figure 4.8 gets stabilized after 110 simulated. While during the first days it moves and oscillates, later it becomes stable due to the size presented and the high caudal value. Since the latter is a big volume of mass injected, and the sizes of the dot presented in the table are small, the resulting effect is a compacted feature. Notice that the size of the resulting vortex is similar to that observed in Figure 4.3.
Figure 4.8: Diagram of BS-1 represented in Potential Vorticity. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/VC. Day 150 of the simulation
2Intensity of a simulated feature can be understood as the combination of caudal effects mixed with the storm size 52 CHAPTER 4. THE SIMULATIONS
The figure above is a particular frame of the final result. The definitive simulation of the vortex has a duration of 200 days.
Figure 4.9: Numerical dissipation of energy during the 200 day simulation process
The Figure 4.9 is provided since the final simulation presented is quite long. The diagram represents the total energy of the simulation, considering potential and kinetic energy. It is observed that no significant dissipation takes place. Third stage: Determination of turbulence conditions The third stage is aimed at providing a turbulence frame. Since the first stage of the process provides a quite stable result, it is thought that in order to get closer to the feature observed, a turbulent background is needed. This turbulent frame has been reached freely, no relevant criteria has been established for its obtaining. The procedure followed for the turbulence background is simple. Eight perturbances located within 36◦ and 43◦, with a space of 1 degree between each one, are injected during 50 days, all together during the same time, so that a lot of material is injected and the system is perturbed during that time. 4.2. STUDY OF STORMS IN THE NORTHERN HEMISPHERE OF SATURN 53
Table 4.3: Turbulence parameters
Perturbance parameters Injection time 50 day Injection interval At the same time Dissipation 0 Latitude coordinates 36◦ N - 43◦ N Velocity -60.0 m/s Caudal 2·109 m3/s σ 15000·103 m Radius 150·103 m Tracer 0
As it can be seen in Table 4.3, the features inserted have a high drifting velocity aimed at helping to the perturbation of the system. Notice that σ and radius values are provided in meters. A frame of the process is presented in Figure 4.10.
Figure 4.10: Diagram of the perturbances intended to create turbulence represented in Potential Vorticity. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 35 of the first simulation
Notice that the turbulence follows a gradual process, form north to south of the do- main. Observe that between 38 and 35 degrees, the perturbances injected are still steady. Since this result is not relevant and the intention is to mix it with the selected features determined in previous stages, no tracer result is presented. Once the result is obtained, an analysis is performed. The simulation’s duration is of 100 days, however, the turbulence is not achieved instantaneously nor lasts for ever. On the one hand it is seen that until day 45 of the simulation, the background is not completely turbulent, there are still southern storms which have not become unstable. Thus, the first decision is that only frames after the day 45 are taken. On the other hand, the activity decrease is also considered. The results provide a peak of activity between 50 and 70 days of simulation, then the turbulence lose intensity and begins to disappear. The second decision is that scenarios corresponding to frames between 50 and 70 days will be used. 54 CHAPTER 4. THE SIMULATIONS
Figure 4.11: Diagram of the perturbances intended to create turbulence represented in Potential Vorticity. Final scenario. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 60 of the first simulation
Nevertheless, with the objective of assuring the previous claims, a continuation of the simulation is run, from day 100 to day 150. Figure 4.12 provides the result of day 125 if the previous 100 days of the first simulation are considered. The results obtained are consistent with the thoughts, despite of having an unstable system in the second test, the turbulence that takes place is not intense enough.
Figure 4.12: Diagram of the perturbances intended to create turbulence represented in Potential Vorticity. Poor turbulent scenario. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 25 of the second simulation
Fourth stage: Analysis of features under turbulence effect This stage presents the result of previous features simulated in first and second stage, but in a turbulent system. As it is mentioned in previous stages, the results obtained are a good approximation of t1 and BS-1, however the intention is to reproduce as accurate as possible REGION 4, and it is thought that carrying this stage out is worthy. In order to carry this stage out, the first thing done is to select an appropriate back- ground. So different simulations are run using day 50 and day 60 of the first turbulence simulation. The simulations are carried out as a continuation starting from the days mentioned above. The article [22] shows that t1 takes 16 Earth days approximately to 4.2. STUDY OF STORMS IN THE NORTHERN HEMISPHERE OF SATURN 55 acquire the turbulent character and that snake-like shape. According to that, simulation of 20 days are selected to be run, a similar result should be obtained within that period of time. So those simulations which continue from 50th day of the turbulence simulation, last up to 70 days. The others which are a continuation starting from day 60 of the turbulence simulation last up to 80 days. In addition to the selection of the day which provides a good scenario of turbulence, the location regarding latitude and longitude has to be defined too. To do so, a set of simulations is prepared. Most of the simulations only add the cluster of storms presented in the first stage. However, a couple of tests include the cluster as well as the vortex intended to be BS-1, and one test aimed at studying the effect of the turbulence over the vortex is performed.
Figure 4.13: Diagram of the cluster under turbulence effects represented in Potential Vorticity. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 71 of the simulation
Figure 4.13 shows the final result of this stage regarding t1 under turbulence effect. This simulation is considered to be the one which parameters provide the best result. Notice that the perturbances cannot be identified because of Potential Vorticity of the rest of the system.
Figure 4.14: Diagram of the cluster under turbulence effects represented with Tracer. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 71 of the simulation
The result obtained with tracer is provided in Figure 4.14 so that it can be identified. It can be seen that the feature is much more altered due to turbulence and instability if it is compared to that presented in Figure 4.7. In addition, the frame provided is that 56 CHAPTER 4. THE SIMULATIONS corresponding to day 11 since the cluster is added to the system, the date corresponds to image e presented in Figure 4.2. Considering that the storm reproduced matches in date regarding to the observations, a very similar dynamics can be perceived as well a common appearance. Moreover, the arrow-head shape is conserved.
Table 4.4: Fourth stage: t1 final parameters under turbulence effects
Cluster parameters Injection time 1 day Injection interval 1.5 days Dissipation 0 Latitude coordinates 39.5◦ N ± 0.5 Velocity -20 m/s 9 3 Caudal1 5·10 m /s 9 3 Caudal2 1·10 m /s ◦ σ1 40 ◦ σ2 0.5 ◦ Radius1 1.5 ◦ Radius2 0.5 Tracer1 1 Tracer2 2
The parameters used for the results presented above are included in Table 4.4. The first double storm is inserted the day 60 as it is mentioned above. Notice that some parameters have changed in comparison with Table 4.1. The reason is obvious, the result obtained with these parameters is better and more accurate than that obtained with the first parameters stated. Notice that the cluster inserted has a less intense nucleus and the bigger storm of each couple is smaller in size. Moving on, when it comes to the simulations which include both storm cluster and vortex, a similar process is carried out. The day of the turbulence simulation selected to insert the cluster is 60th, however the vortex is inserted from the beginning of the simulation on day 0. The results from these test are not such good. The interaction between the vortex and the cluster does not occur as it thought to happen. It is true that there exist an interaction between both features, but in the simulation, unlike the observations, the vortex does not perturb the cluster nor crosses it, in the other way, it is the cluster that overtake the vortex surrounding it from the south leaving it behind without a relevant sign of influence. This result is thought to be direct consequence of the high velocity of the cluster in comparison with the spot. In order to improve the result of simulating together the two features a deeper study should be carried out focused on this fact. 4.2. STUDY OF STORMS IN THE NORTHERN HEMISPHERE OF SATURN 57
Final stage: Improvement of the result This case ends with the stage here presented. After all the process, the relevant results are extracted and fine-tuned, which means that the parameters are slightly changed aimed at providing a result. As it can be seen from previous stages, the most relevant result is the one provided in the previous stage regarding the study of the cluster in a turbulent system. So the results obtained from Figure 4.14 are retaken. As it can be observed in the Figure 4.14, the storm reproduced does have a similar appearance to the real observation but it lacks of zigzag dynamics. The objective of changing the parameters from that simulation is to obtain a more similar result regarding the snake-like shape observed in Figure 4.2, particularly in f image. In order to reach a better result, the one change applied is aimed at simulating more intense double-storms, thus the caudal of the nucleus of the storm couple is increased while its size is kept.
Figure 4.15: Improved result of the cluster under turbulence effects represented with Tracer. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 71 of the simulation
Notice that the result obtained matches with the one thought. The zigzag behaviour has increased in comparison with Figure 4.14. This result suggests that the intensity of the nucleus has strong influence on the appearance of the structure. As well, it can be thought that if the caudal value continues increasing or the size of the storm is reduced with the intention of obtaining one more compact, the zigzag behaviour can increase too. See that the result provided in Figure 4.15 corresponds to the same simulation day of Figure 4.14. In addition, leaving behind the computational aspect, the result is telling that the shape of convective storms that blow in surface depend on the nucleus of it. It has to be considered that the software has limitations already mentioned. If the result in this stage is compared to the corresponding observation image, the observation is fainter and shows less clouds than the simulations. This difference is attributed to thermodynamic effects which take place in the surface of the Gas Giant which are not considered in the software. The same cluster without turbulent background is presented in Figure 4.16 so that the difference of turbulence can be appreciated. 58 CHAPTER 4. THE SIMULATIONS
Figure 4.16: Improved result of the cluster without turbulence effects represented with Tracer. From latitude of 34◦ N to 46◦ N. Resolution of 0.02◦/V. Day 71 of the simulation
In summary, this case has been a long process of studying the dynamics of features observed in Voyager 2 age aimed at reproducing some of them, presented in [22]. The first step was to study the scientific reference and analyse, particularly, t1 which was the first objective. Then, after acquiring an idea of the feature’s nature and dynamics, numerical parameters are established so as to obtain a result which is used later as basic structure. It is observed that the range of latitudes where the cluster may be located goes from 39.1◦ to 41.8◦. Other parameters such the caudal or size are also tested. The, the objective is to reproduce the BS-1. To do so, the feature is also analysed and the paper is read again. The velocity of the spot is calculated and testings provide parameters such location, caudal or size. Finally a turbulent background is determined where the features are reproduced. The final results are successful, a very similar pattern is obtained in the final stage of the case. It has to be remarked that turbulence plays a very important role in this study. Although results without turbulence effects present similar nature, there is no doubt that these results are improved with a turbulence background. See Figure 4.16 and Figure 4.15. If this is applied to the observations, and with the intention of completing the paper [22] used as reference, the computational results show that the storms are very conditioned by a turbulent layer which is not visible, however it still influences the behaviour of the storm once it blows up in the outermost atmospheric layer, according to the results presented for this case. Convective effects are also considered for the evolution of the feature, however since thermodynamics are not computed in the software, the results do not show the effect. The difference between the simulation results and the observations is thought to be mainly caused by the lack of convective effect. Another effect considered for the observed feature dynamics is the advection. Unlike convection, the effect of velocity is computed in the code and can be observed in the results, the cluster is pulled eastwards. 4.3. STUDY OF FEASIBLE ORIGINS OF THE GREAT RED SPOT 59
4.3 Study of feasible origins of the Great Red Spot
The study here presented is intended to provide information about feasible formation mechanisms regarding the Great Red Spot, currently active. The study is aimed at collaborating with Grupo de Ciencias Planetarias UPV/EHU. The intention is to run determined simulations and analyse the results obtained. The great red spot, as chapter 1 states, is the icon of Jupiter and there is not feature similar to it in the Solar System. However, its nature is unknown and the data obtained can be completed. There are older drawings regarding observations which date back to 17th century performed by Galileo Galileo or Robert Hook but it is still unknown if that feature sketched 300 years ago by these astronomers among others, corresponds to the one currently active. With the objective of learning more about the current spot origin a set of simulations is accomplished.
4.3.1 Scientific reference
Different mechanisms for the origin of the dot are suggested. The case presents the result from two different sets of simulations, each of one regarding to the study of a feasible origin of the feature. As well, a reproduction of older observations of the Great Red Spot when it size was bigger is also provided. The first formation mechanism presented is the merger. There exist evidences of active vortex which origin is based on the merge of little storms that have collide and instead of being destroyed or dissolved, a bigger storm has appeared. A relevant example is the formation of Oval BA. It is an anticyclonic storm which drifts in the southern hemisphere of Jupiter, south to The Great Red spot, particularly in the South Tropical Zone. The color which presents varies constantly. It is called the Red Spot Junior because of its behaviour, which has turned to be very similar to that presented by Jupiter’s icon. Figure 4.17 shows the two main Jupiter storms together, the Oval BA southern to the Great Red Spot. See that the Oval BA has a main white color, though it presents a reddish ring. It appeared at the beginning of this century and its color was white, due to the merge of three previous white smaller ovals, DE, FA and BC. These three little storms where formed in 1939 as [23] mentions, and coexisted in 33◦ of latitude until they collide. 60 CHAPTER 4. THE SIMULATIONS
Figure 4.17: Two massive storms captured by Juno 21st December 2018. Instrument: JunoCam. Exracted from [24]
As it can be seen, Figure 4.18 provides another evidence of merger formation mechan- ism. The scenario showed in the image is thought to be produced by Oval BA turbulence. It the image two vortex are captured merging and the Oval BA in the north. Notice that the dynamics are recurring. The Oval BA has interacted with the Great Red Spot but it has managed to do not be dissolved nor destroyed. A possible similar origin could happen for the Great Read Spot.
Figure 4.18: Two storms caught merging in Jupiter South Hemisphere near Oval BA captured by Juno on 26th December 2019. Instrument: JunoCam. Extracted from [24]
The second formation mechanism proposed presents a nature totally different to the one first explained. It is thought, as the author of [25] explains, that the red dot may be derived from high-pressure big cells. These features are longitudinal long stable phenom- ena which have been observed in different opportunities. The phenomena looks like a big 4.3. STUDY OF FEASIBLE ORIGINS OF THE GREAT RED SPOT 61 elongated storm whose circulation is closed. During the simulation work explanation the phenomena will be treat as a ”column”.
Figure 4.19: Diagram of Jupiter high pressure cells taken by the HST in 2007. Extracted from files provided by [25]
An example is provided in Figure 4.19, which include two images, the left one shows the storm itself and the right one give some information about the feature. Notice that the image is inverted, the phenomena takes place in the south of the planet. STB is the South Tropial Belt, STZ is how the South Tropical Zone is labeled and SEB means South Equatorial Belt. The red arrows are aimed at giving an idea of the circulation which governs the pattern. The cell is thought to be contracted and compacted giving rise to the Great Red Spot, as it was observed by Pioneer 10 :
Figure 4.20: Great Red Spot captured by Pioneer 10 in 1974. Extracted from [26]
See the difference of size between the dot presented in chapter 1 and that showed in Figure 4.20. When it was first observed, the Great Red Spot was huge, and that fact lead to think that a possible origin may be caused by the contraction of a giant cell in the southern region. 62 CHAPTER 4. THE SIMULATIONS
The last set of simulations, as it is mentioned above is intended to reproduce a the first Great Red Spot observed in the late 19th century. Figure 4.21 provides a comparison between the current storm and that observed 130 years ago from the Lick Observatory in California.
Figure 4.21: Comparison between the old Great Red Spot, of 1890 (left) and the current one from a image taken in 2014 (right). Extracted from [27]
The last set of simulations takes as reference the Great Red Spot imaged from the Lick Observatory.
4.3.2 Computational Work
The simulation workload has been divided in the three stages explained in the previous section. The path followed for the results obtaining is based on the following schemes provided for each case. It is important to mention that the domain used for the merger and observed set is different from the one used for columns simulations, as well as other parameters mentioned in chapter 3. Another important aspect to be mentioned in comparison with the Saturn case is that the storms are introduced as perturbances, but with the difference that these are in geostrophic equilibrium which means that Coriolis effects counteracts the gravity in order to keep the pressure difference caused by the storm. In this way, the tangential velocity is needed when inserting the vortex parameters. 4.3. STUDY OF FEASIBLE ORIGINS OF THE GREAT RED SPOT 63
Merger simulations (MS): The reference anticyclonic storm used has the following characteristics:
Table 4.5: Merger Reference anticyclonic storm
Perturbance parameters Zonal 3 dimension 15◦ Meridional 4 dimensions 10◦ Mean Latitude 22.5◦ S Tangential velocity 120 m/s
This simulations set is aimed at studying the feasibility of the merge of different vortexes that result in a bigger one. So, the path followed for the merger stage is based on the scheme provided above. Merger simulations scheme
MS1: Simulation of one anticyclonic perturbance (A1) alone. Analysis of its stability
MS2: Simulation of two anticyclonic perturbances (A1, A2) together. The separation between the edges of A1 and A2 is equal to the zonal dimension of the reference storm. Drifting velocities of the vortex have to be slightly different so that they get closer and merge.
MS3: Simulation of three anticyclonic perturbances (A1, A2, A3). The same procedure for SM2 is applied. Once the vortices A1 and A2 have merged in A12, the third storm, A3, is inserted aimed at merging with A12.
MS4: Simulation of four anticyclonic perturbances (a1, A2, A3, A4). The same procedure for SM3 is applied. Once the vortices A3 and A12 have merged in A123, the fourth storm, A4 is inserted aimed at merging with A123.
So the procedure followed is based on the scheme proposed. Firstly, the stability of the reference anticyclonic vortex is studied and the results are successful, it is seen that the storm acquires a steady dynamic in the system. From a set of two simulations, the relevant results are shown here below. Figure 4.22 shows the result of the first step simulation. It can be observed that the system is steady, however a difference between the northern boundary and the southern one is appreciable. The upper limit is more perturbed than the bottom of the domain, and this behaviour is observed during the results of all the Great Red Spot Case. It is attributed to the implementation of the software as well as strong jets in the latitudes of the equator. The feature is injected since the first day.
3Zonal is the adjective given for longitudinal directions 4Meridional is the adjective give for latitudinal directions 64 CHAPTER 4. THE SIMULATIONS
Figure 4.22: MS1 result. Diagram of the reference vortex A1 represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 100 of the simulation
Once the stability of the reference vortex is checked, MS2 is carried out. The results of the previous test, MS1, are used for the continuation of the study. In this step, the intention is to analyse the vortex, A12, resulting from the merge. To do so, two different mechanisms for the approach of the vortex are tested. The first one is based on the variation of tangential velocity, the first vortex has the value presented in Table 4.5 while the second vortex introduced has a lower value of 100 m/s.
Figure 4.23: Diagram of two vortices, A1 and A2, with different tangential velocities represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 150 of the simulation 4.3. STUDY OF FEASIBLE ORIGINS OF THE GREAT RED SPOT 65
Figure 4.23 shows the result obtained from the test above mentioned. It is seen that if the vortices have a different tangential velocity, the features do not merge. The figure shows a result of day 150 of the simulation. Since the results obtained do not match with what it was thought, a different test is carried out. The second process is that proposed in the scheme which consists in the variation of latitude between the two features so as to create a difference in drifting velocity since the zonal wind is not constant along the latitude. In this case, the latitudes proposed for the vortex A2 take the following values: 19◦, 20◦, 21◦ S, 22◦, 23◦ and 24◦ S. A set of simulations is developed and the results are very interesting. The test suggest that depending on the latitude where the two vortices are inserted, three different responses occur. In first place, if the vortex A2 is inserted to the north of A1, when these collide, instead of merge and become one substantially bigger, a great part of the northern vortex A2 is lost, and only a small part of material merges with the southern vortex.
Figure 4.24: Diagram of the moment when A1 and A2 merge represented in Potential Vorticity. A2 is injected at 21◦ S. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 200 of the simulation
The fact is shown in Figure 4.14. A great amount of vortex material is lost during the encounter. As well, the size of the resulting vortex once it is stable is not much bigger than the reference storm. Moreover, the results suggest that if the vortex A2 is injected at the same latitude or close, such 22◦ S or 23◦ S, the result obtained is similar to that presented in Figure 4.23, the vortices remain stable during all the simulation without merging. It can be seen that if there is a difference in drifting velocity of the features, when these are close, the one which is approaching, decreases the velocity and slows down. 66 CHAPTER 4. THE SIMULATIONS
Figure 4.25: Diagram of two vortices, A1 and A2, with different tangential velocities represented in Potential Vorticity. A2 is injected at 23◦ S. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 150 of the simulation
The third hypothesis extracted from this set is that if A2 is inserted in a latitude such 24◦ S, A1 reaches it since it moves faster due to zonal winds. Finally the features merge in a bigger vortex whose dimensions are significantly bigger than those obtained in the two first results and bigger than the reference vortex.
Table 4.6: Final parameters for A1 and A2
Perturbance parameters A1 A2 Zonal dimensions 15◦ 15◦ Meridional dimensions 10◦ 10◦ Mean latitude 22.5◦ S 24◦ S Tangential velocity 120 120
Figure 4.26 shows the resulting vortex of the third test explained above. The vortex A1 is injected at latitude 22.5◦ S, then A2 is injected at 24◦ S. Notice the size of the vortex in comparison with that presented in Figure 4.22. It is 5 degrees bigger in zonal dimension according to the result. The system is quite turbulent and altered due to the merge of the vortices, in addition, it can be noticed again that the northern boundary is more perturbed than that of the south. As well, small features move around the domain such the one that have appeared at latitude 32.5◦ S. However, the system is not extremely turbulent and the vortex has stabilized. 4.3. STUDY OF FEASIBLE ORIGINS OF THE GREAT RED SPOT 67
Figure 4.26: MS2 final result. Diagram of vortex A12, A1 and A2 represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 200 of the simulation
In order to continue with the analysis of this formation method, results from the simulation presented in Figure 4.26 are taken. Consequently, the results obtained from varying the tangential velocities as well as the rest of results from the set where latitude is changed are left behind. Next simulation adds vortex A3 to the result of Figure 4.26. The third vortex is injected at the same latitude of A2, 24◦ S with a tangential velocity equal to that of A1 and A2. Table 4.7 provides the data of the vortex used for the simulation. Observe that as it is mentioned above, the first two, A1 and A2 have the same parameters. The Figure 4.27 shows the diagram of day 300 of the simulation. It is important to remark that MS2 is a continuation of MS1, as well, MS3 continues from the selected results of MS2. It can be seen that A3 has contributed to the grow of the spot, which size is of 27 ◦ approximately. However, it is seen that the growth rate of the storm simulated is not as high as it should. After the analysis by now performed, it is decided to stop and MS4 is not carried out. It is seen that the growth rate of the mechanism is not as high as expected.
Table 4.7: Final parameters for A1, A2 and A3
Perturbance parameters A1 A2 A3 Zonal dimensions 15◦ 15◦ 15◦ Meridional dimensions 10◦ 10◦ 10◦ Mean latitude 22.5◦ S 24◦ S 24◦ S Tangential velocity 120 120 120 68 CHAPTER 4. THE SIMULATIONS
Figure 4.27: MS3 final result. Diagram of vortex A123, A1 and A2 represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 300 of the simulation
Out of the path followed up to this point, a final isolated set of simulations is developed, regarding merger formation mechanism. It consists in the simulation of manifold vortices, 6 specifically with the same characteristics stated in Table 4.5. The first of them placed westward in the domain at latitude 22.5◦ S. The other vortex are placed at the same latitude,n in three different simulations. The first simulation places the rest of vortex at a latitude of 20◦ S. In the second simulation, the 5 vortex are located at latitude of 22.5◦ while in the third one the features are injected at 24◦ S of latitude. Notice that the methodology followed is quite similar to the analysis of MS2. A space of 15 degrees is left between each storm and the characteristics are identical to those presented in Table 4.5. The intention of the test is the same one of the scheme procedure, to analyse if the Great Red Spot can have been originated by the merge of smaller vortex. The main difference of this test regarding the procedure followed until MS3 is that all the vortices are injected at the same moment while the scheme at the beginning of the merger case explanation states that the insertion of the vortices is gradual. The results are quite similar to that obtained in MS2. The simulation of vortices injected north to the first reference vortex ends up with a vortex which size has not increased. The second simulation, which includes all the 6 vortices at latitude 22.5◦ S provides the same result obtained in Figure 4.25. However, an interesting result is obtained from the last test which injects 5 of the 6 vortices at 24◦ S. From this isolated test, Figure 4.28 and Figure 4.29 are provided. Notice that the domain is wider in longitude, the reason is to avoid interference in the system caused by lack of space. Also observe that the size of the resulting storm is of almost 30◦ degrees. This size increase presented in the result of the image below can be attributed 4.3. STUDY OF FEASIBLE ORIGINS OF THE GREAT RED SPOT 69 to the number of vortices simulated.
Figure 4.28: Isolated test final result. Diagram of resultant vortex represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 250 of the simulation
In comparison with the stage MS3, 3 more vortices are added to the analysis. This suggests that this isolated test has a growth rate lower than that obtained following the scheme. Moreover, the resulting system is much more perturbed than that obtained in MS3.
Figure 4.29: Numerical dissipation of energy during a 250 day simulation process
The diagram of Figure 4.29 presents the evolution of the system’s total energy. Notice that there are several loss of energy. This are thought to occur at the moment when vortices merge. As it is stated for MS3, the merge mechanism simulated tends to lose a huge amount of material from the storms and it is reflected in the energy. Notice also the 70 CHAPTER 4. THE SIMULATIONS manifold oscillations recorded in the chart, this response is thought to be caused by the oscillation of the vortex after a merge. In order to see if the system is finally stable and if presents a constant energy, a longer test may be carried out. Columns simulations (CS) This set is aimed at studying the stability of elongated anticyclonic structures, whose characteristics are changed for the different simulations. It does not of simulations does not have a reference pattern for the simulations, unlike the merger case. The simulation parameters are given in the Table 4.8 presented below.
Table 4.8: Parameteres of Column simulation set
Perturbance parameters CS1 CS2 CS3 CS4a CS4b CS4c Zonal dimensions 80◦ 60◦ 50◦ 45◦ 45◦ 45◦ Meridional dimensions 13◦ 13◦ 13◦ 13◦ 13◦ 13◦ Mean latitude 22.5 22.5 22.5 22.5 22.5 22.5 Tangential velocity 50 50 50 50 75 100
As the Table 4.8 shows, different parameters are varied for the simulations. The first one is the zonal size and the second one the tangential velocity. With the different combinations studied it can be stated which of them is the most stable. Moreover, the influence of the zonal size and velocity in the column behaviour can be determined. The most relevant results are presented below. Firstly, the results of the simulation CS1 is presented. It is the best approach obtained for the reproduction of structures such those presented in Figure 4.19.
Figure 4.30: CS1 Result. Diagram of resultant structure represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 75 of the simulation
The image above presents a structure very similar to the cells presented at the begin- ning of the section. Its circulation is closed and it finally gets stable under the conditions of the simulation. In addition to the diagram, a chart providing the evolution of total 4.3. STUDY OF FEASIBLE ORIGINS OF THE GREAT RED SPOT 71 energy is presented, Figure 4.31. From there it can be seen how the energy varies. It can be understood as a record of the simulation. At the beginning its value increases, this growth correspond to insertion of the structure and it division in two parts. Then it gets stable, during the period of simulation when the big division advances to reach the small one. Notice that once both parts have meet again, the energy decreases and finally gets stable. At the end of the simulation, both parts have merged in a bigger one. Any loss of energy due to numerical computation is not seen.
Figure 4.31: CS1 Result. Diagram of resultant structure represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 75 of the simulation
Then all the results of each proposed CS case are analysed and it is seen that definitely size and tangential velocity have a lot to do with the behaviour of the feature. On the one hand the table cases vary from long zonal structures to short ones, on the other hand the velocity variation goes from low to high. It is observed that as long as simulations are carried out, the ease for the feature to get stabilized is higher, that means, the storm reaches stable behaviour when it is shorter faster than with high zonal sizes. However, a similar patter observed for all the cases no matter its size is that the feature is broken at the beginning of the simulation. 72 CHAPTER 4. THE SIMULATIONS
Figure 4.32: CS3 Result. Numerical dissipation of energy during a Diagram of resultant structure represented in Potential Vorticity. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 75 of the simulation
See Figure 4.32, which represents the result on day 75 of the simulation CS3. It shows a small part of the initial inserted structure left behind while the big part is reaching it again. Once both structures have met, these become one stable structure, however this situation is unreal if it is applied to the Gas Giant. See that the domain is a half of total planet’s longitude, in order to achieve such long structures with the software, they have to break and then merge again. The result suggests that from the code, a real reproduction of long structures formation is difficult to performed, but the evolution once the feature has stabilized can be carried out.
The second suggestion is extracted from results corresponding to CS4a and CS4c.A comparison between this cases is carried out aimed at analysing how does the velocity affect to the response of the structure. Notice that the size of the structures corresponding to this cases is equal so the differences observed are completely attributed to tangential velocity.
Figure 4.33: CS4a Result. Diagram of resultant structure represented in Potential Vorti- city. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 140 of the simulation 4.3. STUDY OF FEASIBLE ORIGINS OF THE GREAT RED SPOT 73
Figure 4.34: CS4c Result. Diagram of resultant structure represented in Potential Vorti- city. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 140 of the simulation
It can be seen in the figures above a noticeable difference. It the first one, Figure 4.33 which has a slower tangential velocity, on day 140 of the simulation the structure is still broken. Now, if Figure 4.34 is analysed, a different structure is obtained which is more contracted and compact than that previously presented. It is known from results that the feature presented in Figure 4.34 have not even broken unlike the rest of features. This second structure is shorter, moreover, its shape and size is similar to the Great Red Spot observed from Lick Observatory. This suggests two hypothesis, firstly tangential velocity determines the shape of the column; secondly, if the right value is chosen a good result regarding the observed storm in late 19◦ century can be obtained. From the results obtained it is thought that if the Great Red Spot was derived from a high-pressure cell or column, it could have happened due to an increase in its tangential velocity which can depend of the storm itself or on the zonal wind profile. Unfortunately, there is not a record of an old ind profile of Jupiter. Observed simulations (OS) The last set of simulations presented is intended to the reproduction of the Great Red Spot observed in 1890.
Table 4.9: Parameteres of Column simulation set
Perturbance parameters OS1 OS2 OS3 OS4 Zonal dimensions 33◦ 33◦ 33◦ 33◦ Meridional dimensions 11.5◦ 11.5◦ 11.5◦ 11.5◦ Mean latitude 22.5 22.5 22.5 22.5 Tangential velocity 50 75 100 125
In the same way, the velocity parameter is changed in the different simulations so as to identify which role plays in the stability of a feature such the one presented in left image of Figure 4.21. In this stage of the GRS case, a similar hypothesis is extracted from the 74 CHAPTER 4. THE SIMULATIONS previous one. The most relevant results are presented below. The tests proposed, OS1, OS2, OS3 and OS4, present the same characteristics except for the tangential velocity. Its zonal and meridional size is similar to the ancient Great Red Spot, the latitude selected is the same than that used for the rest of the study and the velocity goes from 50 to 120 m/s. So again, a similar case to the last part of the test of columns is carried out.
Figure 4.35: OS1 Result. Diagram of resultant structure represented in Potential Vorti- city. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 147 of the simulation
After the performance of the simulations and an analysis, it is see that the velocity plays the same role, independently of the size. The first result is presented in Figure 4.35. It shows a quite stable elongated perturbance. Its size is of more than 40◦ wide. Figure 4.36 presents the result obtained from OS4 which is the one whose tangential velocity is higher. Notice that its shape is rounder than that of the previous result. Moreover, after analysing all the simulation days of this case it is noticed that the feature takes less time to get stabilized. If results are compared and taking into consideration the value of tangential velocity, the same hypothesis extracted from columns case is presented. Tangential velocity is a natural mechanism for the storm to control its shape by contraction. So, after the analysis of the four different cases it is decided to fine tune the velocity value. OS1 presents a weak feature, but once the velocity value is increased in OS2 the resulting shape seems to be very contracted. 4.3. STUDY OF FEASIBLE ORIGINS OF THE GREAT RED SPOT 75
Figure 4.36: OS4 Result. Diagram of resultant structure represented in Potential Vorti- city. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 147 of the simulation
After different test simulations, a middle value between OS1 and OS2 of tangential velocity is found. The tangential velocity selected has a value of 58 m/s.
Figure 4.37: OS7 Result. Diagram of resultant structure represented in Potential Vorti- city. From latitude of 40◦ S to 5◦ S. Resolution of 0.04◦/V. Day 131 of the simulation
Figure 4.37 shows the result obtained from OS7. Notice its elongated shape, however is more compact than that of OS1. The feature acquires this shape due to tangential 76 CHAPTER 4. THE SIMULATIONS velocity. Moreover, if it is compared to the left Great Red Spot presented in Figure 4.21 it can be seen that both features are very similar. Thus a successful reproduction is obtained. After reaching the result, since it is a long simulation, the numerical dissipation of energy has to be checked. Furthermore, not only checked but improved. As it is mentioned in chapter 3, there are different ways to improve efficiency regarding numerical loss of energy. Since the domain used in the last simulation presented is such big, to increase the resolution is not considered, so it is decided to used a smaller time step.
Figure 4.38: Caption
As chapter 3 mentions, the time step used in GRS case is 2 s, so it is decided to decrease this value by a factor of 10 in order to see if the improvement is relevant. The results of this test, OS8, are compared with those of the reference test. OS7. Figure 4.38 is provided which shows a diagram of the numerical dissipation of energy of the two tests. It can be seen that the improvement achieved is not relevant. Thus, the reduction of DT in this case is rejected since it has a higher computational cost and does not provide a significant improvement. Once again, after the study of the ancient GRS developed it is seen that tangential velocity can have a strong influence in the nature of the spot. An hypothesis of its actual shrinking can be attributed to an increase of this magnitude. In summary, this case has provided an idea of how could a feature such the sign of Jupiter, GRS, have appeared in the atmosphere of the planet. From the results of the first part of the case which corresponds to merger simulations it is seen that regarding the code, the growth rate of the simulations carried out is low and not enough so as to give rise to the spot, moreover, in the merge of different vortices, an important loss of 4.3. STUDY OF FEASIBLE ORIGINS OF THE GREAT RED SPOT 77 material reflected in the total energy of the system takes place. The results of MS suggest that it may not be a formation mechanism for such storm or that the code limitations do not provide a certain hypothesis. Secondly, after analysing the second formation method based on long structures, columns, the results seem to be closer than those of the first part. A successful res- ult of this structure is obtained with a longitudinal size of 80 degrees approximately. It is seen that energy loss is not a relevant problem for the performance of this case. However, it has to be considered that the formation method for the cells reproduced is far from reality, and only when these are stable the behaviour is that of the observed cells. Fur- thermore, the results also suggest how could a high-pressure cell become the today known Great Red Spot. The mechanism is based on the increase of tangential velocity. However, the hypothesis cannot be affirmed, though the result provides a very good approach. Finally, the last stage is intended to reproduce a storm such the one observed in 1890. The test suggests a similar behaviour of the storm regarding tangential velocity. Finally, after developing a set of 8 sets, a successful value for the parameter of velocity is obtained. Thus, a good reproduction of the phenomena is reached. 78 CHAPTER 4. THE SIMULATIONS
4.4 Validation of software module
The third case performed is intended for the validation of a software module geared for the reproduction of polar dynamics in the Gas Giants. The objective is to carry out simulations and compare the result with other simulations performed. The module was implemented later than the rest of the software by GrETA Degree Student Marc Andr´esCarcasona. Since polar regions present phenomena which dynamics are slightly different from those of the rest of the globe, specific tests have to be developed and analysed independently fro the rest of the software.
4.4.1 Scientific reference
Gas Giants polar dynamics are characterized by the presence of cyclonic vortices bounded by strong winds. The case of Jupiter is different from Saturn’s. Juno space mission has showed recently the tendency of Jupiter north and south pole. Both regions are characterized by the circulation of cyclonic vortices around a central one located at each pole. The north pole’s vortices have a diameter between 4000 and 4600 km, according to [28] while the southern storms’ diameter is between 5600 and 7000 km as [28] says. In the northern region there are 8 vortices surrounding the central, unlike in the south pole that only 5 vortices are drifting around the central storm. The reason of this difference is attributed to the size of the storms in each region.
(a) Jupiter’s South Pole Cyclones in 2016 (b) Jupiter’s South Pole Cyclones in 2019
Figure 4.39: Images captured by Juno orbiter. Instrument: JIRAM. Extracted from [6]
The image composition above presented shows the origin of a sixth vortex in Jupiter’s South Pole. The Figure 4.39a was taken 3 years before the right image, both with an infrared mapper. The sixth vortex has probably continued growing until reaching a size similar to that of the rest. When it comes to Saturn, the most recent information available comes from Cassini space probe. Unlike its big brother, Saturn’s polar region only presents a strong compact central cyclonic vortex around which the polar atmospheric phenomena rotates. The 4.4. VALIDATION OF SOFTWARE MODULE 79 storm is encircled by high-speed winds at 87◦ S and 89◦. In the center of the cyclone a cloud-free central zone, also known as the ”eye” is presented.
Figure 4.40: The northern ”eye” of Saturn seen from Cassini. Instrument: ISS - Narrow Angle Camera. Extracted from [6]
The image above shows the central point of Saturn’s northern cyclone. The image was taken on 2nd April of 2014. Scientists have determined the diameter of the ”eye” to be of 2000 km surrounded by winds of about 150 m/S. Notice that both Jupiter and Saturn share the characteristic of a central powerful storm in each polar region. It is important to mention that currently it is not known which is the origin and nature of the polar vortices observed in the Giant Planets. In order to verify if the results are correct or not and consequently if the software mod- ule is working or not, the results obtained from the simulations are compared with those results presented by Shawn R. Brueshaber, Kunio M. Sayanagi and Timothy E. Dowling in the article [28]. They present a study about polar regions in the Giant Planets aimed a understanding the dynamics and determining which parameters have more influence in the regions. They do it by is a comparison between the all the Outer Planets. The model used for their results is also based on Shallow-Water model implemented with a EPIC code.
4.4.2 Computational Work
The methodology followed in this study so as to obtain the results consists in the insertion of random perturbances each day during a 5000-day simulation so as to see if the response of the system is similar to that presented in the article [28] and to that observed by Cassini and Juno space probes. Firstly, the characteristics of the perturbances are selected after the performance of a 80 CHAPTER 4. THE SIMULATIONS set of tests both for Jupiter and Saturn. It is decided that two different sets of tests for both planets are developed. The difference between the two sets is the intensity of the storms which are going to be injected. The parameters selected for the perturbances of each set for both planets are:
Table 4.10: Set 1. Parameters of perturbances in polar regions
Perturbance parameters Jupiter Saturn Injection time 1 day 1 day Injection interval 1 day 1 day Dissipation 0 0 Latitude coordinates Random Random Velocity 0 m/s 0 m/s Caudal 6·1010 m3/s 1·1011 m3/s σ 0.5◦ 0.5◦ Radius 0.5◦ 0.5◦ Tracer 0 0
Table 4.11: Set 2. Parameters of perturbances in polar regions
Perturbance parameters Jupiter Saturn Injection time 1 day 1 day Injection interval 1 day 1 day Dissipation 0 0 Latitude coordinates Random Random Velocity 0 m/s 0 m/s Caudal 2·1010 m3/s 5·1011 m3/s σ 1◦ 1◦ Radius 1◦ 1◦ Tracer 0 0
Notice that Set 1 of simulations includes values of intensity weaker than those of Set2. In addition to perturbance parameters, it is decided to carry out simulations of 5000 days. No zonal winds are used, the reason is that the software should be able to provide a result which natural dynamics tend to that presented in the reference article, [28], which is the same of the observations. Observe that 5000 perturbances are injected, one per day, which location is random. An important parameter to consider is the dissipation constant for the domain. This constant is also known as the ”sponge” since it absorbs energy from the system and thus, phenomena occurring there disappears. It is an interesting parameters due to the role it plays and where it plays it. It can be established to be active only in a part of the domain. This case uses a ”sponge” which acts for latitudes bigger than 28◦, considering that the domain cap has a radius of 30◦ as it is established in chapter 3. 4.4. VALIDATION OF SOFTWARE MODULE 81
In this case, the layer depth is a parameter to be selected, unlike in the other cases. Since the study is based on the information presented in [28], the depth of Jupiter and Saturn’s fluid layer are determined by information provided in the article.
Jupiter’s layer depth = 8100 m
Saturn’s layer depth = 19157 m
Once the parameters have been selected the simulations are run. The results obtained for the first set are not clear. Set 1 In the case of Jupiter, a polar dynamic is observed to be reproduced since some cyclone (warm colors) and anticyclones (cold colors) are presented, however, in day 5000 of the simulation the little storms injected do not have the character such that observed in figure Figure 4.39b. The features do not converge in a cluster of vortices as it is expected to happen. It is true that the system gets unstable after such amount of days, however, the results are not clear.
Figure 4.41: Set 1. Diagram of Jupiter’s polar region represented with Potential-Vorticity. 82 CHAPTER 4. THE SIMULATIONS
A similar result is obtained with Saturn Set 1 case. Polar dynamics are shown in the figure Figure 4.42, notice the vortices which have grown from small perturbances injected in day 5000 of the simulation. The result also presents cyclones and anticyclones, though more features are cyclonic. Saturn’s case should be easier to be analysed in comparison with real dynamics observed in the planet if the central vortex is considered and indeed it is clear that the result does not clearly show a central powerful cyclone.
Figure 4.42: Set 1. Diagram of Saturn’s polar region represented with Potential Vorticity.
Set 2 The second set presents the result of simulations developed with stronger and bigger features injected, which parameters are presented in Table 4.11 . The procedure followed is the same and the domain characteristics as well as computational are maintained. Every day a feature is inserted randomly in the domain up to 5000 days to be simulated. This set provides a result different from the first one presented. Since the storms injected day by day are more intense, the final system presents grown features which predominant character is cyclonic, though there are several anticyclonic storms. Unlike the case of the Set 1, it can be seen that two big cyclones and one anticyclone have appeared, however, when all the results are analysed it is seen that these features vary fast and do not reach a stable behaviour. Thus, the simulation does not provide a conclusive 4.4. VALIDATION OF SOFTWARE MODULE 83 result in comparison with the information presented in [28].
Figure 4.43: Set 2. Diagram of Jupiter’s polar region represented with Potential Vorticity.
The simulation of Set 2 regarding Saturn are even less decisive than those of Jupiter in this case. It is seen that according to the article and scientific information, the result does not show a main central vortex. Again, the vortices dynamics of the pole are shown in Figure 4.44, obtaining an unstable system. In addition storms obtained are bigger than those of Figure 4.42, though the same patter observed with Jupiter’s simulations occurs. When all the results are analysed, the vortices, despite being bigger, do not have a stable behaviour. The results of Set 1 are less clear than those of Set 2, however none of the results provides clear information. 84 CHAPTER 4. THE SIMULATIONS
Figure 4.44: Set 2. Diagram of Saturn’s polar region represented with Potential Vorticity.
It can be attributed to the computational scheme used for the implementation but it can also be attributed to lack of time in the simulations. The simulations presented in the article [28] last for 20000 days, which is four times the duration of the simulations presented in this case. However, they present a gradual evolution of the system and if the results corresponding to 5000 days provided in the article are compared to those presented here, the difference between the results decreases. Moreover, another cause for the results presented can be the parameters used as well as the lack of zonal wind. Since the tests carried out for the parameters selection has not been as exhaustive as it can, some parameters may be changed in order to obtain a more accurate result. Finally, it is observed that the results in the conditions presented for a polar region study are not clear and a further study should be carried out, and the tests performed should consider more options. 5 Conclusions and follow-up of the study
Throughout the development of the thesis, 3 three different phases are accomplished. The first one involves a deep introduction in the topic of the thesis and the study of the main characteristics of Jupiter and Saturn. The second phase consists in the acquisition of necessary knowledge regarding the computational part of the thesis, which includes familiarisation with an operative system and more important, with the software which is being used. Finally, once the first two stages are completed, 3 different cases about phenomena observed in the Gas Giants are studied by the analysis of several simulations results. The first conclusion which can be drawn from the performance of the project is that it does not matter if the objective to complete is very hard or the conditions are not as good as it could. If one is surrounded by the correct people and one push himself/herself forward no matter what, a good and satisfying result will be obtained. Secondly, I realize during the study performance that there are lots of things to learn. No matter how much information I could think this thesis includes, it is nothing in comparison with what it is available nowadays about the topic. When it comes to the results, firstly, it is seen for the first simulation case about Sat- urn’s convective storms that the morphology that the features present has a lot to do with the turbulence occurring in deeper atmospheric layers. It is seen that the model is quite powerful and representative so as to show main characteristics of convective long-lived storms in Saturn, taking into consideration an important limitation such thermodynamic effects. This part of the thesis is hard to be improved, the results presented are a very good approach of the phenomena observed by Voyager 2 in its visit to the Gas Giant. In order to improve the results, a better knowledge about the atmosphere’s dynamics should be acquired. With regard to the Great Red Spot origin, the results suggest that it may be derived from a big cell rather than from the merge of little vortex such the case of the Oval BA. In the study of columns and the reproduction of the ancient GRS it is seen that the tangential velocity is a clear mechanism for the appearance change of the structures and the results lead to think that if the Great Red spot came from a high-pressure cell a
85 86 CHAPTER 5. CONCLUSIONS AND FOLLOW-UP OF THE STUDY possible mechanism was the increase of its tangential velocity. However, the lack of clarity in the merge case can be attributed to the conditions of the simulation as well as to the Shallow-Water limitations. The continuation of this part of the thesis can be focused on a deeper study about the influence of tangential velocity in the shape of big structures and the comparison of the results with data available as well as a the follow up of merge mechanism with a vortex such the Oval BA in order to see check from a known case if the model is able to provide reliable results of this case. Finally, it is important to emphasise that the last case has been carried out under regular conditions. Since the Marenostrum project awarded has a limited amount of time, the great part of the resources were intended for the first case and the second one. In addition, the performance of this stage has been fast and it is reflected in the poor results. By now, it is seen that the new module aimed at the reproduction of polar dynamics does not provide reliable results. However, a further study should consider the accomplishment of longer simulations in order to get results in the same conditions as the ones which have been used as scientific reference. Moreover, a more accurate study for the selection of parameters should be carried out, taking into consideration the computational scheme and the physical model used in the software. To sum up, the thesis development has supposed a very interesting challenge, a big source of information and learning and the opening of a door which I did not ever consider it was there. It has been a great and different experience of research outside the engineer- ing world. To be able to carry out the thesis with help of the members of UPC TUAREG Team and with such a resource as Marenostrum supercomputer has been incredible. It has also demonstrated that the regular relationship between university professor and student can improved and that if one is thought with passion, one will learn with passion. Bibliography
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[18] Hurricane Structure. [Online]. Available: https://www.unidata.ucar.edu/data/NGCS/lobjects/ chp/structure/. [19] What is Potential Vorticity? - YouTube. [Online]. Available: https://www.youtube.com/watch? v=GEb2I7_9GOY. [20] How Does Potential Vorticity Create Troughs? - YouTube. [Online]. Available: https : / / www . youtube.com/watch?v=_UXACwGHwA0. [21] Technical Information — BSC-CNS. [Online]. Available: https://www.bsc.es/marenostrum/ marenostrum/technical-information. [22] L. A. Sromovsky, H. E. Revercomb, R. J. Krauss and V. E. Suomi, ‘Voyager 2 Observations of Saturn’S Northern Mid-Latitude Cloud Features: Morphology, Motions, and Evolution.’, Journal of Geophysical Research, vol. 88, no. A11, pp. 8650–8666, 1983, issn: 01480227. doi: 10.1029/ JA088iA11p08650. [23] M. H. Wong, I. de Pater, X. Asay-Davis, P. S. Marcus and C. Y. Go, ‘Vertical structure of Jupiter’s Oval BA before and after it reddened: What changed?’, Icarus, vol. 215, no. 1, pp. 211–225, 2011, issn: 00191035. doi: 10.1016/j.icarus.2011.06.032. [Online]. Available: http://dx.doi.org/ 10.1016/j.icarus.2011.06.032. [24] Juno’s Latest Flyby of Jupiter Captures Two Massive Storms — NASA. [Online]. Available: https: //www.nasa.gov/image-feature/jpl/juno-s-latest-flyby-of-jupiter-captures-two- massive-storms. [25] PLANETARY ASTRONOMY — Christophe Pellier. [Online]. Available: https://www.planetary- astronomy-and-imaging.com/en/. [26] Wayback Machine. [Online]. Available: https://web.archive.org/web/20070101114206/http: //spaceprojects.arc.nasa.gov/Space_Projects/pioneer/PNimgs/f14.gif. [27] UK Amateur Recreates the Great Red Spot’s Glory Days - Universe Today. [Online]. Available: https://www.universetoday.com/120765/uk-amateur-recreates-the-great-red-spots- glory-days/. [28] S. R. Brueshaber, K. M. Sayanagi and T. E. Dowling, ‘Dynamical regimes of giant planet polar vortices’, Icarus, vol. 323, no. February, pp. 46–61, 2019, issn: 10902643. doi: 10.1016/j.icarus. 2019.02.001. Environmental awareness
During the development of the thesis, environmental consequences have been taken into consideration too. Since this is a study based on numerical simulations, only con- sequences from the work already performed are considered. There is not a further step of manufacturing or implementation of the result presented in this file. The main environmental impact coming from the accomplishment of this thesis is at- tributed to power usage. During the completion of the study, 750.000 hours of power, approximately, have been used by Marenostrum supercomputer intended for the develop- ment of the simulations. Although the amount of hours used is huge, it is not the main threat regarding pollution. Nevertheless, a good solution for the reduction of power usage pollution would be the use of renewable energy for the supply of the supercomputer.
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