A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities:
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Format: | Abschlussarbeit Buch |
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Köln
Inst. für Geophysik und Meteorologie
2005
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Schriftenreihe: | Mitteilungen aus dem Institut für Geophysik und Meteorologie der Universität zu Köln
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Beschreibung: | VIII, 235 Seiten graphische Darstellungen |
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245 | 1 | 0 | |a A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities |c Thorsten Knetter |
264 | 1 | |a Köln |b Inst. für Geophysik und Meteorologie |c 2005 | |
300 | |a VIII, 235 Seiten |b graphische Darstellungen | ||
336 | |b txt |2 rdacontent | ||
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490 | 1 | |a Mitteilungen aus dem Institut für Geophysik und Meteorologie der Universität zu Köln |v 163 | |
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Datensatz im Suchindex
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adam_text | Contents
1 Introduction 1
2 The solar wind 5
3 Discontinuities 9
3.1 Classification of MHD discontinuities 9
3.1.1 Theory 10
3.1.2 Criteria used in observations 16
3.2 Observations of solar wind discontinuities 18
3.3 Origin and stability of solar wind discontinuities 20
3.4 The importance of solar wind discontinuities 24
4 The Cluster mission 27
4.1 Mission overview and historical background 28
4.2 Orbit and separation strategy 29
4.3 The satellites and their payload 31
4.3.1 Fluxgate magnetometer (FGM) 31
4.3.2 Cluster ion spectrometry (CIS) 32
5 Geometrical considerations of the Cluster array 33
5.1 1-D geometric factors 34
» CONTENTS
5.2 2-D geometric factors and the volumetric tensor 35
5.3 A simple performance study 39
5.4 Evolution of the tetrahedron along one orbit 41
5.5 Constellations used in our analysis 44
6 The set of discontinuities used in this work 53
6.1 Solar wind periods 54
6.2 Search criteria 55
6.3 Application of the search criteria 59
6.4 Avoiding foreshock activity - Final set of DDs 63
6.5 Prevailing solar wind conditions - Coronal holes 70
6.5.1 Solar activity cycle 71
6.5.2 Using ACE as a solar wind monitor 72
6.5.3 Wang-Sheeley model 78
6.5.4 Extreme ultra violet imaging 82
6.5.5 The helium I 1083 nm absorption line 84
6.5.6 DDs in coronal hole flow 85
7 Determination of discontinuity normals 87
7.1 Overview of normal determination techniques 88
7.2 Sonnerup-Cahill minimum variance analysis 90
7.2.1 Description of the method 91
7.2.2 Summary of past error analysis approaches 95
7.3 Cross product method for TDs 99
7.4 Triangulation 99
7.4.1 Description of the method 100
7.4.2 Barycentric formalism - error analysis 103
8 Statistical analysis of discontinuities at 1AU 109
8.1 Single spacecraft analysis 110
8.2 Classification result from triangulation compared to MVA 114
8.3 Reexamination of MVA - consequences 118
CONTENTS m
8.3.1 Parameters affecting the precision of MVA 120
8.3.2 Dependence on the eigenvalue ratio X2/X3 122
8.3.3 Dependence on the spreading angle (0 130
8.3.4 Variability of?i2A3 and co 134
8.3.5 Superposed wave fields - one example 138
8.3.6 New classification criteria for single-spacecraft studies 142
8.4 Consistency tests for triangulation 144
8.5 Surface curvature on the Cluster scale and beyond 147
8.6 Distribution of the discontinuity normals in space 151
8.7 Discontinuity thickness 156
8.8 Using plasma data to identify possible RDs 161
8.8.1 Necessary conditions for RDs to be tested 162
8.8.2 Data coverage and selection of intervals 163
8.8.3 Thermal anisotropy 164
8.8.4 Continuity of pA 166
8.8.5 Polarisation relation 168
8.8.6 Propagation relative to the ambient plasma 175
8.8.7 Properties of possible RDs 179
9 Triangulation error analysis 185
9.1 Method and symmetry considerations 186
9.2 Parameters affecting the error 191
9.3 Histograms and average values of the errors of n and U 196
9.4 Consistency with the assumption of tangential discontinuities 198
10 Discussion and summary 205
A Mathematics of a tetrahedron 215
A.I Area of the sides 215
A.2 Volume of the tetrahedron 216
A.3 Circumscribing sphere 216
A.4 The regular tetrahedron 216
IV CONTENTS
A.5 Calculating the quality factors QGM, QRR and QR% 217
B Tables 219
C MVA error analysis 221
D Generalised formalism of the relative timing technique 223
List of Figures
3.1 Illustration of a TD and an RD 14
3.2 Example of an interplanetary discontinuity 15
3.3 Classification scheme used in this work 17
3.4 Interwoven field lines of a bipolar magnetic region 21
3.5 Illustration of the magnetostatic theorem 21
4.1 GSE coordinate system 29
4.2 Cluster orbit 30
4.3 Spacecraft separations during the Cluster mission 30
5.1 The shape of the polyhedron as a function of E and P 39
5.2 Systematic deformation of a regular tetrahedron 39
5.3 Characteristics of the tetrahedra illustrated in Figure 5.2 40
5.4 Cluster 2 orbit from doy 52 2001 05:30 UT until doy 54 2001 14:40 UT . . . 42
5.5 Tetrahedron characteristics for the orbit shown in Figure 5.4 43
5.6 Extreme cases of the tetrahedra used in our analysis 46
5.7 Spacecraft positions relative to Cl 4 for all DDs 48
5.8 Shape, orientation and location of the tetrahedra used in our analysis 49
5.9 Histograms of£ and P for 2003, and of/?^ and Rce for 2003 and 2001 .... 51
6.1 Overview of days suitable for solar wind analysis with Cluster 55
VI LIST OF FIGURES
6.2 Illustration of the Burlaga-method 57
6.3 Problem of simultaneous identification of a DD at different spacecraft .... 58
6.4 Number of DDs identified by the TS- and the B-method 60
6.5 Relative number of DDs found simultaneously at all four spacecraft as a func¬
tion of the spacecraft separation 61
6.6 Foreshock geometry in the ecliptic plane and illustration of the parameter D . 65
6.7 Six cases describing the evolution of D across a DD 66
6.8 D versus time for two typical days with Cluster in the solar wind 67
6.9 Correlation between D and foreshock waves 69
6.10 Monthly averaged sunspot numbers for cycle 23 through December 2004 . . 72
6.11 Overview of the DDs found and the prevailing solar wind conditions in 2001 . 74
6.12 Same as Figure 6.11 for 2002 75
6.13 Same as Figure 6.12 for 2003 76
6.14 FS I from Figure 6.13 79
6.15 Comparison of the Wang-Sheeley model predictions with the mapped polar¬
ity and velocity of FS I measured at ACE 81
6.16 SOHO images and Coronal hole map from the Hel 1083nm absorption line . 83
6.17 Distributions of the magnetic field and plasma parameters observed with
Cluster in the vicinity of the selected DDs in 2003 86
7.1 Variance ellipsoid 93
7.2 Illustration of the triangulation technique 101
7.3 Determination of the relative timings 102
7.4 Illustration of the barycentric coordinates and the reciprocal vectors 104
7.5 Cone of uncertainty 107
8.1 Histogram of co obtained from Cl 1 Ill
8.2 Classification results 115
8.3 Variance ellipsoids obtained from the magnetic field data of a DD 119
8.4 Distribution of X2/A,3 and kl/ 2 120
8.5 Relationship between X2/X3 and co 121
8.6 Distributions of X2/A,3 and to 122
8.7 Dependence on (X2A3)L 123
LIST OF FIGURES vn
8.8 Dependence of the classification result on (k2/^ i)L 127
8.9 Probability distribution of 9Bn = Z((B),n) and distributions of Bn /Bmax .. 129
8.10 Same as Figure 8.7 for o)L 131
8.11 Dependence of the classification result on (OL 133
8.12 Distributions of A(A,2/X3)/(A2/A.3)max and Aco/(Omav 135
8.13 Average values of AX/Xmax (X =) 2/A3, co) versus average separation .... 137
8.14 1-D discontinuity and superposed fluctuations 139
8.15 Distributions of A( Bn /Bmax) 143
8.16 Consistency tests for the triangulation method 145
8.17 Average angle between normals from single-spacecraft analysis methods . . . 149
8.18 Angles and differences between |Bn /BmaK as a function of separation .... 150
8.19 Distributions of the normals in space 153
8.20 Spatial distribution of normal estimates 155
8.21 Histograms of the discontinuity thickness in s 158
8.22 Histograms of the discontinuity thickness in km 159
8.23 Same as Figure 8.22, but only in the range d = 0 - 1000 km 160
8.24 Intervals to average plasma and field data over 164
8.25 Distributions of T /T± 165
8.26 Distributions of {«} 167
8.27 Distributions of {pA} 167
8.28 Polar scatter diagram of the change in velocity versus ©vb 169
8.29 Distribution of the angle ©vb 169
8.30 Distributions of R 171
8.31 Distributions of RVb 173
8.32 Firehose factor 174
8.33 Histograms of the normal component of the Alfven velocity | V/t-n| 176
8.34 V • n versus U and probability distributions of (V — U) • n 177
8.35 Consistency test for propagating RDs 178
8.36 I (V-U)-n| versus the magnitude of the Alfven speed IV^-nl 179
9.1 Error cone of the triangulation normal for two examples 187
vm LIST OF FIGURES
9.2 Distribution of the angle between the measured normal n and the 1330 nor¬
mals h{, and of the 1331 velocities {/,- 188
9.3 Illustration of the error fan when Ra, Rc and n are perpendicular to each
other. 189
9.4 Histograms of (Sx) 190
9.5 Histograms of Z(n,Rc) and Z(n,Ra) 190
9.6 Histograms of the time differences between the spacecraft 192
9.7 Dependence of the error on U 193
9.8 Dependence of the error on the planarity P 194
9.9 Illustration of the influence of the orientation on the error 195
9.10 Dependence of the error on the orientation of the tetrahedron relative to n . . 196
9.11 Histograms of dn and dU for tmin and 8rmax 197
9.12 Bn /Bmax and QBn with error bars for all DDs 200
9.13 Velocity U in with error bars for tmax 201
9.14 Histograms of Bn /Bmax, (90°—0gn) and -n — U compared to the associ¬
ated error values for 67max and a smaller value of 8/ 203
|
adam_txt |
Contents
1 Introduction 1
2 The solar wind 5
3 Discontinuities 9
3.1 Classification of MHD discontinuities 9
3.1.1 Theory 10
3.1.2 Criteria used in observations 16
3.2 Observations of solar wind discontinuities 18
3.3 Origin and stability of solar wind discontinuities 20
3.4 The importance of solar wind discontinuities 24
4 The Cluster mission 27
4.1 Mission overview and historical background 28
4.2 Orbit and separation strategy 29
4.3 The satellites and their payload 31
4.3.1 Fluxgate magnetometer (FGM) 31
4.3.2 Cluster ion spectrometry (CIS) 32
5 Geometrical considerations of the Cluster array 33
5.1 1-D geometric factors 34
» CONTENTS
5.2 2-D geometric factors and the volumetric tensor 35
5.3 A simple performance study 39
5.4 Evolution of the tetrahedron along one orbit 41
5.5 Constellations used in our analysis 44
6 The set of discontinuities used in this work 53
6.1 Solar wind periods 54
6.2 Search criteria 55
6.3 Application of the search criteria 59
6.4 Avoiding foreshock activity - Final set of DDs 63
6.5 Prevailing solar wind conditions - Coronal holes 70
6.5.1 Solar activity cycle 71
6.5.2 Using ACE as a solar wind monitor 72
6.5.3 Wang-Sheeley model 78
6.5.4 Extreme ultra violet imaging 82
6.5.5 The helium I 1083 nm absorption line 84
6.5.6 DDs in coronal hole flow 85
7 Determination of discontinuity normals 87
7.1 Overview of normal determination techniques 88
7.2 Sonnerup-Cahill minimum variance analysis 90
7.2.1 Description of the method 91
7.2.2 Summary of past error analysis approaches 95
7.3 Cross product method for TDs 99
7.4 Triangulation 99
7.4.1 Description of the method 100
7.4.2 Barycentric formalism - error analysis 103
8 Statistical analysis of discontinuities at 1AU 109
8.1 Single spacecraft analysis 110
8.2 Classification result from triangulation compared to MVA 114
8.3 Reexamination of MVA - consequences 118
CONTENTS m
8.3.1 Parameters affecting the precision of MVA 120
8.3.2 Dependence on the eigenvalue ratio X2/X3 122
8.3.3 Dependence on the spreading angle (0 130
8.3.4 Variability of?i2A3 and co 134
8.3.5 Superposed wave fields - one example 138
8.3.6 New classification criteria for single-spacecraft studies 142
8.4 Consistency tests for triangulation 144
8.5 Surface curvature on the Cluster scale and beyond 147
8.6 Distribution of the discontinuity normals in space 151
8.7 Discontinuity thickness 156
8.8 Using plasma data to identify possible RDs 161
8.8.1 Necessary conditions for RDs to be tested 162
8.8.2 Data coverage and selection of intervals 163
8.8.3 Thermal anisotropy 164
8.8.4 Continuity of pA 166
8.8.5 Polarisation relation 168
8.8.6 Propagation relative to the ambient plasma 175
8.8.7 Properties of possible RDs 179
9 Triangulation error analysis 185
9.1 Method and symmetry considerations 186
9.2 Parameters affecting the error 191
9.3 Histograms and average values of the errors of n and U 196
9.4 Consistency with the assumption of tangential discontinuities 198
10 Discussion and summary 205
A Mathematics of a tetrahedron 215
A.I Area of the sides 215
A.2 Volume of the tetrahedron 216
A.3 Circumscribing sphere 216
A.4 The regular tetrahedron 216
IV CONTENTS
A.5 Calculating the quality factors QGM, QRR and QR% 217
B Tables 219
C MVA error analysis 221
D Generalised formalism of the relative timing technique 223
List of Figures
3.1 Illustration of a TD and an RD 14
3.2 Example of an interplanetary discontinuity 15
3.3 Classification scheme used in this work 17
3.4 Interwoven field lines of a bipolar magnetic region 21
3.5 Illustration of the magnetostatic theorem 21
4.1 GSE coordinate system 29
4.2 Cluster orbit 30
4.3 Spacecraft separations during the Cluster mission 30
5.1 The shape of the polyhedron as a function of E and P 39
5.2 Systematic deformation of a regular tetrahedron 39
5.3 Characteristics of the tetrahedra illustrated in Figure 5.2 40
5.4 Cluster 2 orbit from doy 52 2001 05:30 UT until doy 54 2001 14:40 UT . . . 42
5.5 Tetrahedron characteristics for the orbit shown in Figure 5.4 43
5.6 Extreme cases of the tetrahedra used in our analysis 46
5.7 Spacecraft positions relative to Cl 4 for all DDs 48
5.8 Shape, orientation and location of the tetrahedra used in our analysis 49
5.9 Histograms of£ and P for 2003, and of/?^ and Rce for 2003 and 2001 . 51
6.1 Overview of days suitable for solar wind analysis with Cluster 55
VI LIST OF FIGURES
6.2 Illustration of the Burlaga-method 57
6.3 Problem of simultaneous identification of a DD at different spacecraft . 58
6.4 Number of DDs identified by the TS- and the B-method 60
6.5 Relative number of DDs found simultaneously at all four spacecraft as a func¬
tion of the spacecraft separation 61
6.6 Foreshock geometry in the ecliptic plane and illustration of the parameter D . 65
6.7 Six cases describing the evolution of D across a DD 66
6.8 D versus time for two typical days with Cluster in the solar wind 67
6.9 Correlation between D and foreshock waves 69
6.10 Monthly averaged sunspot numbers for cycle 23 through December 2004 . . 72
6.11 Overview of the DDs found and the prevailing solar wind conditions in 2001 . 74
6.12 Same as Figure 6.11 for 2002 75
6.13 Same as Figure 6.12 for 2003 76
6.14 FS I from Figure 6.13 79
6.15 Comparison of the Wang-Sheeley model predictions with the mapped polar¬
ity and velocity of FS I measured at ACE 81
6.16 SOHO images and Coronal hole map from the Hel 1083nm absorption line . 83
6.17 Distributions of the magnetic field and plasma parameters observed with
Cluster in the vicinity of the selected DDs in 2003 86
7.1 Variance ellipsoid 93
7.2 Illustration of the triangulation technique 101
7.3 Determination of the relative timings 102
7.4 Illustration of the barycentric coordinates and the reciprocal vectors 104
7.5 Cone of uncertainty 107
8.1 Histogram of co obtained from Cl 1 Ill
8.2 Classification results 115
8.3 Variance ellipsoids obtained from the magnetic field data of a DD 119
8.4 Distribution of X2/A,3 and kl/\2 120
8.5 Relationship between X2/X3 and co 121
8.6 Distributions of X2/A,3 and to 122
8.7 Dependence on (X2A3)L 123
LIST OF FIGURES vn
8.8 Dependence of the classification result on (k2/^'i)L 127
8.9 Probability distribution of 9Bn = Z((B),n) and distributions of \Bn\/Bmax . 129
8.10 Same as Figure 8.7 for o)L 131
8.11 Dependence of the classification result on (OL 133
8.12 Distributions of A(A,2/X3)/(A2/A.3)max and Aco/(Omav 135
8.13 Average values of AX/Xmax (X =) 2/A3, co) versus average separation . 137
8.14 1-D discontinuity and superposed fluctuations 139
8.15 Distributions of A(\Bn\/Bmax) 143
8.16 Consistency tests for the triangulation method 145
8.17 Average angle between normals from single-spacecraft analysis methods . . . 149
8.18 Angles and differences between |Bn \/BmaK as a function of separation . 150
8.19 Distributions of the normals in space 153
8.20 Spatial distribution of normal estimates 155
8.21 Histograms of the discontinuity thickness in s 158
8.22 Histograms of the discontinuity thickness in km 159
8.23 Same as Figure 8.22, but only in the range d = 0 - 1000 km 160
8.24 Intervals to average plasma and field data over 164
8.25 Distributions of T\\/T± 165
8.26 Distributions of {«} 167
8.27 Distributions of {pA} 167
8.28 Polar scatter diagram of the change in velocity versus ©vb 169
8.29 Distribution of the angle ©vb 169
8.30 Distributions of R\ 171
8.31 Distributions of RVb 173
8.32 Firehose factor 174
8.33 Histograms of the normal component of the Alfven velocity | V/t-n| 176
8.34 V • n versus U and probability distributions of (V — U) • n 177
8.35 Consistency test for propagating RDs 178
8.36 I (V-U)-n| versus the magnitude of the Alfven speed IV^-nl 179
9.1 Error cone of the triangulation normal for two examples 187
vm LIST OF FIGURES
9.2 Distribution of the angle between the measured normal n and the 1330 nor¬
mals h{, and of the 1331 velocities {/,- 188
9.3 Illustration of the "error fan" when Ra, Rc and n are perpendicular to each
other. 189
9.4 Histograms of (Sx) 190
9.5 Histograms of Z(n,Rc) and Z(n,Ra) 190
9.6 Histograms of the time differences between the spacecraft 192
9.7 Dependence of the error on U 193
9.8 Dependence of the error on the planarity P 194
9.9 Illustration of the influence of the orientation on the error 195
9.10 Dependence of the error on the orientation of the tetrahedron relative to n . . 196
9.11 Histograms of dn and dU for tmin and 8rmax 197
9.12 \Bn\/Bmax and QBn with error bars for all DDs 200
9.13 Velocity U in with error bars for tmax 201
9.14 Histograms of \Bn\/Bmax, (90°—0gn) and \\-n — U\ compared to the associ¬
ated error values for 67max and a smaller value of 8/ 203 |
any_adam_object | 1 |
any_adam_object_boolean | 1 |
author | Knetter, Thorsten 1974- |
author_GND | (DE-588)130424447 |
author_facet | Knetter, Thorsten 1974- |
author_role | aut |
author_sort | Knetter, Thorsten 1974- |
author_variant | t k tk |
building | Verbundindex |
bvnumber | BV035076401 |
classification_rvk | US 6700 |
ctrlnum | (OCoLC)76146779 (DE-599)BVBBV035076401 |
discipline | Physik Geographie |
discipline_str_mv | Physik Geographie |
format | Thesis Book |
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genre | (DE-588)4113937-9 Hochschulschrift gnd-content |
genre_facet | Hochschulschrift |
id | DE-604.BV035076401 |
illustrated | Not Illustrated |
index_date | 2024-07-02T22:05:43Z |
indexdate | 2024-07-09T21:21:39Z |
institution | BVB |
language | English |
oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016744708 |
oclc_num | 76146779 |
open_access_boolean | |
owner | DE-19 DE-BY-UBM DE-83 DE-188 |
owner_facet | DE-19 DE-BY-UBM DE-83 DE-188 |
physical | VIII, 235 Seiten graphische Darstellungen |
publishDate | 2005 |
publishDateSearch | 2005 |
publishDateSort | 2005 |
publisher | Inst. für Geophysik und Meteorologie |
record_format | marc |
series | Mitteilungen aus dem Institut für Geophysik und Meteorologie der Universität zu Köln |
series2 | Mitteilungen aus dem Institut für Geophysik und Meteorologie der Universität zu Köln |
spelling | Knetter, Thorsten 1974- Verfasser (DE-588)130424447 aut A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities Thorsten Knetter Köln Inst. für Geophysik und Meteorologie 2005 VIII, 235 Seiten graphische Darstellungen txt rdacontent n rdamedia nc rdacarrier Mitteilungen aus dem Institut für Geophysik und Meteorologie der Universität zu Köln 163 Zsfassung in dt. Sprache Zugl.: Köln, Univ., Diss., 2005 Sonnenwind (DE-588)4135572-6 gnd rswk-swf Diskontinuität (DE-588)4150167-6 gnd rswk-swf CLUSTER Satellitenmission (DE-588)4845419-9 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Sonnenwind (DE-588)4135572-6 s Diskontinuität (DE-588)4150167-6 s CLUSTER Satellitenmission (DE-588)4845419-9 s DE-188 Mitteilungen aus dem Institut für Geophysik und Meteorologie der Universität zu Köln 163 (DE-604)BV000005283 163 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016744708&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
spellingShingle | Knetter, Thorsten 1974- A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities Mitteilungen aus dem Institut für Geophysik und Meteorologie der Universität zu Köln Sonnenwind (DE-588)4135572-6 gnd Diskontinuität (DE-588)4150167-6 gnd CLUSTER Satellitenmission (DE-588)4845419-9 gnd |
subject_GND | (DE-588)4135572-6 (DE-588)4150167-6 (DE-588)4845419-9 (DE-588)4113937-9 |
title | A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities |
title_auth | A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities |
title_exact_search | A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities |
title_exact_search_txtP | A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities |
title_full | A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities Thorsten Knetter |
title_fullStr | A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities Thorsten Knetter |
title_full_unstemmed | A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities Thorsten Knetter |
title_short | A new perspective of the solar wind micro-structure due to multi-point observations of discontinuities |
title_sort | a new perspective of the solar wind micro structure due to multi point observations of discontinuities |
topic | Sonnenwind (DE-588)4135572-6 gnd Diskontinuität (DE-588)4150167-6 gnd CLUSTER Satellitenmission (DE-588)4845419-9 gnd |
topic_facet | Sonnenwind Diskontinuität CLUSTER Satellitenmission Hochschulschrift |
url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016744708&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
volume_link | (DE-604)BV000005283 |
work_keys_str_mv | AT knetterthorsten anewperspectiveofthesolarwindmicrostructureduetomultipointobservationsofdiscontinuities |