Existence of Purely Alvén Waves in Magnetic Flux Tubes with Arbitrary Cross-Sections
We study linear torsional Alfvén waves in a magnetic flux tube with an arbitrary cross-section. We assume that the equilibrium magnetic field is propagating in the <i>z</i>-direction in Cartesian coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&qu...
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MDPI AG
2022-07-01
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| Series: | Physics |
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| Online Access: | https://www.mdpi.com/2624-8174/4/3/55 |
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| author | Michael S. Ruderman Nikolai S. Petrukhin |
| author_facet | Michael S. Ruderman Nikolai S. Petrukhin |
| author_sort | Michael S. Ruderman |
| collection | DOAJ |
| description | We study linear torsional Alfvén waves in a magnetic flux tube with an arbitrary cross-section. We assume that the equilibrium magnetic field is propagating in the <i>z</i>-direction in Cartesian coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></semantics></math></inline-formula>, and <i>z</i>. The tube cross-section is bounded by a smooth closed curve. Both plasma and magnetic field are homogeneous outside this curve. The magnetic field magnitude is a function of <i>x</i> and <i>y</i>, while the density is a product of two functions: one dependent on <i>z</i> and the other dependent on <i>x</i> and <i>y</i>. As a result, the Alfvén speed is also equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> times a function of <i>z</i>. We define Alfvén waves as waves that do not disturb plasma density. We show that these waves can exist only when the magnetic field magnitude is a function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mn>0</mn></msub></semantics></math></inline-formula>. When the condition of existence of Alfvén waves is satisfied, the waves are polarised in the directions tangent to the level lines of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and orthogonal to the equilibrium magnetic field. We found that the Alfvén wave amplitude has a specific form that depends on a particular coordinate system. |
| first_indexed | 2024-03-09T22:48:25Z |
| format | Article |
| id | doaj.art-66561a539f684201858cc5c05ef49fb9 |
| institution | Directory Open Access Journal |
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| language | English |
| last_indexed | 2024-03-09T22:48:25Z |
| publishDate | 2022-07-01 |
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| spelling | doaj.art-66561a539f684201858cc5c05ef49fb92023-11-23T18:25:34ZengMDPI AGPhysics2624-81742022-07-014386587210.3390/physics4030055Existence of Purely Alvén Waves in Magnetic Flux Tubes with Arbitrary Cross-SectionsMichael S. Ruderman0Nikolai S. Petrukhin1Solar Physics and Space Plasma Research Centre (SP2RC), School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UKHigher School of Economics, Myasnitskaya Ulitsa, 20, Moscow 101000, RussiaWe study linear torsional Alfvén waves in a magnetic flux tube with an arbitrary cross-section. We assume that the equilibrium magnetic field is propagating in the <i>z</i>-direction in Cartesian coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></semantics></math></inline-formula>, and <i>z</i>. The tube cross-section is bounded by a smooth closed curve. Both plasma and magnetic field are homogeneous outside this curve. The magnetic field magnitude is a function of <i>x</i> and <i>y</i>, while the density is a product of two functions: one dependent on <i>z</i> and the other dependent on <i>x</i> and <i>y</i>. As a result, the Alfvén speed is also equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> times a function of <i>z</i>. We define Alfvén waves as waves that do not disturb plasma density. We show that these waves can exist only when the magnetic field magnitude is a function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mn>0</mn></msub></semantics></math></inline-formula>. When the condition of existence of Alfvén waves is satisfied, the waves are polarised in the directions tangent to the level lines of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and orthogonal to the equilibrium magnetic field. We found that the Alfvén wave amplitude has a specific form that depends on a particular coordinate system.https://www.mdpi.com/2624-8174/4/3/55plasmamagnetohydrodynamicsAlfvén wavessolar atmosphere |
| spellingShingle | Michael S. Ruderman Nikolai S. Petrukhin Existence of Purely Alvén Waves in Magnetic Flux Tubes with Arbitrary Cross-Sections Physics plasma magnetohydrodynamics Alfvén waves solar atmosphere |
| title | Existence of Purely Alvén Waves in Magnetic Flux Tubes with Arbitrary Cross-Sections |
| title_full | Existence of Purely Alvén Waves in Magnetic Flux Tubes with Arbitrary Cross-Sections |
| title_fullStr | Existence of Purely Alvén Waves in Magnetic Flux Tubes with Arbitrary Cross-Sections |
| title_full_unstemmed | Existence of Purely Alvén Waves in Magnetic Flux Tubes with Arbitrary Cross-Sections |
| title_short | Existence of Purely Alvén Waves in Magnetic Flux Tubes with Arbitrary Cross-Sections |
| title_sort | existence of purely alven waves in magnetic flux tubes with arbitrary cross sections |
| topic | plasma magnetohydrodynamics Alfvén waves solar atmosphere |
| url | https://www.mdpi.com/2624-8174/4/3/55 |
| work_keys_str_mv | AT michaelsruderman existenceofpurelyalvenwavesinmagneticfluxtubeswitharbitrarycrosssections AT nikolaispetrukhin existenceofpurelyalvenwavesinmagneticfluxtubeswitharbitrarycrosssections |