Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation
Shallow water magnetohydrodynamic (MHD) theory describing incompressible flows of plasma is generalized to the case of compressible flows. A system of MHD equations is obtained that describes the flow of a thin layer of compressible rotating plasma in a gravitational field in the shallow water appro...
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description | Shallow water magnetohydrodynamic (MHD) theory describing incompressible flows of plasma is generalized to the case of compressible flows. A system of MHD equations is obtained that describes the flow of a thin layer of compressible rotating plasma in a gravitational field in the shallow water approximation. The system of quasilinear hyperbolic equations obtained admits a complete simple wave analysis and a solution to the initial discontinuity decay problem in the simplest version of nonrotating flows. In the new equations, sound waves are filtered out, and the dependence of density on pressure on large scales is taken into account that describes static compressibility phenomena. In the equations obtained, the mass conservation law is formulated for a variable that nontrivially depends on the shape of the lower boundary, the characteristic vertical scale of the flow, and the scale of heights at which the variation of density becomes significant. A simple wave theory is developed for the system of equations obtained. All self-similar discontinuous solutions and all continuous centered self-similar solutions of the system are obtained. The initial discontinuity decay problem is solved explicitly for compressible MHD equations in the shallow water approximation. It is shown that there exist five different configurations that provide a solution to the initial discontinuity decay problem. For each configuration, conditions are found that are necessary and sufficient for its implementation. Differences between incompressible and compressible cases are analyzed. In spite of the formal similarity between the solutions in the classical case of MHD flows of an incompressible and compressible fluids, the nonlinear dynamics described by the solutions are essentially different due to the difference in the expressions for the squared propagation velocity of weak perturbations. In addition, the solutions obtained describe new physical phenomena related to the dependence of the height of the free boundary on the density of the fluid. Self-similar continuous and discontinuous solutions are obtained for a system on a slope, and a solution is found to the initial discontinuity decay problem in this case. |
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A. ; Petrosyan, A. S.</creator><creatorcontrib>Klimachkov, D. A. ; Petrosyan, A. S.</creatorcontrib><description>Shallow water magnetohydrodynamic (MHD) theory describing incompressible flows of plasma is generalized to the case of compressible flows. A system of MHD equations is obtained that describes the flow of a thin layer of compressible rotating plasma in a gravitational field in the shallow water approximation. The system of quasilinear hyperbolic equations obtained admits a complete simple wave analysis and a solution to the initial discontinuity decay problem in the simplest version of nonrotating flows. In the new equations, sound waves are filtered out, and the dependence of density on pressure on large scales is taken into account that describes static compressibility phenomena. In the equations obtained, the mass conservation law is formulated for a variable that nontrivially depends on the shape of the lower boundary, the characteristic vertical scale of the flow, and the scale of heights at which the variation of density becomes significant. A simple wave theory is developed for the system of equations obtained. All self-similar discontinuous solutions and all continuous centered self-similar solutions of the system are obtained. The initial discontinuity decay problem is solved explicitly for compressible MHD equations in the shallow water approximation. It is shown that there exist five different configurations that provide a solution to the initial discontinuity decay problem. For each configuration, conditions are found that are necessary and sufficient for its implementation. Differences between incompressible and compressible cases are analyzed. In spite of the formal similarity between the solutions in the classical case of MHD flows of an incompressible and compressible fluids, the nonlinear dynamics described by the solutions are essentially different due to the difference in the expressions for the squared propagation velocity of weak perturbations. In addition, the solutions obtained describe new physical phenomena related to the dependence of the height of the free boundary on the density of the fluid. Self-similar continuous and discontinuous solutions are obtained for a system on a slope, and a solution is found to the initial discontinuity decay problem in this case.</description><identifier>ISSN: 1063-7761</identifier><identifier>EISSN: 1090-6509</identifier><identifier>DOI: 10.1134/S1063776116070098</identifier><language>eng</language><publisher>Moscow: Pleiades Publishing</publisher><subject>Analysis ; Approximation ; APPROXIMATIONS ; Classical and Quantum Gravitation ; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS ; COMPRESSIBILITY ; COMPRESSIBLE FLOW ; CONFIGURATION ; DECAY ; DENSITY ; DISTURBANCES ; Elementary Particles ; Environmental law ; EQUATIONS ; FILTERS ; FLUIDS ; Free boundaries ; GRAVITATIONAL FIELDS ; Gravity (Force) ; INCOMPRESSIBLE FLOW ; MAGNETOHYDRODYNAMICS ; MASS ; MATHEMATICAL SOLUTIONS ; Nonlinear ; Nonlinear dynamics ; NONLINEAR PROBLEMS ; Particle and Nuclear Physics ; PERTURBATION THEORY ; Physics ; Physics and Astronomy ; Quantum Field Theory ; Relativity Theory ; ROTATING PLASMA ; Soft Matter Physics ; Solid State Physics ; SOUND WAVES ; Statistical ; THIN FILMS</subject><ispartof>Journal of experimental and theoretical physics, 2016-09, Vol.123 (3), p.520-539</ispartof><rights>Pleiades Publishing, Inc. 2016</rights><rights>COPYRIGHT 2016 Springer</rights><rights>Copyright Springer Science & Business Media 2016</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c417t-184bab4a92dad83ad47ef5abe72a85de641f4b307ba53107e8305ebd0f01d0c73</citedby><cites>FETCH-LOGICAL-c417t-184bab4a92dad83ad47ef5abe72a85de641f4b307ba53107e8305ebd0f01d0c73</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1134/S1063776116070098$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1134/S1063776116070098$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,314,780,784,885,27924,27925,41488,42557,51319</link.rule.ids><backlink>$$Uhttps://www.osti.gov/biblio/22617178$$D View this record in Osti.gov$$Hfree_for_read</backlink></links><search><creatorcontrib>Klimachkov, D. A.</creatorcontrib><creatorcontrib>Petrosyan, A. S.</creatorcontrib><title>Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation</title><title>Journal of experimental and theoretical physics</title><addtitle>J. Exp. Theor. Phys</addtitle><description>Shallow water magnetohydrodynamic (MHD) theory describing incompressible flows of plasma is generalized to the case of compressible flows. A system of MHD equations is obtained that describes the flow of a thin layer of compressible rotating plasma in a gravitational field in the shallow water approximation. The system of quasilinear hyperbolic equations obtained admits a complete simple wave analysis and a solution to the initial discontinuity decay problem in the simplest version of nonrotating flows. In the new equations, sound waves are filtered out, and the dependence of density on pressure on large scales is taken into account that describes static compressibility phenomena. In the equations obtained, the mass conservation law is formulated for a variable that nontrivially depends on the shape of the lower boundary, the characteristic vertical scale of the flow, and the scale of heights at which the variation of density becomes significant. A simple wave theory is developed for the system of equations obtained. All self-similar discontinuous solutions and all continuous centered self-similar solutions of the system are obtained. The initial discontinuity decay problem is solved explicitly for compressible MHD equations in the shallow water approximation. It is shown that there exist five different configurations that provide a solution to the initial discontinuity decay problem. For each configuration, conditions are found that are necessary and sufficient for its implementation. Differences between incompressible and compressible cases are analyzed. In spite of the formal similarity between the solutions in the classical case of MHD flows of an incompressible and compressible fluids, the nonlinear dynamics described by the solutions are essentially different due to the difference in the expressions for the squared propagation velocity of weak perturbations. In addition, the solutions obtained describe new physical phenomena related to the dependence of the height of the free boundary on the density of the fluid. Self-similar continuous and discontinuous solutions are obtained for a system on a slope, and a solution is found to the initial discontinuity decay problem in this case.</description><subject>Analysis</subject><subject>Approximation</subject><subject>APPROXIMATIONS</subject><subject>Classical and Quantum Gravitation</subject><subject>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS</subject><subject>COMPRESSIBILITY</subject><subject>COMPRESSIBLE FLOW</subject><subject>CONFIGURATION</subject><subject>DECAY</subject><subject>DENSITY</subject><subject>DISTURBANCES</subject><subject>Elementary Particles</subject><subject>Environmental law</subject><subject>EQUATIONS</subject><subject>FILTERS</subject><subject>FLUIDS</subject><subject>Free boundaries</subject><subject>GRAVITATIONAL FIELDS</subject><subject>Gravity (Force)</subject><subject>INCOMPRESSIBLE FLOW</subject><subject>MAGNETOHYDRODYNAMICS</subject><subject>MASS</subject><subject>MATHEMATICAL SOLUTIONS</subject><subject>Nonlinear</subject><subject>Nonlinear dynamics</subject><subject>NONLINEAR PROBLEMS</subject><subject>Particle and Nuclear Physics</subject><subject>PERTURBATION THEORY</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Quantum Field Theory</subject><subject>Relativity Theory</subject><subject>ROTATING PLASMA</subject><subject>Soft Matter Physics</subject><subject>Solid State Physics</subject><subject>SOUND WAVES</subject><subject>Statistical</subject><subject>THIN FILMS</subject><issn>1063-7761</issn><issn>1090-6509</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp1kcFq3DAQhk1poWmSB-hN0FMPTke2bMnHENo0EFpImrMYS-O1glfaSlqSffvKbKENpeggMfN94memqt5zuOC8FZ_uOfStlD3nPUiAQb2qTjgMUPcdDK_Xd9_Wa_9t9S6lRwBQDQwn1fwt-MV5wsjyTCEeWJjYFjeecpgPNgZ78Lh1hk1LeEprE5kJ212klNy4UKnvnWXOrzpLMy6FY0-YKTLc7WJ4dlvMLviz6s2ES6Lz3_dp9fDl84-rr_Xt9-ubq8vb2gguc82VGHEUODQWrWrRCklThyPJBlVnqRd8EmMLcsSu5SBJtdDRaGECbsHI9rT6cPw3pOx0Mi6TmU3wnkzWTdNzyaX6Q5WEP_eUsn4M--hLMM2VAtWB6JtCXRypDS6knZ9CjmjKsVRGEjxNrtQvO4AelBi6Inx8IRQm03Pe4D4lfXN_95LlR9bEkFKkSe9imVU8aA563an-Z6fFaY5OKqzfUPwr9n-lX0IDoys</recordid><startdate>20160901</startdate><enddate>20160901</enddate><creator>Klimachkov, D. A.</creator><creator>Petrosyan, A. S.</creator><general>Pleiades Publishing</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>ISR</scope><scope>OTOTI</scope></search><sort><creationdate>20160901</creationdate><title>Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation</title><author>Klimachkov, D. A. ; Petrosyan, A. S.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c417t-184bab4a92dad83ad47ef5abe72a85de641f4b307ba53107e8305ebd0f01d0c73</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Analysis</topic><topic>Approximation</topic><topic>APPROXIMATIONS</topic><topic>Classical and Quantum Gravitation</topic><topic>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS</topic><topic>COMPRESSIBILITY</topic><topic>COMPRESSIBLE FLOW</topic><topic>CONFIGURATION</topic><topic>DECAY</topic><topic>DENSITY</topic><topic>DISTURBANCES</topic><topic>Elementary Particles</topic><topic>Environmental law</topic><topic>EQUATIONS</topic><topic>FILTERS</topic><topic>FLUIDS</topic><topic>Free boundaries</topic><topic>GRAVITATIONAL FIELDS</topic><topic>Gravity (Force)</topic><topic>INCOMPRESSIBLE FLOW</topic><topic>MAGNETOHYDRODYNAMICS</topic><topic>MASS</topic><topic>MATHEMATICAL SOLUTIONS</topic><topic>Nonlinear</topic><topic>Nonlinear dynamics</topic><topic>NONLINEAR PROBLEMS</topic><topic>Particle and Nuclear Physics</topic><topic>PERTURBATION THEORY</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Quantum Field Theory</topic><topic>Relativity Theory</topic><topic>ROTATING PLASMA</topic><topic>Soft Matter Physics</topic><topic>Solid State Physics</topic><topic>SOUND WAVES</topic><topic>Statistical</topic><topic>THIN FILMS</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Klimachkov, D. A.</creatorcontrib><creatorcontrib>Petrosyan, A. S.</creatorcontrib><collection>CrossRef</collection><collection>Gale In Context: Science</collection><collection>OSTI.GOV</collection><jtitle>Journal of experimental and theoretical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Klimachkov, D. A.</au><au>Petrosyan, A. S.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation</atitle><jtitle>Journal of experimental and theoretical physics</jtitle><stitle>J. Exp. Theor. Phys</stitle><date>2016-09-01</date><risdate>2016</risdate><volume>123</volume><issue>3</issue><spage>520</spage><epage>539</epage><pages>520-539</pages><issn>1063-7761</issn><eissn>1090-6509</eissn><abstract>Shallow water magnetohydrodynamic (MHD) theory describing incompressible flows of plasma is generalized to the case of compressible flows. A system of MHD equations is obtained that describes the flow of a thin layer of compressible rotating plasma in a gravitational field in the shallow water approximation. The system of quasilinear hyperbolic equations obtained admits a complete simple wave analysis and a solution to the initial discontinuity decay problem in the simplest version of nonrotating flows. In the new equations, sound waves are filtered out, and the dependence of density on pressure on large scales is taken into account that describes static compressibility phenomena. In the equations obtained, the mass conservation law is formulated for a variable that nontrivially depends on the shape of the lower boundary, the characteristic vertical scale of the flow, and the scale of heights at which the variation of density becomes significant. A simple wave theory is developed for the system of equations obtained. All self-similar discontinuous solutions and all continuous centered self-similar solutions of the system are obtained. The initial discontinuity decay problem is solved explicitly for compressible MHD equations in the shallow water approximation. It is shown that there exist five different configurations that provide a solution to the initial discontinuity decay problem. For each configuration, conditions are found that are necessary and sufficient for its implementation. Differences between incompressible and compressible cases are analyzed. In spite of the formal similarity between the solutions in the classical case of MHD flows of an incompressible and compressible fluids, the nonlinear dynamics described by the solutions are essentially different due to the difference in the expressions for the squared propagation velocity of weak perturbations. In addition, the solutions obtained describe new physical phenomena related to the dependence of the height of the free boundary on the density of the fluid. Self-similar continuous and discontinuous solutions are obtained for a system on a slope, and a solution is found to the initial discontinuity decay problem in this case.</abstract><cop>Moscow</cop><pub>Pleiades Publishing</pub><doi>10.1134/S1063776116070098</doi><tpages>20</tpages></addata></record> |
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subjects | Analysis Approximation APPROXIMATIONS Classical and Quantum Gravitation CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS COMPRESSIBILITY COMPRESSIBLE FLOW CONFIGURATION DECAY DENSITY DISTURBANCES Elementary Particles Environmental law EQUATIONS FILTERS FLUIDS Free boundaries GRAVITATIONAL FIELDS Gravity (Force) INCOMPRESSIBLE FLOW MAGNETOHYDRODYNAMICS MASS MATHEMATICAL SOLUTIONS Nonlinear Nonlinear dynamics NONLINEAR PROBLEMS Particle and Nuclear Physics PERTURBATION THEORY Physics Physics and Astronomy Quantum Field Theory Relativity Theory ROTATING PLASMA Soft Matter Physics Solid State Physics SOUND WAVES Statistical THIN FILMS |
title | Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation |
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