Material derivatives of higher dimension in geophysical fluid dynamics
The familiar operator D/Dt in fluid dynamics defines the material derivative for a fluid particle with dimension zero. In this paper we define and use "macroscopic" or multidimensional material derivatives D1/Dt, D2/Dt and D3/Dt. They are the material derivatives of infinitesimal propertie...
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| Format: | Article |
| Language: | English |
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Borntraeger
2004-12-01
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| Series: | Meteorologische Zeitschrift |
| Online Access: | http://dx.doi.org/10.1127/0941-2948/2004/0013-0499 |
| _version_ | 1797320682269310976 |
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| author | Heinz Fortak |
| author_facet | Heinz Fortak |
| author_sort | Heinz Fortak |
| collection | DOAJ |
| description | The familiar operator D/Dt in fluid dynamics defines the material derivative for a fluid particle with dimension zero. In this paper we define and use "macroscopic" or multidimensional material derivatives D1/Dt, D2/Dt and D3/Dt. They are the material derivatives of infinitesimal properties of the fluid having dimensions, i.e. when particles build a line, a surface area, or a volume. Simple rules between the three operators are presented that avoid complicated calculations in fluid dynamics. For example, these operators are invariant with respect to solid rotations of coordinate systems. We rewrite a number of equations of fluid dynamics in terms of these operators and show that simple identities involving these operators already contain the structure of known vorticity theorems, especially those given by Hans Ertel. One application deals with the circulation of eddy velocities in atmospheric turbulence, showing that this circutation maybe an almost material invariant with time. Further possible applications (e.g., in electrodynamics of fluids and in radiation hydrodynamics) are also suggested. |
| first_indexed | 2024-03-08T04:46:31Z |
| format | Article |
| id | doaj.art-29d6cec391a84a7d829c043b5c36b2ad |
| institution | Directory Open Access Journal |
| issn | 0941-2948 |
| language | English |
| last_indexed | 2024-03-08T04:46:31Z |
| publishDate | 2004-12-01 |
| publisher | Borntraeger |
| record_format | Article |
| series | Meteorologische Zeitschrift |
| spelling | doaj.art-29d6cec391a84a7d829c043b5c36b2ad2024-02-08T08:44:37ZengBorntraegerMeteorologische Zeitschrift0941-29482004-12-0113649951010.1127/0941-2948/2004/0013-049953390Material derivatives of higher dimension in geophysical fluid dynamicsHeinz FortakThe familiar operator D/Dt in fluid dynamics defines the material derivative for a fluid particle with dimension zero. In this paper we define and use "macroscopic" or multidimensional material derivatives D1/Dt, D2/Dt and D3/Dt. They are the material derivatives of infinitesimal properties of the fluid having dimensions, i.e. when particles build a line, a surface area, or a volume. Simple rules between the three operators are presented that avoid complicated calculations in fluid dynamics. For example, these operators are invariant with respect to solid rotations of coordinate systems. We rewrite a number of equations of fluid dynamics in terms of these operators and show that simple identities involving these operators already contain the structure of known vorticity theorems, especially those given by Hans Ertel. One application deals with the circulation of eddy velocities in atmospheric turbulence, showing that this circutation maybe an almost material invariant with time. Further possible applications (e.g., in electrodynamics of fluids and in radiation hydrodynamics) are also suggested.http://dx.doi.org/10.1127/0941-2948/2004/0013-0499 |
| spellingShingle | Heinz Fortak Material derivatives of higher dimension in geophysical fluid dynamics Meteorologische Zeitschrift |
| title | Material derivatives of higher dimension in geophysical fluid dynamics |
| title_full | Material derivatives of higher dimension in geophysical fluid dynamics |
| title_fullStr | Material derivatives of higher dimension in geophysical fluid dynamics |
| title_full_unstemmed | Material derivatives of higher dimension in geophysical fluid dynamics |
| title_short | Material derivatives of higher dimension in geophysical fluid dynamics |
| title_sort | material derivatives of higher dimension in geophysical fluid dynamics |
| url | http://dx.doi.org/10.1127/0941-2948/2004/0013-0499 |
| work_keys_str_mv | AT heinzfortak materialderivativesofhigherdimensioningeophysicalfluiddynamics |