Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-s82fj Total loading time: 0.278 Render date: 2022-09-30T23:11:36.028Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow

Published online by Cambridge University Press:  14 December 2007

TIFFANY A. SHAW
Affiliation:
Department of Physics, University of Toronto, Toronto, ON, Canada, M5S 1A7
THEODORE G. SHEPHERD
Affiliation:
Department of Physics, University of Toronto, Toronto, ON, Canada, M5S 1A7

Abstract

Wave-activity conservation laws are key to understanding wave propagation in inhomogeneous environments. Their most general formulation follows from the Hamiltonian structure of geophysical fluid dynamics. For large-scale atmospheric dynamics, the Eliassen–Palm wave activity is a well-known example and is central to theoretical analysis. On the mesoscale, while such conservation laws have been worked out in two dimensions, their application to a horizontally homogeneous background flow in three dimensions fails because of a degeneracy created by the absence of a background potential vorticity gradient. Earlier three-dimensional results based on linear WKB theory considered only Doppler-shifted gravity waves, not waves in a stratified shear flow. Consideration of a background flow depending only on altitude is motivated by the parameterization of subgrid-scales in climate models where there is an imposed separation of horizontal length and time scales, but vertical coupling within each column. Here we show how this degeneracy can be overcome and wave-activity conservation laws derived for three-dimensional disturbances to a horizontally homogeneous background flow. Explicit expressions for pseudoenergy and pseudomomentum in the anelastic and Boussinesq models are derived, and it is shown how the previously derived relations for the two-dimensional problem can be treated as a limiting case of the three-dimensional problem. The results also generalize earlier three-dimensional results in that there is no slowly varying WKB-type requirement on the background flow, and the results are extendable to finite amplitude. The relationship between pseudoenergy and pseudomomentum , where c is the horizontal phase speed in the direction of symmetry associated with , has important applications to gravity-wave parameterization and provides a generalized statement of the first Eliassen–Palm theorem.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abarbanel, H. D. I., Holm, D. D., Marsden, J. E. & Ratiu, T. S. 1986 Nonlinear stability analysis of stratified fluid equilibria. Phil. Trans. R. Soc. Lond. A 318, 349409.CrossRefGoogle Scholar
Andrews, D. G., Holton, J. R. & Leovy, C. B. 1987 Middle Atmosphere Dynamics. Academic.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978 On wave-action and its relatives. J. Fluid Mech. 89, 647664.CrossRefGoogle Scholar
Becker, E. 2004 Direct heating rates associated with gravity wave saturation. J. Atmos. Solar–Terres. Phys. 66, 683696.CrossRefGoogle Scholar
Bretherton, F. P. 1966 The propagation of groups of internal gravity waves in a shear flow. Q. J. R. Met. Soc. 92, 466480.CrossRefGoogle Scholar
Eliassen, A. & Palm, E. 1961 On the transfer of energy by mountain waves. Geofysiske Publ. 22, 94101.Google Scholar
Hines, C. O. & Reddy, C. A. 1967 On the propagation of atmospheric gravity waves through regions of wind shear. J. Geophys. Res. 72, 10151034.CrossRefGoogle Scholar
Lindzen, R. S. 1990 Dynamics in Atmospheric Physics: Lecture Notes for an Introductory Graduate-level Course. Cambridge University Press.CrossRefGoogle Scholar
Lipps, F. B. & Hemler, R. S. 1982 On the anelastic approximation for deep convection. J. Atmos. Sci. 39, 21922210.2.0.CO;2>CrossRefGoogle Scholar
Lübken, F.-J. 1997 Seasonal variation of turbulent energy dissipation rates at high latitudes as determined by in situ measurements of neutral density fluctuations. J. Geophys. Res. 102, 1344113456.CrossRefGoogle Scholar
Scinocca, J. F. & Shepherd, T. G. 1992 Nonlinear wave-activity conservation laws and Hamiltonian structure for the two-dimensional anelastic equations. J. Atmos. Sci. 49, 527.2.0.CO;2>CrossRefGoogle Scholar
Shepherd, T. G. 1990 Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287338.CrossRefGoogle Scholar
Shepherd, T. G. 2003 Hamiltonian dynamics. In Encyclopedia of Atmospheric Sciences (ed. Holton, J. R. et al. ), pp. 929938. Academic.CrossRefGoogle Scholar
Vanneste, J. & Shepherd, T. G. 1998 On the group-velocity property for wave-activity conservation laws. J. Atmos. Sci. 55, 10631068.2.0.CO;2>CrossRefGoogle Scholar
10
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *