Nonlinear dynamics of hydrostatic internal gravity waves

Samuel N. Stechmann, Andrew J. Majda, Boualem Khouider

Research output: Contribution to journalArticle

Abstract

Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden-Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbolicity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations.

Original languageEnglish (US)
Pages (from-to)407-432
Number of pages26
JournalTheoretical and Computational Fluid Dynamics
Volume22
Issue number6
DOIs
StatePublished - Nov 2008

Fingerprint

Gravity waves
gravity waves
hydrostatics
convection
wind shear
partial differential equations
Partial differential equations
Madden-Julian Oscillation
asymmetry
shear
shallow water
profiles
Nonlinear equations
nonlinear equations
S waves
bursts
Numerical methods
discontinuity
nonlinearity
Convection

Keywords

  • Computational methods for nonconservative PDE
  • Convectively generated gravity waves
  • Internal bores
  • Internal gravity waves
  • Nonconservative PDE
  • Stratified fluids
  • Tropical atmospheric dynamics

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Fluid Flow and Transfer Processes
  • Engineering(all)
  • Computational Mechanics

Cite this

Nonlinear dynamics of hydrostatic internal gravity waves. / Stechmann, Samuel N.; Majda, Andrew J.; Khouider, Boualem.

In: Theoretical and Computational Fluid Dynamics, Vol. 22, No. 6, 11.2008, p. 407-432.

Research output: Contribution to journalArticle

Stechmann, Samuel N. ; Majda, Andrew J. ; Khouider, Boualem. / Nonlinear dynamics of hydrostatic internal gravity waves. In: Theoretical and Computational Fluid Dynamics. 2008 ; Vol. 22, No. 6. pp. 407-432.
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