An Exact Finite-Amplitude Wave on a Helmholtz Velocity Profile in an Infinite Boussinesq Fluid

1980 ◽  
Vol 37 (12) ◽  
pp. 2793-2795 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Finite Amplitude ◽  
Amplitude Wave ◽  
Boussinesq Fluid

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

Nonlinear aspects of an internal gravity wave co-existing with an unstable mode associated with a Helmholtz velocity profile

1976 ◽  
Vol 76 (1) ◽  
pp. 65-84 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Wave ◽  
Unstable Mode ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Linear Stability Theory ◽  
Jump Discontinuity ◽  
Weakly Nonlinear ◽  
Boussinesq Fluid

Recently Lindzen (1974) has proposed a model of a shear-layer instability which allows unstable modes to co-exist with radiating internal gravity waves. The model is an infinite, continuously stratified, Boussinesq fluid, with a simple jump discontinuity in the velocity profile. Linear stability theory shows that the model is stable for wavenumbers k such that k2 < N2/2U2, where N is the Brunt—Väisälä frequency and 2U is the change in velocity across the discontinuity. For N2/2U2 < k2 < N2/U2 an unstable mode may co-exist with an internal gravity wave. This paper examines the weakly nonlinear aspects of this model for wavenumbers k close to the critical wavenumber N/2½U. An equation governing the evolution of the amplitude of the interfacial displacement is derived. It is shown that the interface may support a stable finite amplitude internal gravity wave.

On parametric instabilities of finite-amplitude internal gravity waves

1982 ◽  
Vol 119 ◽  
pp. 367-377 ◽  
Author(s):  
J. Klostermeyer
Keyword(s):  
Gravity Wave ◽  
Internal Gravity Wave ◽  
Maximum Growth ◽  
Numerical Approach ◽  
Internal Gravity Waves ◽  
Resonant Interaction ◽  
Finite Amplitude ◽  
Interaction Diagram ◽  
Parametric Instabilities ◽  
Boussinesq Fluid

The equations describing parametric instabilities of a finite-amplitude internal gravity wave in an inviscid Boussinesq fluid are studied numerically. By improving the numerical approach, discarding the concept of spurious roots and considering the whole range of directions of the Floquet vector, Mied's work is generalized to its full complexity. In the limit of large disturbance wavenumbers, the unstable disturbances propagate in the directions of the two infinite curve segments of the related resonant-interaction diagram. They can therefore be classified into two families which are characterized by special propagation directions. At high wavenumbers the maximum growth rates converge to limits which do not depend on the direction of the Floquet vector. The limits are different for both families; the disturbance waves propagating at the smaller angle to the basic gravity wave grow at the larger rate.

Finite-Amplitude Equilibration of Baroclinic Waves on a Jet

2010 ◽  
Vol 67 (2) ◽  
pp. 434-451 ◽  
Author(s):  
Sukyoung Lee
Keyword(s):  
Wave Breaking ◽  
Lower Layer ◽  
Finite Amplitude ◽  
Primary Role ◽  
Ekman Pumping ◽  
Amplitude Wave ◽  
Wave Source ◽  
Model Calculations ◽  
Baroclinic Waves ◽  
Pv Gradient

Abstract A two-layer quasigeostrophic model is used to study the equilibration of baroclinic waves. In this model, if the background flow is relaxed toward a jetlike profile, a finite-amplitude baroclinic wave solution can be realized in both supercritical and subcritical regions of the model’s parameter space. Analyses of the model equations and numerical model calculations indicate that the finite-amplitude wave equilibration hinges on the breaking of Rossby waves before they reach their critical latitude. This “jetward” wave breaking results in an increase in the upper-layer wave generation and a reduction in the vertical phase tilt. This change in the phase tilt has a substantial impact on the Ekman pumping, as it weakens the damping on the lower-layer wave for some parameter settings and enables the Ekman pumping to serve as a source of wave growth at other settings. Together, these processes can account for the O(1)-amplitude wave equilibration. From a potential vorticity (PV) perspective, the wave breaking reduces the meridional scale of the upper-layer eddy PV flux, which destabilizes the mean flow. This is followed by a strengthening of the lower-layer eddy PV flux, which weakens the lower-layer PV gradient and constrains the growth of the lower-layer eddy PV. The same jetward wave breaking focuses the upper-layer PV flux toward the jet center where the upper-layer PV gradient is greatest. This results in an intensification of the upper-layer eddy PV relative to lower-layer eddy PV. Because of this large ratio, the upper-layer eddy PV plays the primary role in inducing the upper- and lower-layer eddy streamfunction fields, decreasing the vertical phase tilt. As a result, the Ekman pumping on the eddies is weakened, and for some parameter settings the Ekman pumping can even act as a wave source, contributing toward O(1)-amplitude wave equilibration. By reducing the horizontal shear of the zonal wind, the same wave breaking process weakens the barotropic decay, which also contributes to the wave amplification.

Local Finite-Amplitude Wave Activity as a Diagnostic for Rossby Wave Packets

2018 ◽  
Vol 146 (12) ◽  
pp. 4099-4114 ◽  
Author(s):  
Paolo Ghinassi ◽  
Georgios Fragkoulidis ◽  
Volkmar Wirth
Keyword(s):  
Rossby Wave ◽  
Model Simulation ◽  
Wave Packets ◽  
Meridional Wind ◽  
Finite Amplitude ◽  
Wave Activity ◽  
Amplitude Wave ◽  
High Impact Weather ◽  
Finite Amplitude Wave Activity ◽  
Isentropic Coordinates

AbstractUpper-tropospheric Rossby wave packets (RWPs) are important dynamical features, because they are often associated with weather systems and sometimes act as precursors to high-impact weather. The present work introduces a novel diagnostic to identify RWPs and to quantify their amplitude. It is based on the local finite-amplitude wave activity (LWA) of Huang and Nakamura, which is generalized to the primitive equations in isentropic coordinates. The new diagnostic is applied to a specific episode containing large-amplitude RWPs and compared with a more traditional diagnostic based on the envelope of the meridional wind. In this case, LWA provides a more coherent picture of the RWPs and their zonal propagation. This difference in performance is demonstrated more explicitly in the framework of an idealized barotropic model simulation, where LWA is able to follow an RWP into its fully nonlinear stage, including cutoff formation and wave breaking, while the envelope diagnostic yields reduced amplitudes in such situations.

Finite-amplitude convection in the presence of an unsteady shear flow

1995 ◽  
Vol 287 ◽  
pp. 225-249 ◽  
Author(s):  
Philip Hall
Keyword(s):  
Shear Flow ◽  
Critical Size ◽  
Low Frequency ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Present Calculation ◽  
Wide Range ◽  
Prandtl Numbers ◽  
Boussinesq Fluid

The effect of an unsteady shear flow on the planform of convection in a Boussinesq fluid heated from below is investigated. In the absence of the shear flow it is well-known, if non-Boussinesq effects can be neglected, that convection begins in the form of a supercritical bifurcation to rolls. Subcritical convection in the form of say hexagons can be induced by non-Boussinesq behaviour which destroys the symmetry of the basic state. Here it is found that the symmetry breaking effects associated with an unsteady shear flow are not sufficient to cause subcritical convection so the problem reduces to the determination of how the orientations of roll cells are modified by an unsteady shear flow. Recently Kelly & Hu (1993) showed that such a flow has a significant stabilizing effect on the linear stability problem and that, for a wide range of Prandtl numbers, the effect is most pronounced in the low-frequency limit. In the present calculation it is shown that the stabilizing effects found by Kelly & Hu (1993) do survive for most frequencies when nonlinear effects and imperfections are taken into account. However a critical size of the frequency is identified below which the Kelly & Hu (1993) conclusions no longer carry through into the nonlinear regime. For frequencies of size comparable with this critical size it is shown that the convection pattern changes in time. The cell pattern is found to be extremely complicated and straight rolls exist only for part of a period.

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