scholarly journals Singularity Formation and Global Existence of Classical Solutions for One-Dimensional Rotating Shallow Water System

2018 ◽  
Vol 50 (3) ◽  
pp. 2486-2508 ◽  
Author(s):  
Bin Cheng ◽  
Peng Qu ◽  
Chunjing Xie
Keyword(s):  
Global Existence ◽  
Shallow Water ◽  
Water System ◽  
Classical Solutions ◽  
Singularity Formation ◽  
One Dimensional ◽  
Shallow Water System ◽  
Rotating Shallow Water

Ageostrophic instability in rotating shallow water

2012 ◽  
Vol 712 ◽  
pp. 327-353 ◽  
Author(s):  
Peng Wang ◽  
James C. McWilliams ◽  
Ziv Kizner
Keyword(s):  
Shallow Water ◽  
Gravity Wave ◽  
Shear Wave ◽  
Water System ◽  
Wave Mode ◽  
Shear Instability ◽  
Mean Flow ◽  
Shallow Water System ◽  
Rotating Shallow Water ◽  
Wave Modes

AbstractLinear instabilities, both momentum-balanced and unbalanced, in several different $ \overline{u} (y)$ shear profiles are investigated in the rotating shallow water equations. The unbalanced instabilities are strongly ageostrophic and involve inertia–gravity wave motions, occurring only for finite Rossby ($\mathit{Ro}$) and Froude ($\mathit{Fr}$) numbers. They serve as a possible route for the breakdown of balance in a rotating shallow water system, which leads the energy to cascade towards small scales. Unlike previous work, this paper focuses on general shear flows with non-uniform potential vorticity, and without side or equatorial boundaries or vanishing layer depth (frontal outcropping). As well as classical shear instability among balanced shear wave modes (i.e. B–B type), two types of ageostrophic instability (B–G and G–G) are found. The B–G instability has attributes of both a balanced shear wave mode and an inertia–gravity wave mode. The G–G instability occurs as a sharp resonance between two inertia–gravity wave modes. The criterion for the occurrence of the ageostrophic instability is associated with the second stability condition of Ripa (1983), which requires a sufficiently large local Froude number. When $\mathit{Ro}$ and especially $\mathit{Fr}$ increase, the balanced instability is suppressed, while the ageostrophic instabilities are enhanced. The profile of the mean flow also affects the strength of the balanced and ageostrophic instabilities.

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