Scaling Estimation for Growth Rate and Horizontal Wavelength of Charney-Type Submesoscale Baroclinic Instabilities (C-SBCIs)

Abstract

The Charney-type submesoscale baroclinic instabilities (C-SBCIs) originating from the mean ocean state are ubiquitous in the global ocean, characterized by a vertical structure that is surface-intensified and depth-decaying. In a companion study, we examined the geographic distribution and seasonal variation of C-SBCIs, focusing on growth rates, horizontal wavelengths, and Charney depth (or vertical scale). The Charney depth, defined as the depth range of quasi-geostrophic potential vorticity gradient necessary for the instability, serves as an important indicator for energy conversion. In the linear stage, phase change and lateral and vertical eddy buoyancy fluxes are significant above this depth but negligible below it. Based on this, a scaling formula for the growth rate of C-SBCIs is derived using the available potential energy averaged from the surface to the Charney depth; although a scaling formula for the horizontal wavelength of C-SBCIs is derived using the stratification averaged from the surface to the Charney depth. These scaling formulas are analogous to those for the Eady-type instabilities but rely on a self-selected Charney depth instead of the prescribed vertical scale in the Eady model, enhancing their applicability to the complex real ocean state. Additionally, the mechanism underlying phase speeds of C-SBCIs is investigated, which is predominantly controlled by the mean flow averaged from the surface to the Charney depth. The newly derived scaling formulas for the growth rates and horizontal wavelengths of C-SBCIs offer a potential framework for parameterizing the temporal and spatial scale of submesoscale turbulence associated with submesoscale eddies.