AOSN-II Rossby radii

This page describes a preliminary quick comparison of the Rossby radii of deformation in a few vertical layers produced by a data-driven primitive equation simulation of the AOSN-II experiment.

Methodology

A simple formula for the Rossby radius of deformation is chosen1, namely L_D = f^-1 Ñ D, where:

• L_D is the Rossby radius of deformation
• f is the Coriolis parameter
• Ñ is the representative Brunt-Väisälä frequency
• D is the vertical length scale.

1. Ñ is the vertical mean of the Brunt-Väisälä frequency, N.

   Taking Ñ to be the vertical mean (in the integrated sense) of the Brunt-Väisälä over the range [z_bot  z_top], the product ÑD is simply the integral of the Brunt-Väisälä over that same range. The Rossby radius can then be written as:

   	∫	ztop
   LD = f-1			N dz
   		zbot

   where the integral is numerically evaluated using a midpoint quadrature.

2. Ñ is the vertical root-mean-square of the Brunt-Väisälä frequency, N.

   In this case the integrand is N², which lends itself to analytic methods. Specifically, use the formula N² = - (g/ρ_θ) dρ_θ/dz where:

   • g is the acceleration due to gravity
   • ρ_θ is the potential density

   Assuming g is independent of z, an exact integral can be computed, resulting in a Rossby radius of: L_D = f^-1  sqrt{ g D  ln[ρ_θ(z_bot)/ρ_θ(z_top) ]}

When implementing these methods:

1. At each horizontal point the Rossby radius is only computed where data is available over the entire range [z_bot  z_top]
2. In unstable regions, N² < 0, we arbitrarily reset N² to zero².
3. To get the expected units for the Rossby radius, z_top, z_bot & dz must be expressed in kilometers and, in the root-mean-square case, g must be expressed in km/s².

Results

To the right is a table indicating exactly which layers were selected, the observed ranges of Rossby radii (outliers removed) and links to the corresponding animations of the Rossby radii. One quick observation, the mixed layer Rossby radii are dominated by the diurnal heating. The radii increase in the local afternoon3 as the heating sets up a surface temperature gradient. This sensitivity coupled with the small surface radii suggest that the mixed-layer Rossby radii may not be well defined. In fact, an on-going issue is the examination of the choice of depth ranges as well as the decision to use constant depth ranges rather than ranges that depend on the local hydrography.

¹This formula was obtained from:

Pedlosky, J. (1987) "Geophysical Fluid Dynamics". Springer-Verlag. New York. 710pp

²It might be instructive to simply not calculate the Rossby radius in regions with instabilities.

³Interpreting the term "local afternoon" requires the additional datum that the simulation starts at 0000Z. This corresponds to 1600 PST.