Density ratio

The density ratio of a column of seawater is a measure of the relative contributions of temperature and salinity in determining the density gradient.^[1] At a density ratio of 1, temperature and salinity are said to be compensated: their density signatures cancel, leaving a density gradient of zero. The formula for the density ratio, ${\displaystyle R_{\rho }}$, is:

${\displaystyle R_{\rho }={\frac {\alpha d\theta /dz}{\beta dS/dz}}}$

where

• θ is the potential temperature
• S is the salinity
• z is the vertical coordinate (with subscript denoting differentiation by z)
• ρ is the density
• α = −ρ⁻¹∂ρ/∂θ is the thermal expansion coefficient
• β = ρ⁻¹∂ρ/∂S is the haline contraction coefficient

When a water column is "doubly stable"—both temperature and salinity contribute to the stable density gradient—the density ratio is negative (a doubly unstable water column would also have a negative density ratio but does not commonly occur). When either the temperature- or salinity-induced stratification is statically unstable, while the overall density stratification is statically stable, double-diffusive instability exists in the water column.^[2][3] Double-diffusive instability can be separated into two different regimes of statically stable density stratification: a salt fingering regime (warm salty overlying cool fresh) when the density ratio is greater than 1,^[4] and a diffusive convection regime (cool fresh overlying warm salty) when the density ratio is between 0 and 1.^[5]

Density ratio may also be used to describe thermohaline variability over a non-vertical spatial interval, such as across a front in the mixed layer.^[6]