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. 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, $R_{\rho }$, is:

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

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. 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, and a diffusive convection regime (cool fresh overlying warm salty) when the density ratio is between 0 and 1. 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.

Diffusive density ratio

In place of the density ratio, sometimes the diffusive density ratio $R_{\rho }^{*}$ is used, which is defined as

$R_{\rho }^{*}=\frac{1}{R_{\rho }}=\frac{\alpha dS/dz}{\beta d\theta /dz}$

Turner Angle

If the signs of both the numerator and denominator are reversed, the density ratio remains unchanged. A related quantity which avoids this ambiguity as well as the infinite values possible when the denominator vanishes is the Turner angle, $Tu$, which was introduced by Barry Ruddick and named after Stewart Turner. It is defined by

$Tu=\frac{3\pi }{4}-\mathrm{a}\mathrm{r}\mathrm{g}(\beta \frac{dS}{dz}+i\alpha \frac{d\theta }{dz}).$

The Turner angle is related to the density ratio by

$R_{\rho }=-\tan (Tu+\frac{\pi }{4}).$

See also

• Spice (oceanography)
• Double diffusive convection