Turner angle

The Turner angle Tu, introduced by Ruddick(1983) and named after J. Stewart Turner, is a parameter used to describe the local stability of an inviscid water column as it undergoes double-diffusive convection. The temperature and salinity attributes, which generally determine the water density, both respond to the water vertical structure. By putting these two variables in orthogonal coordinates, the angle with the axis can indicate the importance of the two in stability. Turner angle is defined as:

${\displaystyle Tu(\deg )=\tan ^{-1}\left(\alpha {\frac {\partial \theta }{\partial z}}-\beta {\frac {\partial S}{\partial z}},\alpha {\frac {\partial \theta }{\partial z}}+\beta {\frac {\partial S}{\partial z}}\right)}$

where tan⁻¹ is the four-quadrant arctangent; α is the coefficient of thermal expansion; β is the equivalent coefficient for the addition of salinity, sometimes referred to as the "coefficient of saline contraction"; θ is potential temperature; and S is salinity. The relation between Tu and stability is as shown

• If −45° < Tu < 45°, the column is statically stable.
• If −90° < Tu < −45°, the column is unstable to diffusive convection.
• If 45° < Tu < 90°, the column is unstable to salt fingering.
• If −90° > Tu or Tu > 90°, the column is statically unstable to Rayleigh–Taylor instability.