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]

Diffusive density ratio

In place of the density ratio, sometimes the diffusive density ratio ${\displaystyle R_{\rho }^{*}}$ is used, which is defined as ^[7]

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

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. ^{[8] [9]} It is defined by

${\displaystyle Tu=135^{\circ }-\mathrm {arg} (\beta dS/dz+i\alpha d\theta /dz).}$

The Turner angle is related to the density ratio by

${\displaystyle R_{\rho }=-\tan(Tu+45^{\circ }).}$

See also

• Spice (oceanography)
• Double diffusive convection

References

1. You, Yuzhu. "A global ocean climatological atlas of the Turner angle: implications for double-diffusion and water-mass structure." Deep Sea Research Part I: Oceanographic Research Papers 49.11 (2002): 2075-2093.
2. van der Boog, Carine G.; Dijkstra, Henk A.; Pietrzak, Julie D.; Katsman, Caroline A. (2021-02-24). "Double-diffusive mixing makes a small contribution to the global ocean circulation". Communications Earth & Environment. 2 (1): 1–9. doi: 10.1038/s43247-021-00113-x . ISSN   2662-4435.
3. Stern, Melvin E. (1960). "The "Salt-Fountain" and Thermohaline Convection". Tellus. 12 (2): 172–175. doi: 10.3402/tellusa.v12i2.9378 . ISSN   0040-2826.
4. Sirevaag, Anders; Fer, Ilker (2012). "Vertical heat transfer in the Arctic Ocean: The role of double-diffusive mixing". Journal of Geophysical Research: Oceans. 117 (C7): 1–16. doi:10.1029/2012JC007910.
5. Kelley, D. E.; Fernando, H. J. S.; Gargett, A. E.; Tanny, J.; Özsoy, E. (2003-03-01). "The diffusive regime of double-diffusive convection". Progress in Oceanography. Double-Diffusion in Oceanography. 56 (3): 461–481. doi:10.1016/S0079-6611(03)00026-0. ISSN   0079-6611.
6. Rudnick, Daniel L., and Raffaele Ferrari. "Compensation of horizontal temperature and salinity gradients in the ocean mixed layer." Science 283.5401 (1999): 526-529.
7. Radko, T. (2013). Double-diffusive convection. Cambridge University Press.
8. Ruddick, B. (1983). A practical indicator of the stability of the water column to double-diffusive activity. Deep Sea Research Part A. Oceanographic Research Papers, 30(10), 1105-1107.
9. American Meteorological Society Glossary