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]

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

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, ${\displaystyle Tu}$, which was introduced by Barry Ruddick and named after Stewart Turner. ^{[8] [9]} It is defined by

${\displaystyle Tu={\frac {3\pi }{4}}-\mathrm {arg} \left(\beta {\frac {dS}{dz}}+i\alpha {\frac {d\theta }{dz}}\right).}$

The Turner angle is related to the density ratio by

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

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): 46. Bibcode:2021ComEE...2...46V. 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. Bibcode:2012JGRC..117.7010S. 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. Bibcode:2003PrOce..56..461K. 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