Density ratio explained

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,

R _{\rho} , is:R _{\rho=} (\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

R ^{*} _{\rho} is used, which is defined as^[7] R ^{* = (1)/(R _{\rho})} _{\rho} = (\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 byTu = (3 \pi)/(4) - arg \left(\beta (dS)/(dz) + i\alpha (d\theta)/(dz) \right).

The Turner angle is related to the density ratio by

R _{\rho} =- \tan\left(Tu + (\pi)/(4) \right).

See also

• Spice (oceanography)
• Double diffusive convection

Notes and 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 . en . 2 . 1 . 46 . 10.1038/s43247-021-00113-x . 2662-4435. free . 2021ComEE...2...46V .
3. Stern . Melvin E. . 1960 . The "Salt-Fountain" and Thermohaline Convection . Tellus . en . 12 . 2 . 172–175 . 10.3402/tellusa.v12i2.9378 . 0040-2826. free .
4. Sirevaag . Anders . Fer . Ilker . 2012 . Vertical heat transfer in the Arctic Ocean: The role of double-diffusive mixing . Journal of Geophysical Research: Oceans . en . 117 . C7 . 1–16 . 10.1029/2012JC007910. 2012JGRC..117.7010S .
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 . en . 56 . 3 . 461–481 . 10.1016/S0079-6611(03)00026-0 . 2003PrOce..56..461K . 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. http://glossary.ametsoc.org/wiki/Turner_angle American Meteorological Society Glossary