en>B.Andersohn: minor edit: clarified existing reference/ quote

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minor edit: clarified existing reference/ quote

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The '''neutral density''' ( <math> \gamma^n\, </math> ) is a variable used in [[oceanography]], introduced in 1997 by David R. Jackett and Trevor J. McDougall.<ref name="ReferenceA">Jackett, David R., Trevor J. McDougall, 1997: A Neutral Density Variable for the World's Oceans. J. Phys. Oceanogr., 27, 237–263</ref>
It is function of the three state variables ([[salinity]], [[temperature]], and [[pressure]]) and the geographical location ([[longitude]] and [[latitude]]) and it has the typical units of [[density]] (M/V).
The level surfaces of <math> \gamma^n\, </math> form the “neutral density surfaces”, which are the most natural layer interfaces stratifying the [[deep sea|deep ocean]] circulation, along which the strong lateral mixing in the ocean occurs.
These surfaces are used in the analyses of ocean data and to perform models of the ocean circulation.
The formation of neutral density surfaces from a given hydrographic observation requires only a call to a computational code (available for [[Matlab]] and [[Fortran]]), that contains the computational [[algorithm]] developed by Jackett and McDougall.

== Mathematical expression ==

A neutral density surface is the surface along which a given [[water mass]] will move, remaining neutrally [[buoyancy|buoyant]].<ref name="ReferenceA"/>

McDougall and Jackett <ref>McDougall, T. J. and D. R. Jackett, 1988: On the helical nature of neutral surfaces. Progress in Oceanography, Vol. 20, Pergamon, 153–183</ref> demonstrated that the normal to the neutral surfaces is in the direction of <math> \beta \nabla S - \alpha \nabla \theta </math>, where S is the [[salinity]], <math> \theta \, </math> is the [[potential temperature]], <math> \alpha \, </math> the [[thermal expansion]] coefficient and <math> \beta \, </math> the saline [[concentration]] coefficient.
Thus, neutral surfaces are defined as the surfaces everywhere perpendicular to the vector <math> \rho (\beta \nabla S - \alpha \nabla \theta) </math>. For such a surface to exist, its [[Hydrodynamical helicity|helicity]] H must be zero;<ref name=autogenerated3>Jackett, David R., Trevor J. McDougall, 1997: A Neutral Density Variable for the World's Oceans. J. Phys. Oceanogr., 27, page 238</ref> if this condition is respected, a scalar <math> \gamma^n\, </math> exists and it is the one which satisfies the following formula:<ref name=autogenerated3 />
:<math> \nabla\gamma^n\ = b \rho (\beta \nabla S - \alpha \nabla \theta);</math> {{EquationRef|1}}
where b is an integrating scalar factor, which is function of space.

This formula represents a coupled system of first-order [[partial differential equations]], that has to be solved to obtain the desired value of <math> \gamma^n\, </math>.
The solutions of ({{EquationNote|1}} ) can be obtained by using [[Numerical partial differential equations|numerical techniques]].

In the real ocean, the condition of [[helicity]] equal to zero is not generally satisfied exactly. Therefore, and because of the non-linear terms in the equation of state, it is impossible to create analytically a [[Well-defined]] neutral density surface.<ref>Klocker et all., 2007, “Diapycnal motion due to neutral helicity”</ref> There will always be flow through the calculated surfaces, because of the presence of a neutral [[helicity]].

Therefore it is possible to obtain only a best-fit approximate neutral surface, through which there is no flow of major proportions and along which it is generally accepted that flow takes place.
<math> \gamma^n\, </math> is a [[Well-defined]] function and Jackett and McDougall demonstrated that the [[inaccuracy]] due to the not exact neutrality is below the present instrumentation error in density.<ref>Jackett, David R., Trevor J. McDougall, 1997: A Neutral Density Variable for the World's Oceans. J. Phys. Oceanogr., 27, page 239</ref> Neutral density surfaces stay within a few tens meters of an ideal surface anywhere in the world.<ref name=autogenerated1>{{cite web|url=http://oceanworld.tamu.edu/resources/ocng_textbook/chapter06/chapter06_05.htm |title=Introduction to Physical Oceanography : Chapter 6 - Temperature, Salinity, and Density - Density, Potential Temperature, and Neutral Density |publisher=Oceanworld.tamu.edu |date= |accessdate=2012-11-16}}</ref>

For how <math> \gamma^n\, </math> has been defined, neutral density surfaces can be considered the continuous analog of the commonly used [[potential density]] surfaces, which are defined over various discrete values of pressures (see for example <ref>Montgomery, R. B., 1938: Circulation in the upper layers of the southern North Atlantic, Pap. Phys. Oceanogr. Meteor., 6(2), 55 pp.</ref> and <ref name=autogenerated2>Reid, J. L., 1994: On the total geostrophic circulation of the North Atlantic Ocean: Flow patterns, tracers and transports. Progress in Oceanography,Vol. 33, Pergamon, 1–92</ref>).

== Spatial dependence ==
Given the spatial dependence of the neutral density, its calculation requires the knowledge of the spatial distribution of temperature and [[salinity]] in the ocean. Therefore the definition of <math> \gamma^n\, </math> has to be linked with a global hydrographic dataset, based on the climatology of the world’s ocean (see [[World Ocean Atlas]] and <ref>Levitus, S. (1982) Climatological Atlas of the World Ocean, NOAA Professional Paper No. 13, U.S. Govt. Printing Office, 173 pp., -ftp://ftp.nodc.noaa.gov/pub/data.nodc/woa/PUBLICATIONS/levitus_atlas_1982.pdf</ref>).
In this way, the solution of ({{EquationNote|1}} ) provides values of <math> \gamma^n\, </math> for a referenced global dataset.
The solution of the system for a high resolution dataset would be computationally very expensive. In this case, the original dataset can be sub-sampled and ({{EquationNote|1}} ) can be solved over a more limited set of data.

== Algorithm for the computation of neutral surfaces using <math> \gamma^n\, </math> ==
Jackett and McDougall constructed the variable <math> \gamma^n\, </math> using the data in the “Levitus dataset”.<ref>Levitus, S. (1982) Climatological Atlas of the World Ocean, NOAA Professional Paper No. 13, U.S. Govt. Printing Office, 173 pp. - ftp://ftp.nodc.noaa.gov/pub/data.nodc/woa/PUBLICATIONS/levitus_atlas_1982.pdf</ref>
As this dataset consist of measurements of S and T at 33 standard depth levels at a 1° resolution, the solution of ({{EquationNote|1}} ) for such a large dataset would be computationally very expensive. Therefore, they sub-sampled the data of the original dataset onto a 4°x4° grid and solved ({{EquationNote|1}} ) on the nodes of this grid.
The authors suggested to solve this system by using a combination of the [[method of characteristics]] in nearly 85% of the ocean (the characteristic surfaces of ({{EquationNote|1}} ) are neutral surfaces along which <math> \gamma^n\, </math> is constant) and the [[finite difference method|finite differences method]] in the remaining 15%.
The output of these calculations is a global dataset labeled with values of <math> \gamma^n\, </math>.
The field of <math> \gamma^n\, </math> values resulting from the solution of the differential system ({{EquationNote|1}} ) satisfies ({{EquationNote|1}} ) an order of magnitude better (on average) than the present instrumentation error in [[density]].<ref>Jackett, David R., Trevor J. McDougall, 1997: A Neutral Density Variable for the World's Oceans. J. Phys. Oceanogr., 27, page 248</ref>

The labeled dataset is then used to assign <math> \gamma^n\, </math> values to any arbitrary hydrographic data at new locations, where values are measured as a function of depth by [[interpolation]] to the four closest points in the Levitus atlas.

== Practical computation of <math> \gamma^n\, </math> ==
The formation of neutral density surfaces from a given hydrographic observation requires only a call to a computational code that contains the [[algorithm]] developed by Jackett and McDougall.

The Neutral Density code comes as a package of [[Matlab]] or as a [[Fortran]] routine. It enables the user to fit neutral density surfaces to arbitrary hydrographic data and just 2 [[Megabyte|MBytes]] of storage are required to obtain an accurately pre-labelled world ocean.

Then, the code permits to [[interpolation|interpolate]] the labeled data in terms of spatial location and [[hydrography]]. By taking a [[Weighted mean|weighted average]] of the four closest casts from the labeled data set, it enables to assign <math> \gamma^n\, </math> values to any arbitrary hydrographic data.

Another function provided in the code, given a vertical profile of labeled data and <math> \gamma^n\, </math> surfaces, finds the positions of the specified <math> \gamma^n\, </math> surfaces within the [[water column]], together with [[error bar]]s.

The complete code is available through the World Wide Web at http://www.teos-10.org/preteos10_software/ . The code comes with documentation in the form of [[README|Readme]] files.

== Advantages of using the neutral density variable ==
Comparisons between the approximated neutral surfaces obtained by using the variable <math> \gamma^n\, </math> and the previous commonly used methods to obtain discretely referenced neutral surfaces (see for example Reid (1994),<ref name=autogenerated2 /> that proposed to approximate neutral surfaces by a linked sequence of [[potential density]] surfaces referred to a discrete set of reference pressures) have shown an improvement of [[Accuracy and precision|accuracy]] (by a factor of about 5) <ref>Jackett, David R., Trevor J. McDougall, 1997: A Neutral Density Variable for the World's Oceans. J. Phys. Oceanogr., 27, page 252</ref> and an easier and computationally less expensive [[algorithm]] to form neutral surfaces.
A neutral surface defined using <math> \gamma^n\, </math> differs only slightly from an ideal neutral surface. In fact, if a parcel moves around a gyre on the neutral surface and returns to its starting location, its depth at the end will differ by around 10m from the depth at the start.<ref name=autogenerated1 /> If [[potential density]] surfaces are used, the difference can be hundreds of meters, a far larger error.<ref name=autogenerated1 />

== References ==
{{Reflist|2}}

== External links ==
*[http://www.teos-10.org/preteos10_software/neutral_density.html TEOS-10, Thermodynamic Equation Of Seawater]
*Jackett, David R., Trevor J. McDougall, 1997: [http://journals.ametsoc.org/doi/pdf/10.1175/1520-0485%281997%29027%3C0237%3AANDVFT%3E2.0.CO%3B2 A Neutral Density Variable for the World's Oceans. J. Phys. Oceanogr.], 27, 237–263.
*[http://woce.nodc.noaa.gov/wdiu/wocedocs/newsltr/news19/news19.pdf World Climate Research Programme (WOCW)], International Newsletter, June 1995.
*Andreas Klocker, Trevor J. McDougall, David R. Jackett, 2007, “[http://web.mit.edu/aklocker/www/files/pdf_files/amos_2007_2.pdf Diapycnal motion due to neutral helicity]”).
*Oceanworld TAMU, http://oceanworld.tamu.edu/resources/ocng_textbook/chapter06/chapter06_05.htm
*Rui Xin Huang, 2010: [http://www.whoi.edu/science/po/people/rhuang/Ocean%20Garden/NeutrSur.pdf Is the neutral surface really neutral?]
*NOAA, U.S. Department of Commerce, 1982: Climatological Atlas of the World Ocean,ftp://ftp.nodc.noaa.gov/pub/data.nodc/woa/PUBLICATIONS/levitus_atlas_1982.pdf
*Thermodynamic Equation Of Seawater (TEOS), [http://www.teos-10.org/pubs/Getting_Started.pdf Oceanographic Toolbox, 2011: Getting started with TEOS-10 and the Gibbs Seawater (GSW)].

[[Category:Oceanography]]
[[Category:Variables]]
[[Category:Oceans]]
[[Category:Hydrography]]

Infrastructure-based development - Revision history

en>B.Andersohn: minor edit: clarified existing reference/ quote