Neutral Variables!

There are a number of densities floating around in the oceanographic world and it can be hard to understand the difference between them. This article aims to be a straightforward primer on oceanographic density for those with some background in physical oceanography. It is not meant to be overly rigorous as there is a lot of literature on the subject.

The in-situ density of seawater is the mass per unit volume of seawater as a function of its salinity, temperature, and pressure (depth). In-situ density is experimentally determined in a lab, and software toolboxes like TEOS-10 (The Equation Of State 10) offer extremely accurate functional forms of in-situ density which anyone can use. Case closed then right? Wrong! While in-situ density gives us the mass per unit volume of any water parcel in the world ocean, oceanographers want more than density out of their density variable. To understand why, we're going to take a step back and look at the broader picture.

The ocean is on average 4 kilometers deep and has basins which are thousands of kilometers wide. The ocean is a thin film that lies on the crust of the earth. This may lead you to believe that the ocean is constantly mixing vertically from top to bottom, but the opposite is true. Beneath the top layer of the ocean, which is mixed by winds and weather, seawater mixes very little vertically. Once water slips beneath the surface it will often maintain the properties that it had when it was last at the surface.

Water resists being mixed vertically because it is stratified by density. Cold salty water which is formed at the poles, sinks to the bottom of the water column and becomes even more dense as it is crushed by 1000s of dbar of water above it. Super warm water formed in the tropics sits on top. Much like a cup that contains oil and water, it is hard to mix waters which are so different. Water, of course, does mix vertically in the interior of the ocean, but making the assumption that that mixing is minimal is a powerful one. It allows physical oceanographers to break the ocean into layers and decompose its complex three-dimensional circulation into a set of two-dimensional circulations on those layers. These surfaces over which water travels through but doesn't mix across are called neutral surfaces. In a perfect world surfaces of constant density would be neutral surfaces.

Let's now return to in-situ density. Consider two adjacent points in the ocean x₀ and x₁, which have the same in-situ density, but different water properties (s₀, t₀, p₀) and (s₁, t₁, p₁) respectively. These points have the same in-situ density (mass over volume), but does that mean they lie on the same neutral surface? Well, if the two points are on the same neutral surface, then it should be that they can mix without any work. Let's pretend that both parcels of water mix at their mid-point. Without any loss of generality, let's assume that the water parcel at x₀ is at a lower pressure than the water at x₁. This means to meet at their midpoint, the parcel that began at x₀ must rise, and the parcel that began at x₁ must sink. As the x₀ parcel rises, it is being crushed by less water above it, it expands, and becomes less dense. As the x₁ parcel sinks, it is being crushed by more water making it shrink and become more dense. By the time they meet in the middle, the parcel which began at x₀ is significantly less dense than the one at x₁. This is illustrated in the gif below. We can consider the difference in the densities of these two water parcels at their midpoint a measure of their neutrality. If these two water parcels lay on the same neutral surface, that difference would be 0.

So what does that little thought experiment tell us really? It tells us that lines of constant in-situ density are not perfectly neutral surfaces, in fact, they are not even close.

Next up is the measure of density which is ?the most widely? used. Potential density is the density of a water parcel if it were moved to a reference pressure. Potential density approaches the problems with surfaces of constant in-situ density and says, "The real problem with in-situ density is the effect of pressure on density. Let's use the exact same equation of state, but let's treat water parcels like they are at a reference pressure instead of their actual pressure!" As a result, surfaces of constant potential density are far more neutral than surfaces of in-situ density near the reference pressure chosen. However, this does not actually solve the problems outlined in the previous section, as two parcels of water which lie on the same neutral surface do not rise/descend to some arbirtrary pressure to mix. They meet somewhere in between. In addition, the pressure of water parcels is important, and ignoring it can have some weird effects! Below is a section in the Atlantic ocean of Potential Density with a reference pressure of 0 dbar (\(\sigma_0\)). Circled in red is a region where water of higher potential density lies above water of lower potential density. This inversion of density, with less dense water above more dense water , is an artifact of our reference pressure being at the surface. If we were to look at the same chart but with a reference closer to that region (e.g. 4000 dbar) that inversion would disappear. Unfortunately, choosing a deeper reference may result in a density inversion closer to the surface. It's density whack-a-mole!

OK. We've seen how surfaces of constant in-situ density and potential density aren't very neutral, but how do we construct a better one? The answer may appear simple; construct a surface that is perfectly neutral. The thing stopping us is called helicity. But first, a definition:

Helicity. Blech. But we are working oceanographers and helicity will not defeat us! We still want a density variable whose surfaces of constant density are well defined and as neutral as possible so we just have to make some tradeoffs! Enter Neutral Density! Neutral density was made in 1997 by Jacket and Macdougall (3) and here's a summarized (and simplified) recipe to create it.

Thanks for reading! I hope you got something out of this write up. The coolest part about density and neutral densities to me is the underlying tension between the oceanographer's desire for a measure of density which splits the ocean into well-defined surfaces upon which water travels throughout the ocean, and the reality that those don't really exist. Throw in the fact that the more complex density variables are often more computationally expensive than the simpler less neutral ones, and it becomes clear that an oceanographer has to make a careful choice about what density variable is best for each problem they are working on. Feel free to reach out to me at garrettdreyfus at gmail dot com.

Here are some one-sentence descriptions of other density variables for fun.

Thanks to Geoff Stanley and Andrew Stewart for reading this over and providing corrections where I went wrong!

1. McDougall, T. J., and D. R. Jackett. 1988. On the helical nature of neutral trajectories in the ocean. Progress in Oceanography 20: 153–183. 10.1016/0079-6611(88)90001-8
2. Stanley, G. J. 2019. Neutral surface topology. Ocean Modelling 138: 88–106. 10.1016/j.ocemod.2019.01.008
3. Jackett, D. R., and T. J. Mcdougall. 1997. A Neutral Density Variable for the World’s Oceans. JOURNAL OF PHYSICAL OCEANOGRAPHY 27: 27.