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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Potential density</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>The <b>potential density</b> of a fluid parcel at <a href="Pressure" title="Pressure">pressure</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> is the <a href="Density" title="Density">density</a> that the parcel would acquire if <a href="Adiabatic_process" title="Adiabatic process">adiabatically</a> brought to a reference pressure <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation>
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</math></span><img src="./671bd891701e0d6cfa6da0114a5dd64233b58709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}" loading="lazy"></span>, often 1 <a href="Bar_(unit)" title="Bar (unit)">bar</a> (100 <a href="Kilopascal" class="mw-redirect" title="Kilopascal">kPa</a>). Whereas density changes with changing pressure, potential density of a fluid parcel is <a href="Conservation_law_(physics)" class="mw-redirect" title="Conservation law (physics)">conserved</a> as the pressure experienced by the parcel changes (provided no mixing with other parcels or net heat flux occurs). The concept is used in <a href="Oceanography" title="Oceanography">oceanography</a> and (to a lesser extent) <a href="Atmospheric_science" title="Atmospheric science">atmospheric science</a>.
</p><p>Potential density is a dynamically important property: for static stability potential density must decrease upward. If it doesn't, a fluid parcel displaced upward finds itself lighter than its neighbors, and continues to move upward; similarly, a fluid parcel displaced downward would be heavier than its neighbors. This is true even if the density of the fluid decreases upward. In stable conditions (potential density decreasing upward) motion along surfaces of constant potential density (<a href="Isopycnal" title="Isopycnal">isopycnals</a>) is energetically favored over flow across these surfaces (diapycnal flow), so most of the motion within a 3-D geophysical fluid takes place along these 2-D surfaces.
</p><p>In oceanography, the symbol <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\theta }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho _{\theta }}</annotation>
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</math></span><img src="./bafd6599b104157fb82927b6079ebe9ea5f9a41d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.205ex; height:2.176ex;" alt="{\displaystyle \rho _{\theta }}" loading="lazy"></span> is used to denote <i>potential density</i>, with the reference pressure <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
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<annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation>
</semantics>
</math></span><img src="./671bd891701e0d6cfa6da0114a5dd64233b58709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}" loading="lazy"></span> taken to be the pressure at the ocean surface. The corresponding <i>potential density anomaly</i> is denoted by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\theta }=\rho _{\theta }-1000}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
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</msub>
<mo>=</mo>
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mn>1000</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{\theta }=\rho _{\theta }-1000}</annotation>
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</math></span><img src="./87cc1f59bd0010d397e91db580af85d67ab64556.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.125ex; height:2.676ex;" alt="{\displaystyle \sigma _{\theta }=\rho _{\theta }-1000}" loading="lazy"></span> <a href="Kilogram" title="Kilogram">kg</a>/m<sup>3</sup>. Because the <a href="Compressibility" title="Compressibility">compressibility</a> of <a href="Seawater" title="Seawater">seawater</a> varies with <a href="Salinity" title="Salinity">salinity</a> and <a href="Temperature" title="Temperature">temperature</a>, the reference pressure must be chosen to be near the actual pressure to keep the definition of potential density dynamically meaningful. Reference pressures are often chosen as a whole multiple of 100 bar; for water near a pressure of 400 bar (40 <a href="Megapascal" class="mw-redirect" title="Megapascal">MPa</a>), say, the reference pressure 400 bar would be used, and the potential density anomaly symbol would be written <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
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<annotation encoding="application/x-tex">{\displaystyle \sigma _{4}}</annotation>
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</math></span><img src="./85f7f23444f3d4f86e7eb823b536155f747704bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="{\displaystyle \sigma _{4}}" loading="lazy"></span>.
Surfaces of constant potential density (relative to and in the vicinity of a given reference pressure) are used in the analyses of ocean data and to construct models of <a href="Ocean_circulation" class="mw-redirect" title="Ocean circulation">ocean currents</a>. <a href="Neutral_density" title="Neutral density">Neutral density</a> surfaces, defined using another variable called <a href="Neutral_density" title="Neutral density">neutral density</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{n}}">
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<annotation encoding="application/x-tex">{\displaystyle \gamma ^{n}}</annotation>
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</math></span><img src="./3a0ae8e8ad15b0116ca891b3108c50bb68fcdd9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.498ex; height:2.843ex;" alt="{\displaystyle \gamma ^{n}}" loading="lazy"></span>), can be considered the continuous analog of these potential density surfaces.
</p><p>Potential density adjusts for the effect of <a href="Compressibility" title="Compressibility">compression</a> in two ways:
</p>
<ul><li>The effect of a parcel's change in volume due to a change in pressure (as pressure increases, volume decreases).</li>
<li>The effect of the parcel's change in temperature due to adiabatic change in pressure (as pressure increases, temperature increases).</li></ul>
<p>A parcel's density may be calculated from an <a href="Equation_of_state" title="Equation of state">equation of state</a>:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\rho (P,T,S_{1},S_{2},...)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ρ<!-- ρ --></mi>
<mo>=</mo>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">(</mo>
<mi>P</mi>
<mo>,</mo>
<mi>T</mi>
<mo>,</mo>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho =\rho (P,T,S_{1},S_{2},...)}</annotation>
</semantics>
</math></span></span>
where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> is temperature, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> is pressure, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation>
</semantics>
</math></span><img src="./9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}" loading="lazy"></span> are other <a href="Flow_tracer" title="Flow tracer">tracers</a> that affect density (e.g. <a href="Salinity" title="Salinity">salinity</a> of <a href="Seawater" title="Seawater">seawater</a>). The potential density would then be calculated as:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\theta }=\rho (P_{0},\theta ,S_{1},S_{2},...)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>=</mo>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho _{\theta }=\rho (P_{0},\theta ,S_{1},S_{2},...)}</annotation>
</semantics>
</math></span></span>
where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
</semantics>
</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> is the <a href="Potential_temperature" title="Potential temperature">potential temperature</a> of the fluid parcel for the same reference pressure <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation>
</semantics>
</math></span><img src="./671bd891701e0d6cfa6da0114a5dd64233b58709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Potential_energy" title="Potential energy">Potential energy</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFJohn_M._Wallace_and_Peter_V._Hobbs2006" class="citation book cs1">John M. Wallace and Peter V. Hobbs (2006). <i>Atmospheric Science, An Introductory Survey, Second Edition</i>. Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-732950-1</bdi>.</cite></li>
<li><cite id="CITEREFRobert_H._Stewart2002" class="citation book cs1">Robert H. Stewart (2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121205102538/http://oceanworld.tamu.edu/resources/ocng_textbook/chapter06/chapter06_05.htm"><i>Introduction to Physical Oceanography</i></a>. Archived from <a rel="nofollow" class="external text" href="http://oceanworld.tamu.edu/resources/ocng_textbook/chapter06/chapter06_05.htm">the original</a> on 2012-12-05<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-11-14</span></span>.</cite></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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