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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Geopotential height</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Hauteur_du_g%C3%A9opotentiel" title="Hauteur du géopotentiel – French" lang="fr" hreflang="fr" data-title="Hauteur du géopotentiel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geopotenti%C3%ABle_hoogte" title="Geopotentiële hoogte – Dutch" lang="nl" hreflang="nl" data-title="Geopotentiële hoogte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Geopotensiell_h%C3%B8yde" title="Geopotensiell høyde – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Geopotensiell høyde" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Geopotensiell_h%C3%B8gd" title="Geopotensiell høgd – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Geopotensiell høgd" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wysoko%C5%9B%C4%87_geopotencja%C5%82u" title="Wysokość geopotencjału – Polish" lang="pl" hreflang="pl" data-title="Wysokość geopotencjału" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a 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searchaux" style="display:none">Type of altitude above mean sea level</div> <p><b>Geopotential height</b> or <b>geopotential altitude</b> is a <a href="/wiki/Vertical_coordinate" class="mw-redirect" title="Vertical coordinate">vertical coordinate</a> referenced to <a href="/wiki/Earth" title="Earth">Earth</a>'s <a href="/wiki/Mean_sea_level" class="mw-redirect" title="Mean sea level">mean sea level</a> (assumed zero <a href="/wiki/Geopotential" title="Geopotential">geopotential</a>) that represents the <a href="/wiki/Work_(physics)" title="Work (physics)">work</a> involved in lifting one <a href="/wiki/Unit_of_mass" class="mw-redirect" title="Unit of mass">unit of mass</a> over one <a href="/wiki/Unit_of_length" title="Unit of length">unit of length</a> through a hypothetical <a href="/wiki/Space_(mathematics)" title="Space (mathematics)">space</a> in which the <a href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">acceleration of gravity</a> is assumed constant.<sup id="cite_ref-NASA_1-0" class="reference"><a href="#cite_note-NASA-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI units</a>, a geopotential height difference of one <a href="/wiki/Metre" title="Metre">meter</a> implies the vertical transport of a parcel of one <a href="/wiki/Kilogram" title="Kilogram">kilogram</a>; adopting the <a href="/wiki/Standard_gravity" title="Standard gravity">standard gravity</a> value (9.80665 <a href="/wiki/Metres_per_second_squared" class="mw-redirect" title="Metres per second squared">m/s<sup>2</sup></a>), it corresponds to a constant work or <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> difference of 9.80665 <a href="/wiki/Joule" title="Joule">joules</a>. </p><p>Geopotential height differs from geometric height (as given by a <a href="/wiki/Tape_measure" title="Tape measure">tape measure</a>) because <a href="/wiki/Earth%27s_gravity" class="mw-redirect" title="Earth's gravity">Earth's gravity</a> is not constant, varying markedly with altitude and latitude; thus, a 1-m geopotential height difference implies a different <a href="/wiki/Vertical_distance" class="mw-redirect" title="Vertical distance">vertical distance</a> in <a href="/wiki/Physical_space" class="mw-redirect" title="Physical space">physical space</a>: "the unit-mass must be lifted higher at the equator than at the pole, if the same amount of work is to be performed".<sup id="cite_ref-Bjerknes_1910_p._13_2-0" class="reference"><a href="#cite_note-Bjerknes_1910_p._13-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It is a useful concept in <a href="/wiki/Meteorology" title="Meteorology">meteorology</a>, <a href="/wiki/Climatology" title="Climatology">climatology</a>, and <a href="/wiki/Oceanography" title="Oceanography">oceanography</a>; it also remains a historical convention in aeronautics as the altitude used for calibration of aircraft <a href="/wiki/Pressure_altitude" title="Pressure altitude">barometric altimeters</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geopotential_height&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i><a href="/wiki/Geopotential" title="Geopotential">Geopotential</a></i> is the <a href="/wiki/Gravitational_energy" title="Gravitational energy">gravitational potential energy</a> per unit mass at elevation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (Z)=\int _{0}^{Z}\ g(\phi ,Z)\,dZ}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msubsup> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (Z)=\int _{0}^{Z}\ g(\phi ,Z)\,dZ}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c9c4a01ef508094dc38e2cabdb771d8798cfd86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.427ex; height:6.176ex;" alt="{\displaystyle \Phi (Z)=\int _{0}^{Z}\ g(\phi ,Z)\,dZ}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\phi ,Z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(\phi ,Z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd6a1cdedc748491d4f1d891ce54b3a5b8e41a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.025ex; height:2.843ex;" alt="{\displaystyle g(\phi ,Z)}"></span> is the acceleration due to <a href="/wiki/Gravity" title="Gravity">gravity</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is <a href="/wiki/Latitude" title="Latitude">latitude</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> is the geometric elevation.<sup id="cite_ref-NASA_1-1" class="reference"><a href="#cite_note-NASA-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><b>Geopotential height</b> may be obtained from normalizing geopotential by the acceleration of gravity: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {H}={\frac {\Phi }{g_{0}}}\ ={\frac {1}{g_{0}}}\int _{0}^{Z}\ g(\phi ,Z)\,dZ}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Φ</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msubsup> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {H}={\frac {\Phi }{g_{0}}}\ ={\frac {1}{g_{0}}}\int _{0}^{Z}\ g(\phi ,Z)\,dZ}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/038c8aa2dd77a2a9eaef27d057b843b3f81ea886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.388ex; height:6.176ex;" alt="{\displaystyle {H}={\frac {\Phi }{g_{0}}}\ ={\frac {1}{g_{0}}}\int _{0}^{Z}\ g(\phi ,Z)\,dZ}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d13273b9af4564fa2c421c96d039c414db8628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{0}}"></span> = 9.80665 m/s<sup>2</sup>, the <a href="/wiki/Standard_gravity" title="Standard gravity">standard gravity</a> at mean sea level.<sup id="cite_ref-https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf_4-0" class="reference"><a href="#cite_note-https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Expressed in differential form, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {g_{0}}\ {dH}={g}\ {dZ}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>H</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {g_{0}}\ {dH}={g}\ {dZ}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b84fd59ac853658bfe9c29a950263af32bd7e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.715ex; height:2.509ex;" alt="{\displaystyle {g_{0}}\ {dH}={g}\ {dZ}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Role_in_planetary_fluids">Role in planetary fluids</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geopotential_height&action=edit&section=2" title="Edit section: Role in planetary fluids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Geopotential height plays an important role in atmospheric and oceanographic studies. The differential form above may be substituted into the <a href="/wiki/Hydrostatic_equilibrium" title="Hydrostatic equilibrium">hydrostatic equation</a> and <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a> in order to relate pressure to ambient temperature and geopotential height for measurement by barometric altimeters regardless of latitude or geometric elevation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {dP}={-g}\ {\rho }\ {dZ}={-g_{0}}\ {\rho }\ {dH}={\frac {-g_{0}\ P}{R\ T}}\ {dH}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>P</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>g</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>Z</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>H</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <mi>P</mi> </mrow> <mrow> <mi>R</mi> <mtext> </mtext> <mi>T</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dP}={-g}\ {\rho }\ {dZ}={-g_{0}}\ {\rho }\ {dH}={\frac {-g_{0}\ P}{R\ T}}\ {dH}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b9b50aaf454ba31d141862d69db7b9c5145e9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.048ex; height:5.509ex;" alt="{\displaystyle {dP}={-g}\ {\rho }\ {dZ}={-g_{0}}\ {\rho }\ {dH}={\frac {-g_{0}\ P}{R\ T}}\ {dH}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dP}{P}}=-{\frac {g_{0}}{R\ T}}\ {dH}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mi>P</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mi>R</mi> <mtext> </mtext> <mi>T</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dP}{P}}=-{\frac {g_{0}}{R\ T}}\ {dH}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/492779aa15fdeac2a212fc95294413d05cb3c81f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.381ex; height:5.509ex;" alt="{\displaystyle {\frac {dP}{P}}=-{\frac {g_{0}}{R\ T}}\ {dH}}"></span></dd></dl> <p>where <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" 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}"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" 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}"></span> are ambient pressure and temperature, respectively, as functions of geopotential height, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the specific <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a>. For the subsequent <a href="/wiki/Definite_integral" class="mw-redirect" title="Definite integral">definite integral</a>, the simplification obtained by assuming a constant value of gravitational acceleration is the sole reason for defining the geopotential altitude.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Usage">Usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geopotential_height&action=edit&section=3" title="Edit section: Usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:NAM_500_MB.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/NAM_500_MB.PNG/330px-NAM_500_MB.PNG" decoding="async" width="330" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/NAM_500_MB.PNG/495px-NAM_500_MB.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6b/NAM_500_MB.PNG 2x" data-file-width="640" data-file-height="480" /></a><figcaption>Geopotential height analysis on the <a href="/wiki/North_American_Mesoscale_Model" title="North American Mesoscale Model">North American Mesoscale Model</a> (NAM) at 500 hPa.</figcaption></figure> <p><a href="/wiki/Geophysics" title="Geophysics">Geophysical</a> sciences such as meteorology often prefer to express the horizontal <a href="/wiki/Pressure-gradient_force" title="Pressure-gradient force">pressure gradient force</a> as the gradient of <a href="/wiki/Geopotential" title="Geopotential">geopotential</a> along a constant-pressure surface, because then it has the properties of a <a href="/wiki/Conservative_force" title="Conservative force">conservative force</a>. For example, the <a href="/wiki/Primitive_equations" title="Primitive equations">primitive equations</a> that <a href="/wiki/Numerical_weather_prediction" title="Numerical weather prediction">weather forecast models</a> solve use <a href="/wiki/Hydrostatic_pressure" class="mw-redirect" title="Hydrostatic pressure">hydrostatic pressure</a> as a vertical coordinate, and express the slopes of those pressure surfaces in terms of geopotential height. </p><p>A plot of geopotential height for a single pressure level in the atmosphere shows the troughs and ridges (<a href="/wiki/High-pressure_area" title="High-pressure area">highs</a> and <a href="/wiki/Low-pressure_area" title="Low-pressure area">lows</a>) which are typically seen on upper air charts. The geopotential thickness between pressure levels – difference of the 850 <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">hPa</a> and 1000 hPa geopotential heights for example – is proportional to mean <a href="/wiki/Virtual_temperature" title="Virtual temperature">virtual temperature</a> in that layer. Geopotential height contours can be used to calculate the <a href="/wiki/Geostrophic_wind" title="Geostrophic wind">geostrophic wind</a>, which is faster where the contours are more closely spaced and tangential to the geopotential height contours.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2016)">citation needed</span></a></i>]</sup> </p><p>The United States <a href="/wiki/National_Weather_Service" title="National Weather Service">National Weather Service</a> defines geopotential height as: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>"...roughly the height above sea level of a pressure level. For example, if a station reports that the 500 mb [i.e. <a href="/wiki/Millibar" class="mw-redirect" title="Millibar">millibar</a>] height at its location is 5600 m, it means that the level of the atmosphere over that station at which the atmospheric pressure is 500 mb is 5600 meters above sea level. This is an estimated height based on temperature and pressure data."<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></p></blockquote> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geopotential_height&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Atmospheric_model" title="Atmospheric model">Atmospheric model</a></li> <li><a href="/wiki/Above_mean_sea_level" class="mw-redirect" title="Above mean sea level">Above mean sea level</a></li> <li><a href="/wiki/Dynamic_height" title="Dynamic height">Dynamic height</a>, a similar quantity used in geodesy, based on a slightly different gravity value</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geopotential_height&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-NASA-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-NASA_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-NASA_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMinznerReberJacchiaHuang1976" class="citation web cs1">Minzner, R. A.; Reber, C. A.; Jacchia, L. G.; Huang, F. T.; Cole, A. E.; Kantor, A. J.; Keneshea, T. J.; Zimmerman, S. P.; Forbes, J. M. (May 1976). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170307211228/https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf">"NASA Technical Report R-459: Defining Constants, Equations, and Abbreviated Tables of the 1976 Standard Atmosphere"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-03-07.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=NASA+Technical+Report+R-459%3A+Defining+Constants%2C+Equations%2C+and+Abbreviated+Tables+of+the+1976+Standard+Atmosphere&rft.date=1976-05&rft.aulast=Minzner&rft.aufirst=R.+A.&rft.au=Reber%2C+C.+A.&rft.au=Jacchia%2C+L.+G.&rft.au=Huang%2C+F.+T.&rft.au=Cole%2C+A.+E.&rft.au=Kantor%2C+A.+J.&rft.au=Keneshea%2C+T.+J.&rft.au=Zimmerman%2C+S.+P.&rft.au=Forbes%2C+J.+M.&rft_id=https%3A%2F%2Fntrs.nasa.gov%2Farchive%2Fnasa%2Fcasi.ntrs.nasa.gov%2F19760017709.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeopotential+height" class="Z3988"></span></span> </li> <li id="cite_note-Bjerknes_1910_p._13-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bjerknes_1910_p._13_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBjerknes1910" class="citation book cs1"><a href="/wiki/V._Bjerknes" class="mw-redirect" title="V. Bjerknes">Bjerknes, V.</a> (1910). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ub5XAAAAYAAJ&pg=PA13"><i>Dynamic Meteorology and Hydrography: Part [1]-2, [and atlas of plates]</i></a>. Carnegie Institution of Washington publication. Carnegie Institution of Washington. p. 13<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamic+Meteorology+and+Hydrography%3A+Part+%5B1%5D-2%2C+%5Band+atlas+of+plates%5D&rft.series=Carnegie+Institution+of+Washington+publication&rft.pages=13&rft.pub=Carnegie+Institution+of+Washington&rft.date=1910&rft.aulast=Bjerknes&rft.aufirst=V.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dub5XAAAAYAAJ%26pg%3DPA13&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeopotential+height" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson2007" class="citation book cs1">Anderson, John (2007). <i>Introduction to Flight</i>. McGraw-Hill Science/Engineering/Math. p. 109.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Flight&rft.pages=109&rft.pub=McGraw-Hill+Science%2FEngineering%2FMath&rft.date=2007&rft.aulast=Anderson&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeopotential+height" class="Z3988"></span> </span> </li> <li id="cite_note-https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMinznerReberJacchiaHuang1976" class="citation web cs1">Minzner, R. A.; Reber, C. A.; Jacchia, L. G.; Huang, F. T.; Cole, A. E.; Kantor, A. J.; Keneshea, T. J.; Zimmerman, S. P.; Forbes, J. M. (May 1976). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170307211228/https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf">"NASA Technical Report R-459: Defining Constants, Equations, and Abbreviated Tables of the 1976 Standard Atmosphere"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-03-07.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=NASA+Technical+Report+R-459%3A+Defining+Constants%2C+Equations%2C+and+Abbreviated+Tables+of+the+1976+Standard+Atmosphere&rft.date=1976-05&rft.aulast=Minzner&rft.aufirst=R.+A.&rft.au=Reber%2C+C.+A.&rft.au=Jacchia%2C+L.+G.&rft.au=Huang%2C+F.+T.&rft.au=Cole%2C+A.+E.&rft.au=Kantor%2C+A.+J.&rft.au=Keneshea%2C+T.+J.&rft.au=Zimmerman%2C+S.+P.&rft.au=Forbes%2C+J.+M.&rft_id=https%3A%2F%2Fntrs.nasa.gov%2Farchive%2Fnasa%2Fcasi.ntrs.nasa.gov%2F19760017709.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeopotential+height" class="Z3988"></span> </span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson2007" class="citation book cs1">Anderson, John (2007). <i>Introduction to Flight</i>. McGraw-Hill Science/Engineering/Math. p. 116.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Flight&rft.pages=116&rft.pub=McGraw-Hill+Science%2FEngineering%2FMath&rft.date=2007&rft.aulast=Anderson&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeopotential+height" class="Z3988"></span> </span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.weather.gov/glossary/index.php?letter=h">"Height"</a>. <i>NOAA's National Weather Service Glossary</i>. NOAA National Weather Service<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-03-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=NOAA%27s+National+Weather+Service+Glossary&rft.atitle=Height&rft_id=http%3A%2F%2Fwww.weather.gov%2Fglossary%2Findex.php%3Fletter%3Dh&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeopotential+height" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geopotential_height&action=edit&section=6" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Hofmann-Wellenhof, B. and Moritz, H. "Physical Geodesy", 2005. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-211-23584-1" title="Special:BookSources/3-211-23584-1">3-211-23584-1</a>.</li> <li>Eskinazi, S. "Fluid Mechanics and Thermodynamics of our Environment", 1975. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-242540-5" title="Special:BookSources/0-12-242540-5">0-12-242540-5</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geopotential_height&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Geopotential_height" class="extiw" title="commons:Category:Geopotential height">Geopotential height</a> at Wikimedia Commons</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Geopotential_height&oldid=1248224933">https://en.wikipedia.org/w/index.php?title=Geopotential_height&oldid=1248224933</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Atmospheric_dynamics" title="Category:Atmospheric dynamics">Atmospheric dynamics</a></li><li><a href="/wiki/Category:Vertical_position" title="Category:Vertical position">Vertical position</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_February_2016" title="Category:Articles with unsourced statements from February 2016">Articles with unsourced statements from February 2016</a></li><li><a href="/wiki/Category:Commons_category_link_from_Wikidata" title="Category:Commons category link from Wikidata">Commons category link from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 28 September 2024, at 09:54<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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