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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Equivalent potential temperature</span></span>

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<p><b>Equivalent potential temperature</b>, commonly referred to as <b>theta-e</b> <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 \left(\theta _{e}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(\theta _{e}\right)}</annotation>

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</math></span><img src="./5aa2e40020036d865ff1cc22add2f75134bfb64e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.898ex; height:2.843ex;" alt="{\displaystyle \left(\theta _{e}\right)}" loading="lazy"></span>, is a quantity that is conserved during changes to an air parcel's pressure (that is, during vertical motions in the <a href="Earth's_atmosphere" class="mw-redirect" title="Earth's atmosphere">atmosphere</a>), even if water vapor <a href="Condense" class="mw-redirect" title="Condense">condenses</a> during that pressure change. It is therefore more conserved than the ordinary <a href="Potential_temperature" title="Potential temperature">potential temperature</a>, which remains constant only for unsaturated vertical motions (pressure changes).

</p><p><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 _{e}}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{e}}</annotation>

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</math></span><img src="./930299aaf5570dad3cfeb4dbca8d9623e9d739cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.089ex; height:2.509ex;" alt="{\displaystyle \theta _{e}}" loading="lazy"></span> is the <a href="Temperature" title="Temperature">temperature</a> a parcel of air would reach if all the water vapor in the parcel were to <a href="Condense" class="mw-redirect" title="Condense">condense</a>, releasing its <a href="Latent_heat" title="Latent heat">latent heat</a>, and the parcel was brought <a href="Adiabatic" class="mw-redirect" title="Adiabatic">adiabatically</a> to a standard reference pressure, usually 1000 <a href="Hectopascal" class="mw-redirect" title="Hectopascal">hPa</a> (1000 <a href="Millibar" class="mw-redirect" title="Millibar">mbar</a>) which is roughly equal to <a href="Atmospheric_pressure" title="Atmospheric pressure">atmospheric pressure</a> at <a href="Sea_level" title="Sea level">sea level</a>.

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<div class="mw-heading mw-heading2"><h2 id="Use_in_estimating_atmospheric_stability">Use in estimating atmospheric stability</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Stability_of_incompressible_fluid">Stability of incompressible fluid</h3></div>

<p>Like a ball balanced on top of a hill, <a href="Density" title="Density">denser</a> <a href="Fluid" title="Fluid">fluid</a> lying above less dense fluid would be dynamically unstable: overturning motions (<a href="Convection" title="Convection">convection</a>) can lower the center of gravity, and thus will occur spontaneously, rapidly producing a <a href="Stable_stratification" class="mw-redirect" title="Stable stratification">stable stratification</a> (see also <a href="Stratification_(water)" title="Stratification (water)">stratification (water)</a>) which is thus the observed condition almost all the time. The condition for stability of an incompressible fluid is that <i>density decreases monotonically with height</i>.

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<div class="mw-heading mw-heading3"><h3 id="Stability_of_compressible_air:_Potential_temperature">Stability of compressible air: Potential temperature</h3></div>

<p>If a fluid is <a href="Compressible" class="mw-redirect" title="Compressible">compressible</a> like air, the criterion for dynamic stability instead involves <a href="Potential_density" title="Potential density">potential density</a>, the density of the fluid at a fixed reference pressure. For an ideal gas (see <i><a href="Gas_laws" title="Gas laws">gas laws</a></i>), the stability criterion for an air column is that <i><a href="Potential_temperature" title="Potential temperature">potential temperature</a> increases monotonically with height</i>.

</p><p>To understand this, consider dry convection in the atmosphere, where the vertical variation in pressure is substantial and adiabatic temperature change is important: As a parcel of air moves upward, the ambient pressure drops, causing the parcel to expand. Some of the <a href="Internal_energy" title="Internal energy">internal energy</a> of the parcel is used up in doing the <a href="Work_(physics)" title="Work (physics)">work</a> required to expand against the atmospheric pressure, so the temperature of the parcel drops, even though it has not lost any heat. Conversely, a sinking parcel is compressed and becomes warmer even though no heat is added.

</p><p>Air at the top of a mountain is usually colder than the air in the valley below, but the arrangement is not unstable: if a parcel of air from the valley were somehow lifted up to the top of the mountain, when it arrived it would be even colder than the air already there, due to adiabatic cooling; it would be heavier than the ambient air, and would sink back toward its original position. Similarly, if a parcel of cold mountain-top air were to make the trip down to the valley, it would arrive warmer and lighter than the valley air, and would float back up the mountain.

</p><p>So cool air lying on top of warm air can be stable, as long as the temperature decrease with height is less than the <a href="Adiabatic_lapse_rate" class="mw-redirect" title="Adiabatic lapse rate">adiabatic lapse rate</a>; the dynamically important quantity is not the temperature, but the <a href="Potential_temperature" title="Potential temperature">potential temperature</a>—the temperature the air would have if it were brought adiabatically to a reference pressure. The air around the mountain is stable because the air at the top, due to its lower pressure, has a higher potential temperature than the warmer air below.

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<div class="mw-heading mw-heading3"><h3 id="Effects_of_water_condensation:_Equivalent_potential_temperature">Effects of water condensation: Equivalent potential temperature</h3></div>

<p>A rising parcel of air containing water vapor, if it rises far enough, reaches its <a href="Lifted_condensation_level" class="mw-redirect" title="Lifted condensation level">lifted condensation level</a>: it becomes saturated with water vapor (see <i><a href="Clausius%E2%80%93Clapeyron_relation" title="Clausius–Clapeyron relation">Clausius–Clapeyron relation</a></i>). If the parcel of air continues to rise, water vapor condenses and releases its <a href="Latent_heat_of_vaporisation" class="mw-redirect" title="Latent heat of vaporisation">latent heat</a> to the surrounding air, partially offsetting the adiabatic cooling. A saturated parcel of air therefore cools less than a dry one would as it rises (its temperature changes with height at the <a href="Moist_adiabatic_lapse_rate" class="mw-redirect" title="Moist adiabatic lapse rate">moist adiabatic lapse rate</a>, which is smaller than the <a href="Dry_adiabatic_lapse_rate" class="mw-redirect" title="Dry adiabatic lapse rate">dry adiabatic lapse rate</a>). Such a saturated parcel of air can achieve <a href="Buoyancy" title="Buoyancy">buoyancy</a>, and thus accelerate further upward, a runaway condition (instability) even if potential temperature increases with height. The sufficient condition for an air column to be absolutely stable, even with respect to saturated convective motions, is that the <i>equivalent potential temperature must increase monotonically with height.</i>

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<div class="mw-heading mw-heading2"><h2 id="Formula">Formula</h2></div>

<p>The definition of the equivalent potential temperature is:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \theta _{e}=T\left({\frac {p_{0}}{p}}\right)^{R_{d}/(c_{pd}+r_{t}c)}H^{-r_{v}R_{v}/(c_{pd}+r_{t}c)}\exp \left[{\frac {L_{v}r_{v}}{(c_{pd}+r_{t}c)T}}\right]}</annotation>

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</math></span><img src="./14bbef50e1887ebfb0524235a3940b163316de8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.647ex; height:6.843ex;" alt="{\displaystyle \theta _{e}=T\left({\frac {p_{0}}{p}}\right)^{R_{d}/(c_{pd}+r_{t}c)}H^{-r_{v}R_{v}/(c_{pd}+r_{t}c)}\exp \left[{\frac {L_{v}r_{v}}{(c_{pd}+r_{t}c)T}}\right]}" loading="lazy"></span></dd></dl>

<p>Where:

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</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 the temperature [K] of air at 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}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>,</li>

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<annotation encoding="application/x-tex">{\displaystyle p_{0}}</annotation>

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</math></span><img src="./2b969ada68a88e2aeba9a2d2096abaf1fd53c21d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{0}}" loading="lazy"></span> is a reference pressure that is taken as 1000 hPa,</li>

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> is the pressure at the point,</li>

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<annotation encoding="application/x-tex">{\displaystyle R_{d}}</annotation>

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</math></span><img src="./3548b4345af65927fd175ef9e10cccb0b984a096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.856ex; height:2.509ex;" alt="{\displaystyle R_{d}}" loading="lazy"></span> 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 R_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{v}}</annotation>

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</math></span><img src="./40da33633288f9598f0c6f560681e91bcd8fcee2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle R_{v}}" loading="lazy"></span> are the <a href="Gas_constant#Specific_gas_constant" title="Gas constant">specific gas constants</a> of dry air and of water vapour, respectively,</li>

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<annotation encoding="application/x-tex">{\displaystyle c_{pd}}</annotation>

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</math></span><img src="./bfca997b14acdfe9b75c038bb5b426220ed87b3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.926ex; height:2.343ex;" alt="{\displaystyle c_{pd}}" loading="lazy"></span> 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 c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

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</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> are the <a href="Specific_heat_capacity" title="Specific heat capacity">specific heat capacities</a> of dry air and of liquid water, respectively,</li>

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<annotation encoding="application/x-tex">{\displaystyle r_{t}}</annotation>

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</math></span><img src="./fb555a4a6332d0b3c8f786c87eccda2e940936d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.875ex; height:2.009ex;" alt="{\displaystyle r_{t}}" loading="lazy"></span> 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 r_{v}}">

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</math></span><img src="./6351a9c12f3cd743f37050ddb9a0d0adfc33f190.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.078ex; height:2.009ex;" alt="{\displaystyle r_{v}}" loading="lazy"></span> are the total water and water vapour <a href="Mixing_ratio" title="Mixing ratio">mixing ratios</a>, respectively,</li>

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> is the <a href="Relative_humidity" class="mw-redirect" title="Relative humidity">relative humidity</a>,</li>

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<annotation encoding="application/x-tex">{\displaystyle L_{v}}</annotation>

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</math></span><img src="./67d3252ec5dbd18467fc109498966e5abb43123b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.509ex;" alt="{\displaystyle L_{v}}" loading="lazy"></span> is the <a href="Latent_heat" title="Latent heat">latent heat</a> of vapourisation of water.</li></ul>

<p>A number of approximate formulations are used for calculating equivalent potential temperature, since it is not easy to compute integrations along motion of the parcel. Bolton (1980) <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> gives review of such procedures with estimates of error. His best approximation formula is used when accuracy is needed:

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<annotation encoding="application/x-tex">{\displaystyle \theta _{e}=\theta _{L}\exp \left[\left({\frac {3036}{T_{L}}}-1.78\right)r\left(1+0.448r\right)\right]}</annotation>

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</math></span><img src="./04e3ce234795bc5ef57d6b9cbe27bd9e0bfffe70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.886ex; height:6.176ex;" alt="{\displaystyle \theta _{e}=\theta _{L}\exp \left[\left({\frac {3036}{T_{L}}}-1.78\right)r\left(1+0.448r\right)\right]}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \theta _{L}=T\left({\frac {p_{0}}{p-e}}\right)^{\kappa _{d}}\left({\frac {T}{T_{L}}}\right)^{0.28r}}</annotation>

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</math></span><img src="./27529bfcec27a787a8d3d618fb11782eaf61b672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.432ex; height:6.509ex;" alt="{\displaystyle \theta _{L}=T\left({\frac {p_{0}}{p-e}}\right)^{\kappa _{d}}\left({\frac {T}{T_{L}}}\right)^{0.28r}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle T_{L}={\frac {1}{{\frac {1}{T_{d}-56}}+{\frac {\log _{e}(T/T_{d})}{800}}}}+56}</annotation>

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</math></span><img src="./d4ecc3489c6f7654e7614b00207fdc5cc4aa2f26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:29.034ex; height:7.676ex;" alt="{\displaystyle T_{L}={\frac {1}{{\frac {1}{T_{d}-56}}+{\frac {\log _{e}(T/T_{d})}{800}}}}+56}" loading="lazy"></span></dd></dl>

<p>Where:

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<annotation encoding="application/x-tex">{\displaystyle \theta _{L}}</annotation>

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</math></span><img src="./38899584b25f0bcc5e09e5935605db83e9cd5315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.442ex; height:2.509ex;" alt="{\displaystyle \theta _{L}}" loading="lazy"></span> is (dry) potential temperature [K] at the <a href="Lifted_condensation_level" class="mw-redirect" title="Lifted condensation level">lifted condensation level</a> (LCL),</li>

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<annotation encoding="application/x-tex">{\displaystyle T_{L}}</annotation>

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</math></span><img src="./1ee8de51b033092db57d6f739ecc540e753eca3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.709ex; height:2.509ex;" alt="{\displaystyle T_{L}}" loading="lazy"></span> is (approximated) temperature [K] at LCL,</li>

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<annotation encoding="application/x-tex">{\displaystyle T_{d}}</annotation>

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</math></span><img src="./70cadcc969636b990957852f205868f9f7178840.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.45ex; height:2.509ex;" alt="{\displaystyle T_{d}}" loading="lazy"></span> is dew point temperature at 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}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>,</li>

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<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

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</math></span><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> is the water vapor pressure (to obtain <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 _{L}}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{L}}</annotation>

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</math></span><img src="./38899584b25f0bcc5e09e5935605db83e9cd5315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.442ex; height:2.509ex;" alt="{\displaystyle \theta _{L}}" loading="lazy"></span> for dry air),</li>

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<annotation encoding="application/x-tex">{\displaystyle \kappa _{d}=R_{d}/c_{pd}}</annotation>

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</math></span><img src="./291c26f855eb013d1372704be95431eddc5c0825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.474ex; height:3.009ex;" alt="{\displaystyle \kappa _{d}=R_{d}/c_{pd}}" loading="lazy"></span> is the ratio of the specific gas constant to the specific heat of dry air at constant pressure (0.2854),</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> is mixing ratio of water vapor mass per mass [kg/kg] (sometimes value is given in [g/kg]<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and that should be divided by 1000).</li></ul>

<p>A little more theoretical formula is commonly used in literature like Holton (1972) <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> when theoretical explanation is important:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{e}\approx \theta _{L}\exp \left[{\frac {r_{s}(T_{L})L_{v}(T_{L})}{c_{pd}T_{L}}}\right]}</annotation>

</semantics>

</math></span><img src="./136839664e1778d0c95eaed07ed3bfe6ed1917a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.561ex; height:6.343ex;" alt="{\displaystyle \theta _{e}\approx \theta _{L}\exp \left[{\frac {r_{s}(T_{L})L_{v}(T_{L})}{c_{pd}T_{L}}}\right]}" loading="lazy"></span></dd></dl>

<p>Where:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{s}(T_{L})}</annotation>

</semantics>

</math></span><img src="./46f1543686a7d941e3b6dd35764370a5f9ed0ff9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.57ex; height:2.843ex;" alt="{\displaystyle r_{s}(T_{L})}" loading="lazy"></span> is saturated mixing ratio of water at 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 T_{L}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{L}}</annotation>

</semantics>

</math></span><img src="./1ee8de51b033092db57d6f739ecc540e753eca3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.709ex; height:2.509ex;" alt="{\displaystyle T_{L}}" loading="lazy"></span>, the temperature at the saturation level of the air,</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{v}(T_{L})}</annotation>

</semantics>

</math></span><img src="./ab1e9744e4c2f1d1c19a5f6100b4207b86db809c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.131ex; height:2.843ex;" alt="{\displaystyle L_{v}(T_{L})}" loading="lazy"></span> is <a href="Latent_heat" title="Latent heat">latent heat</a> of evaporation at 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 T_{L}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{L}}</annotation>

</semantics>

</math></span><img src="./1ee8de51b033092db57d6f739ecc540e753eca3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.709ex; height:2.509ex;" alt="{\displaystyle T_{L}}" loading="lazy"></span> (2406 kJ/kg {at 40 °C} to 2501 kJ/kg {at 0 °C}), and</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{pd}}</annotation>

</semantics>

</math></span><img src="./bfca997b14acdfe9b75c038bb5b426220ed87b3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.926ex; height:2.343ex;" alt="{\displaystyle c_{pd}}" loading="lazy"></span> is specific heat of dry air at constant pressure (1005.7 J/(kg·K)).</li></ul>

<p>Further more simplified formula is used (in, for example, Stull 1988<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> §13.1 p. 546) for simplicity, if it is desirable to avoid computing <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_{L}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{L}}</annotation>

</semantics>

</math></span><img src="./1ee8de51b033092db57d6f739ecc540e753eca3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.709ex; height:2.509ex;" alt="{\displaystyle T_{L}}" loading="lazy"></span>:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{e}=T_{e}\left({\frac {p_{0}}{p}}\right)^{\kappa _{d}}\approx \left(T+{\frac {L_{v}}{c_{pd}}}r\right)\left({\frac {p_{0}}{p}}\right)^{\frac {R_{d}}{c_{pd}}}}</annotation>

</semantics>

</math></span><img src="./d0836f66da42156dbf80ba5a91112d503252025c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.889ex; height:7.676ex;" alt="{\displaystyle \theta _{e}=T_{e}\left({\frac {p_{0}}{p}}\right)^{\kappa _{d}}\approx \left(T+{\frac {L_{v}}{c_{pd}}}r\right)\left({\frac {p_{0}}{p}}\right)^{\frac {R_{d}}{c_{pd}}}}" loading="lazy"></span></dd></dl>

<p>Where:

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{e}}</annotation>

</semantics>

</math></span><img src="./2db970f307b2a8900a1e3644dcec78d60e3a8a9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.356ex; height:2.509ex;" alt="{\displaystyle T_{e}}" loading="lazy"></span> = <a href="Equivalent_temperature" title="Equivalent temperature">equivalent temperature</a></li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{d}}</annotation>

</semantics>

</math></span><img src="./3548b4345af65927fd175ef9e10cccb0b984a096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.856ex; height:2.509ex;" alt="{\displaystyle R_{d}}" loading="lazy"></span> = specific gas constant for air (287.04 J/(kg·K))</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Usage">Usage</h2></div>

<p>This applies on the <a href="Synoptic_scale_meteorology" title="Synoptic scale meteorology">synoptic scale</a> for characterisation of air masses. For instance, in a study of the <a href="North_American_Ice_Storm_of_1998" class="mw-redirect" title="North American Ice Storm of 1998">North American Ice Storm of 1998</a>, professors Gyakum (<a href="McGill_University" title="McGill University">McGill University</a>, <a href="Montreal" title="Montreal">Montreal</a>) and Roebber (<a href="University_of_Wisconsin-Milwaukee" class="mw-redirect" title="University of Wisconsin-Milwaukee">University of Wisconsin-Milwaukee</a>) have demonstrated that the air masses involved originated from high Arctic at an altitude of 300 to 400 hPa the previous week, went down toward the surface as they moved to the Tropics, then moved back up along the <a href="Mississippi_Valley" class="mw-redirect" title="Mississippi Valley">Mississippi Valley</a> toward the <a href="St._Lawrence_Valley" class="mw-redirect" title="St. Lawrence Valley">St. Lawrence Valley</a>. The back trajectories were evaluated using the constant equivalent potential temperatures.<sup id="cite_ref-Gya_7-0" class="reference"><a href="#cite_note-Gya-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>In the <a href="Mesoscale_meteorology" title="Mesoscale meteorology">mesoscale</a>, equivalent potential temperature is also a useful measure of the static stability of the unsaturated atmosphere. Under normal, stably stratified conditions, the potential temperature increases with height,

</p>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \theta _{e}}{\partial z}}>0}</annotation>

</semantics>

</math></span><img src="./16875e5b5e08c5e9260c3fc34bab78a80b9f8cd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.504ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \theta _{e}}{\partial z}}>0}" loading="lazy"></span></dd></dl>

<p>and vertical motions are suppressed. If the equivalent potential temperature decreases with height,

</p>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \theta _{e}}{\partial z}}<0}</annotation>

</semantics>

</math></span><img src="./96dc4be3ed65b5ddc2013783664971bc8b87dd52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.504ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \theta _{e}}{\partial z}}<0}" loading="lazy"></span></dd></dl>

<p>the atmosphere is unstable to vertical motions, and <a href="Atmospheric_convection" title="Atmospheric convection">convection</a> is likely. Situations in which the equivalent potential temperature decreases with height, indicating instability in saturated air, are quite common.

</p>

<ul><li><a href="Meteorology" title="Meteorology">Meteorology</a></li>

<li><a href="Moist_static_energy" title="Moist static energy">Moist static energy</a></li>

<li><a href="Potential_temperature" title="Potential temperature">Potential temperature</a></li>

<li><a href="Weather_forecasting" title="Weather forecasting">Weather forecasting</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li>M K Yau and R.R. Rogers, <i>Short Course in Cloud Physics, Third Edition</i>, published by Butterworth-Heinemann, January 1, 1989, 304 pages. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780750632157</bdi> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7506-3215-1</bdi></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><div class="reflist">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite id="CITEREFEmanuel1994" class="citation book cs1">Emanuel, Kerry (1994). <i>Atmospheric Convection</i>. Oxford University Press.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://glossary.ametsoc.org/wiki/Equivalent_potential_temperature">"Equivalent potential temperature"</a>. <i>AMS Glossary of Meteorology</i>. American Meteorological Society<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-03</span></span>.</cite></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">D Bolton, 1980: <i>The Computation of Equivalent Potential Temperature</i>. Mon. Wea. Rev., Vol. 108, pp.1046-1053.</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFMet_Office" class="citation web cs1"><a href="Met_Office" title="Met Office">Met Office</a>. <a rel="nofollow" class="external text" href="https://www.wmo.int/pages/prog/www/IMOP/meetings/Upper-Air/ET-IOC-3/Doc5.pps#374,7,Data%20processing%20procedure">"Data processing procedure"</a>. <i>E-<a href="AMDAR" class="mw-redirect" title="AMDAR">AMDAR</a> Evaluation</i>. <a href="World_Meteorological_Organisation" class="mw-redirect" title="World Meteorological Organisation">World Meteorological Organisation</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2009-08-02</span></span>.</cite></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">J R Holton, <i>An Introduction to Dynamical Meteorology</i>. Academic Press, 1972, 319 pages.</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">R B Stull, <i>An Introduction to Boundary Layer Meteorology</i>, Kluwer, 1988, 666 pages, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9027727694</bdi>.</span>

</li>

<li id="cite_note-Gya-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gya_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGyakumRoebber2001" class="citation journal cs1">Gyakum, John R.; Roebber, Paul J. (December 2001). <a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0493%282001%29129%3C2983%3ATISAOA%3E2.0.CO%3B2">"The 1998 Ice Storm, Analysis of a Planetary-Scale Event"</a>. <i><a href="Monthly_Weather_Review" title="Monthly Weather Review">Monthly Weather Review</a></i>. <b>129</b> (12). American Meteorological Society: <span class="nowrap">2983–</span>2997. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001MWRv..129.2983G">2001MWRv..129.2983G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0493%282001%29129%3C2983%3ATISAOA%3E2.0.CO%3B2">10.1175/1520-0493(2001)129<2983:TISAOA>2.0.CO;2</a></span>.</cite>.</span>

</li>

</ol></div></div>

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</style><div id="Meteorological_data_and_variables132" style="font-size:114%;margin:0 4em">Meteorological data and variables</div></th></tr><tr><th scope="row" class="navbox-group" style="background-color: skyblue;width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adiabatic_process" title="Adiabatic process">Adiabatic processes</a></li>

<li><a href="Advection" title="Advection">Advection</a></li>

<li><a href="Buoyancy" title="Buoyancy">Buoyancy</a></li>

<li><a href="Lapse_rate" title="Lapse rate">Lapse rate</a></li>

<li><a href="Lightning" title="Lightning">Lightning</a></li>

<li><a href="Solar_radiation" class="mw-redirect" title="Solar radiation">Surface solar radiation</a></li>

<li><a href="Surface_weather_analysis" title="Surface weather analysis">Surface weather analysis</a></li>

<li><a href="Visibility" title="Visibility">Visibility</a></li>

<li><a href="Vorticity" title="Vorticity">Vorticity</a></li>

<li><a href="Wind_shear" title="Wind shear">Wind shear</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="background-color: skyblue;width:1%"><a href="Condensation" title="Condensation">Condensation</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cloud" title="Cloud">Cloud</a></li>

<li><a href="Cloud_condensation_nuclei" title="Cloud condensation nuclei">Cloud condensation nuclei (CCN)</a></li>

<li><a href="Convective_condensation_level" title="Convective condensation level">Convective condensation level (CCL)</a></li>

<li><a href="Lifting_condensation_level" title="Lifting condensation level">Lifting condensation level (LCL)</a></li>

<li><a href="Precipitable_water" title="Precipitable water">Precipitable water</a></li>

<li><a href="Precipitation" title="Precipitation">Precipitation</a></li>

<li><a href="Water_vapor" title="Water vapor">Water vapor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="background-color: skyblue;width:1%"><a href="Atmospheric_convection" title="Atmospheric convection">Convection</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convective_available_potential_energy" title="Convective available potential energy">Convective available potential energy (CAPE)</a></li>

<li><a href="Convective_inhibition" title="Convective inhibition">Convective inhibition (CIN)</a></li>

<li><a href="Convective_instability" title="Convective instability">Convective instability</a></li>

<li><a href="Convective_momentum_transport" title="Convective momentum transport">Convective momentum transport</a></li>

<li><a href="Conditional_symmetric_instability" title="Conditional symmetric instability">Conditional symmetric instability</a></li>

<li><a href="Convective_temperature" title="Convective temperature">Convective temperature (<i>T</i><sub>c</sub>)</a></li>

<li><a href="Equilibrium_level" title="Equilibrium level">Equilibrium level (EL)</a></li>

<li><a href="Free_convective_layer" title="Free convective layer">Free convective layer (FCL)</a></li>

<li><a href="Hydrodynamical_helicity#Meteorology" title="Hydrodynamical helicity">Helicity</a></li>

<li><a href="K-index_(meteorology)" title="K-index (meteorology)">K Index</a></li>

<li><a href="Level_of_free_convection" title="Level of free convection">Level of free convection (LFC)</a></li>

<li><a href="Lifted_index" title="Lifted index">Lifted index (LI)</a></li>

<li><a href="Maximum_parcel_level" title="Maximum parcel level">Maximum parcel level (MPL)</a></li>

<li><a href="Bulk_Richardson_number" title="Bulk Richardson number">Bulk Richardson number (BRN)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="background-color: skyblue;width:1%"><a href="Temperature" title="Temperature">Temperature</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dew_point" title="Dew point">Dew point (<i>T</i><sub>d</sub>)</a></li>

<li><a href="Dew_point_depression" title="Dew point depression">Dew point depression</a></li>

<li><a href="Dry-bulb_temperature" title="Dry-bulb temperature">Dry-bulb temperature</a></li>

<li><a href="Equivalent_temperature" title="Equivalent temperature">Equivalent temperature (<i>T</i><sub>e</sub>)</a></li>

<li><a href="Forest_fire_weather_index" title="Forest fire weather index">Forest fire weather index</a></li>

<li><a href="Haines_Index" title="Haines Index">Haines Index</a></li>

<li><a href="Heat_index" title="Heat index">Heat index</a></li>

<li><a href="Humidex" title="Humidex">Humidex</a></li>

<li><a href="Humidity" title="Humidity">Humidity</a></li>

<li><a href="Relative_humidity" class="mw-redirect" title="Relative humidity">Relative humidity (RH)</a></li>

<li><a href="Mixing_ratio" title="Mixing ratio">Mixing ratio</a></li>

<li><a href="Potential_temperature" title="Potential temperature">Potential temperature (<i>θ</i>)</a></li>

<li><a href="Sea_surface_temperature" title="Sea surface temperature">Sea surface temperature (SST)</a></li>

<li><a href="Temperature_anomaly" title="Temperature anomaly">Temperature anomaly</a></li>

<li><a href="Thermodynamic_temperature" title="Thermodynamic temperature">Thermodynamic temperature</a></li>

<li><a href="Vapor_pressure" title="Vapor pressure">Vapor pressure</a></li>

<li><a href="Virtual_temperature" title="Virtual temperature">Virtual temperature</a></li>

<li><a href="Wet-bulb_temperature" title="Wet-bulb temperature">Wet-bulb temperature</a></li>

<li><a href="Wet-bulb_globe_temperature" title="Wet-bulb globe temperature">Wet-bulb globe temperature</a></li>

<li><a href="Wet-bulb_potential_temperature" title="Wet-bulb potential temperature">Wet-bulb potential temperature</a></li>

<li><a href="Wind_chill" title="Wind chill">Wind chill</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="background-color: skyblue;width:1%"><a href="Pressure" title="Pressure">Pressure</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Atmospheric_pressure" title="Atmospheric pressure">Atmospheric pressure</a></li>

<li><a href="Baroclinity" title="Baroclinity">Baroclinity</a></li>

<li><a href="Barotropic" class="mw-redirect" title="Barotropic">Barotropicity</a></li>

<li><a href="Pressure_gradient" title="Pressure gradient">Pressure gradient</a></li>

<li><a href="Pressure-gradient_force" title="Pressure-gradient force">Pressure-gradient force (PGF)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="background-color: skyblue;width:1%"><a href="Velocity" title="Velocity">Velocity</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Maximum_potential_intensity" title="Maximum potential intensity">Maximum potential intensity</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-03" href="https://en.wikipedia.org/wiki/?title=Equivalent_potential_temperature&oldid=1298623987">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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