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An air parcel is a small material volume followed through an atmospheric flow. The elementary parcel model assumes rapid pressure adjustment to its environment but negligible heat and material exchange during the displacement. Its potential temperature is conserved during reversible dry adiabatic process at fixed composition.

For dry atmospheric thermodynamics, choose potential temperature as the materially conserved scalar in Ertel's theorem. The displayed density-normalized scalar has units $\mathrm{K}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-1}$. Its conservation in adiabatic inviscid motion depends on conservative body forces and a composition-fixed equation of state. It differs in normalization from the Boussinesq approximation convention using buoyancy without the variable density factor. Large-scale balanced weather can be organized by its distribution, with potential-vorticity inversion additionally requiring balance and boundary information.

Dry potential temperature is conserved for a fixed-composition ideal gas undergoing a reversible adiabatic change. Adiabatic means no heat passes through the boundary; it does not by itself prohibit internal entropy production. With constant specific heat capacity at constant pressure,Positive irreversible entropy production can therefore increase $\theta$ even when the parcel receives no boundary heat.

For example, dissipation of mechanical energy by viscosity heats the gas, and a shock changes its entropy. Mixing also requires care because the resulting parcel is not simply the original unmixed material following one reversible path. This qualification prevents treating all unheated atmospheric motion as exactly conserving dry potential temperature.

Mountain blocking can prevent low-level upstream air from crossing the barrier while air from higher levels passes over and descends. In stable stratification that elevated source air has higher potential temperature; adiabatic descent can bring it into a lee valley warmer than the blocked low-level air. This source-air change permits a dry foehn without requiring condensation and rainout along the arriving parcel's path.

The entropy form of dry potential temperature follows from its parcel definition. The potential temperature is the temperature a parcel would reach if brought reversibly and adiabatically to a reference pressure $p_0$. For dry air of fixed composition, treated as an ideal gas with constant specific heat capacity,Per unit mass, the first law of thermodynamics, together with $\alpha =R_{d}T/p$ for specific volume, gives $\delta q=c_{p}dT-\alpha dp$. The reversible form of the Second law of thermodynamics then givesHence dry reversible adiabatic motion has $D\theta /Dt=0$. Pressure changes cause expansion cooling or compression warming, while the pressure factor in theta compensates exactly. Adiabaticity alone is insufficient if irreversible mixing or viscous dissipation produces entropy. Condensation or freezing supplies latent heating, so ordinary dry potential temperature need not be conserved in a moist adiabatically moving parcel.

A foehn wind is relatively warm, dry downslope air on a mountain's lee side. In the classical condensation pathway, ascending windward air expands and cools at the dry adiabatic lapse rate until the lifting condensation level. Condensation then releases latent heat, reducing the cooling rate to the moist adiabatic lapse rate. Atmospheric precipitation removes condensate, and the descending parcel warms by compression at approximately the dry rate. Its relative humidity falls.

For equal starting and finishing heights, complete condensate removal and a constant moist lapse rate above condensation height $z_c$, the precipitation-driven foehn warming isHere $H>z_c$ is the parcel's maximum height and ${\Gamma }_d>{\Gamma }_m$. The warming is adiabatic compression during descent, not heating by friction. Without condensation or a difference in source air, a reversible dry parcel returning to its original height would regain its original temperature.

Precipitation is not essential to every foehn event. Isentropic foehn drawdown can bring air of higher potential temperature from aloft down the lee slope; turbulent mixing and extra solar heating beneath clear skies can also contribute. The sketch shows the classical rainout pathway rather than asserting a unique mechanism.^[ref]

For dry atmospheric Ertel potential vorticity, use absolute vorticity ${\mathbf{\omega }}_a=\nabla \times \mathbf{u}+2\mathbf{\Omega }$ and potential temperature $\theta =T(p_0/p{)}^{R_d/c_p}$:The conservation statement is Ertel's theorem for smooth, inviscid, adiabatic flow with conservative body forces and a thermodynamic equation of state $p=p(\rho ,\theta )$. Stretching of ${\mathbf{\omega }}_a/\rho$ cancels deformation of $\nabla \theta$ in their material derivative; the remaining baroclinic triple product vanishes because $\nabla p$ lies in the span of $\nabla \rho ,\nabla \theta$. The atmospheric form includes $1/\rho$; the commonly used Boussinesq approximation version has a different normalization.

In a predominantly vertical stratification, $q_E\simeq (f+\zeta ){\partial }_z\theta /\rho$. Thus rotation and stratification jointly label air masses. Material conservation explains how parcels deform and stretch while preserving a useful scalar, supports Rossby waves, and helps diagnose sharp fronts, jets and stratospheric intrusions.

Under an appropriate balance model, potential-vorticity inversion recovers associated wind and temperature fields from a PV distribution together with stratification and boundary data. PV alone does not specify every component of an arbitrary atmospheric flow. Potential-vorticity generation by heating and friction limits conservative parcel reasoning near strong heating, surface stress or mixing.^[ref]

The isentropic potential vorticity is Ertel potential vorticity expressed on constant potential temperature surfaces:Here ${\zeta }_{\theta }$ is the vertical curl of horizontal velocity differentiated on an isentropic surface, and ${\sigma }_{\theta }$ is mass per unit area per unit potential-temperature interval. For smooth inviscid adiabatic motion, $Dq/Dt=0$. Conserved anomalies are advected and deformed, explaining much balanced large-scale weather variability. Diagnosing velocity from PV additionally needs balance, stratification and boundary information. Strong latent/radiative heating, friction and irreversible mixing generate or modify PV, limiting a purely conservative explanation.^[ref]

Let ${\mathbf{U}}_h=\Gamma z{\mathbf{u}}_h(x,y)$ denote the actual horizontal velocity field. With constant nonzero Coriolis parameter $f_0$, geostrophic balance and hydrostatic balance, using pressure per reference density $\pi$, giveDifferentiate the first equation in $z$ to obtain thermal-wind balance:Taking the dot product with ${\mathbf{U}}_h$ provesThus the horizontal velocity is tangent to the horizontal buoyancy contours at every height. For a fluid whose thermal equation of state makes horizontal buoyancy contours coincide with horizontal temperature contours, there is no horizontal advection across those contours. In a dry terrestrial atmosphere, use potential temperature as the materially conserved thermal variable. A salinity contribution to buoyancy would require separate treatment before interpreting the result as a statement about temperature.

The usual statement about net meridional heat transport also follows explicitly. Introduce a horizontal streamfunction $\Psi$ such that ${\mathbf{u}}_h=(-{\Psi }_y,{\Psi }_x)$. Integration of thermal-wind balance givesand henceFor a zonally periodic function $\Psi$, or vanishing endpoint contributions, the zonal integral is zero. Likewise, the anomalous eddy buoyancy flux isThe required correlation between meridional velocity and thermal anomaly is absent. A nonzero vertical phase tilt, as in the next part, permits that correlation.

Qualification of the wording. The result does not say that the local heat flux vector $b{\mathbf{U}}_h$ vanishes, or that an arbitrary open-domain integral vanishes without boundary conditions. For a concrete counterexample to that literal reading, take ${\mathbf{u}}_h=(0,1)$ and $b=f_0\Gamma x+N^{2}z$. These satisfy the displayed thermal-wind balance, but $vb=\Gamma z(f_0\Gamma x+N^{2}z)$ is generally nonzero. What the hypotheses alone guarantee is zero transport across horizontal buoyancy contours; the vanishing zonally integrated heat transport uses the stated endpoint condition.

Let $b$ denote buoyancy in a stably stratified Boussinesq approximation. Geostrophic balance and hydrostatic balance imply the thermal wind relation $f_0{U}_z=-{\bar{b}}_y$. Consequently the specified vertical shear requiresFor northern-hemisphere $f_0>0$ and $\Lambda >0$, the mean potential temperature decreases northward and increases upward: ${\bar{\Theta }}_y=-({\Theta }_0/g)f_0\Lambda$, ${\bar{\Theta }}_z=({\Theta }_0/g)N^2$. The corresponding mean isentropes slope upward toward the north, with $dz/dy=f_0\Lambda /N^2$. Stable potential-temperature stratification does not require the actual temperature to increase upward; the dry adiabatic lapse also enters that comparison.

In the rigid-lid constant-shear baroclinic model, perturbation quasi-geostrophic potential vorticity isThe mean interior potential-vorticity gradient vanishes on this constant-$f_0$, constant-$N$, linear-shear basic state. Linear advection of that perturbation therefore becomes $({\partial }_t+\Lambda z{\partial }_x)q^{\prime }=0$, the supplied interior equation. It retains slow balanced motion and omits friction, planetary-vorticity gradients and nonlinear disturbance advection.

Substitution of ${\psi }^{\prime }=\Psi (z)\cos (ly)e^{ik(x-ct)}$ givesA possible branch has zero interior perturbation potential vorticity, soBoth supplied applications have $f_0>0$, so this equals the source's expression. For an unstable mode the nonzero imaginary part of the phase speed prevents a real critical level $\Lambda z=c$; for real speeds the displayed solution remains a valid zero-interior-PV branch away from, and smoothly through, any critical level. Taking the real part gives the physical disturbance.

The rigid lids enforce $w^{\prime }=0$. Linearized adiabatic buoyancy conservation isUse the geostrophic conventions $v^{\prime }={\psi }_x^{\prime }$ and $b^{\prime }=f_0{\psi }_z^{\prime }$. At a lid this gives $({\partial }_t+\Lambda z{\partial }_x){\psi }_z^{\prime }-\Lambda {\psi }_x^{\prime }=0$, henceThese are the no-normal-flow condition expressed through boundary buoyancy advection, not conditions of zero horizontal velocity. Boundary temperature anomalies can interact even though this solution has no interior perturbation PV, allowing baroclinic instability to extract mean available potential energy.

Potential temperature and a parcel. For a dry ideal gas of fixed composition, the first law of thermodynamics per unit mass isA reversible adiabatic process has $\delta q=0$. Taking $c_p,R_d$ constant gives $d\ln T=\kappa d\ln p$, where $\kappa =R_d/c_p$. Integrating to a fixed reference pressure $p_0$ defines the potential temperatureIt is the temperature a parcel would have after reversible dry adiabatic process taking it to $p_0$. Equivalently $ds=c_{p}d\ln \Theta$ under these assumptions, making potential temperature a thermodynamic entropy label useful for diagnosing stratification and air-mass motion. Latent heating, radiation, mixing and variable composition require modifying the conservation statement.

An air parcel is a small material volume followed with the flow. The parcel model treats it as internally approximately uniform, large compared with molecular scales and small compared with environmental variation scales. Rapid acoustic wave adjustment makes its pressure approximately equal to the surrounding pressure, while slower heat and mass exchange can be neglected during an idealized displacement. It is not a rigid container or an isolated molecule.

At the parcel's height, environmental hydrostatic pressure supplies an upward pressure force ${\rho }_{e}g$ per unit volume. Dividing by parcel density gives the upward accelerationThe last steps use equal pressure, equal gas constants and the same reference pressure. Moisture-dependent density requires virtual temperature or its potential-temperature counterpart. Nonhydrostatic pressure perturbations and entrainment are excluded from this elementary buoyancy expression.

The convective available potential energy is the positive buoyancy work per unit mass between the level of free convection and the equilibrium level:One sums positive-buoyancy layers when there is more than one. Negative work below free convection contributes convective inhibition rather than positive CAPE. In a lossless parcel estimate, conversion of CAPE to vertical kinetic energy gives $w_{\max }\simeq \sqrt{2\mathrm{CAPE}}$; pressure work beyond this model, mixing and precipitation can change that speed.

Linear stability. Let $\xi$ denote displacement from a neutrally buoyant reference altitude. Dry motion through an adiabatic process retains ${\Theta }_p={\Theta }_e(0)$, while ${\Theta }_e(\xi )={\Theta }_e(0)+\xi {\Theta }_e^{\prime }(0)+O({\xi }^2)$. The linear vertical momentum equation is thereforeFor $N_0^2>0$, $\xi =C\cos N_{0}t+D\sin N_{0}t$: stable buoyancy oscillations with period $2\pi /N_0$. For $N_0^2=0$, $\xi =C+Dt$: neutral stratification with no restoring force. For $N_0^2<0$, growing and decaying exponentials occur, indicating convective instability. The growing solution is generic unless the initial data select only the decaying branch. These are local small-displacement results with an approximately constant environmental gradient.

Lapse-rate estimate. Define $\Gamma =-dT/dz$. Hydrostatic balance and the ideal gas law imply $p^{\prime }/p=-g/(R_{d}T)$, soIn the free-parcel-oscillator interpretation, identifying the ridge encounter interval with the buoyancy period gives $P=10^4/20=500\mathrm{s}$ and $N_0=2\pi /P=0.01257{\mathrm{s}}^{-1}$. With $T=293.15\mathrm{K}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$ and $c_p\simeq 1004\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$,The frequency identification is an extra modeling assumption. A stationary mountain gravity wave generally obeys ${\hat{\omega }}^2=N_0^2{k}_h^2/(k_h^2+m_z^2)$ without rotation, with $\hat{\omega }=U{k}_h$. Ridge spacing fixes $\hat{\omega }$, not the vertical wavenumber $m_z$. For example, $m_z=k_h$ allows the same spacing and wind with $N_0=\sqrt{2}\hat{\omega }$, and a lapse rate about $0.3\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$. Thus $5.1$ is the requested parcel-period estimate, not a uniquely identifiable lapse rate for every possible mountain-wave field; a resolved wave field needs its vertical structure as well.

Linearize about a uniform horizontal flow $U$ and a hydrostatic, vertically stratified basic state. The basic-flow material derivative is $D_0={\partial }_t+U{\partial }_x$. The first two equations are the horizontal and vertical perturbation momentum balances: pressure gradients accelerate the disturbance, while the vertical equation retains the buoyancy force $-g{\rho }^{\prime }/{\rho }_0$ after the basic hydrostatic balance is subtracted.

The Boussinesq approximation neglects fractional density variations in inertia and enforces incompressibility, $u_x+w_z=0$, but retains them in buoyancy. Advecting the stable background density gradient gives $D_0{\rho }^{\prime }+wd\bar{\rho }/dz=0$. With $N^2=-(g/{\rho }_0)d\bar{\rho }/dz>0$, this is the fourth supplied equation. A displaced parcel then has a restoring buoyancy tendency.

Small amplitude allows products of perturbations to be discarded; two-dimensional motion removes $y$ dependence and transverse flow. Rotation, viscosity, diffusion, diabatic heating and mean shear are neglected. ${\rho }_0$ is the reference inertial density, whereas the small background density gradient still sets $N$. In a compressible real atmosphere the appropriate stability measure involves potential temperature; this Boussinesq system is a local idealization that filters acoustic waves rather than a full compressible atmospheric model.

Write $\kappa =R_d/c_p$, where $R_d$ is the specific gas constant and $c_p$ the specific heat capacity at constant pressure. Conservation of potential temperature on a dry adiabatic process givesLet $Q=(1-{\alpha }_c)S$ be the absorbed stellar irradiance, and let $T_0=(Q/\sigma {)}^{1/4}$, with $\sigma$ the Stefan-Boltzmann constant. The transparent gas contributes no thermal radiation. For a cloud with emissivity $e$, Kirchhoff's law of thermal radiation gives thermal absorptivity $e$; assume thermal transmissivity $1-e$ and negligible thermal reflection, as in the usual one-layer grey body model. The outgoing radiant flux is the transmitted surface emission plus upward cloud emission:ThusNo factor of four is needed: $S$ is the incident irradiance of the specified column, rather than a stellar-beam irradiance that still needs a spherical average.

For fixed ${\alpha }_c,p_c,p_s$, increasing $e$ monotonically warms the ground when $0<q<1$. At $e=0$, $T_g=T_0$; at $e=1$, $T_c=T_0$ and $T_g=T_0/q$. A cloud at the surface, $q=1$, produces no such warming. Comparing with a genuinely cloud-free column also requires including its different albedo: relative to $(S/\sigma {)}^{1/4}$, the ratio is $[(1-{\alpha }_c)/(1-e+e{q}^4){]}^{1/4}$. Reflection can outweigh the greenhouse effect.

The governing balance here is the whole-column radiation budget. Imposing an additional purely radiative surface balance would incorrectly remove the convective heat transfer that the prescribed dry adiabatic process may require.

Hotter Archean mantle would generally produce thicker, more magnesium-rich oceanic crust, together with a more strongly depleted residual mantle. Greater melt production, not merely thermal expansion of an unchanged modern column, is the principal reason for the increased crustal thickness.

At a spreading ridge, upwelling peridotite undergoes adiabatic decompression melting. A higher mantle potential temperature makes the ascending mantle intersect the solidus at greater pressure and depth. It therefore melts over a longer decompression interval and generally reaches a larger integrated melt fraction. With comparable melt focusing, more melt is supplied per unit area of newly formed oceanic crust.

Modern normal oceanic crust is about 6–7 km thick and contains basaltic lavas, a sheeted dike complex, and gabbro with mafic cumulates at depth. An illustrative reconstruction for hot ambient Archean mantle gives about 25–35 km of oceanic crust, rather than a universal fixed thickness. Its primary liquids are more magnesium-rich than typical modern primary ridge liquids; their crystallization produces more olivine- and pyroxene-rich cumulates. The extracted liquid and its crystal products must be distinguished: the entire final oceanic crust need not be a homogeneous komatiite layer.^[ref]

The residual peridotite loses more melt components, becoming more refractory and commonly harzburgite-rich. Thick buoyant basaltic crust, its later eclogitization, and the depleted underlying mantle can change the mechanics of subduction. A hotter mantle does not by itself establish how fast plates moved or whether a globally connected modern plate tectonics regime operated.

Komatiites, with very magnesium-rich primary liquids and high inferred eruption temperatures, provide evidence for hot melting, but may sample especially hot mantle plumes. Estimates of ambient mantle conditions also use less exceptional non-arc basalts. Variable source composition, water content, cooling, melt extraction and later alteration all limit the precision of a single inferred Archean thickness. The robust prediction is a larger melting column and melt yield, thicker igneous oceanic crust, and a more depleted residue.

Specify what “onset” means. I would distinguish the first local, perhaps short-lived subduction event from persistent, self-sustaining recycling in a connected global network of tectonic plates. Modern plate tectonics combines relatively rigid moving lithosphere, localized boundary deformation, seafloor spreading and sustained convergence. A sinking drip or a single imposed descending sheet is not sufficient evidence of that global regime. Different processes may become established at different times, so a transition interval may be a better hypothesis than an instantaneous worldwide switch.

Construct competing physical models. Solve conservation of mass, momentum and energy for evolving mantle convection and lithosphere, including temperature- and stress-dependent rheology, brittle yielding, plastic deformation, melting, compositional buoyancy and relevant phase changes. Use geodynamic modelling to compare a stagnant lid, episodic overturn, drip-dominated recycling and a sustained mobile lid. In a simplified incompressible creeping-flow framework,Here $\mathbf{u}$ is velocity, $p$ pressure, $\eta$ effective viscosity, $\dot{\mathbf{ϵ}}$ the strain-rate tensor, $\rho$ density, $\mathbf{g}$ gravitational acceleration, $c_p$ specific heat capacity, $k$ thermal conductivity, $H$ radiogenic power per unit mass, and $\Phi$ dissipative heating per unit volume. These equations are a framework: compressibility, latent heat and chemical transport may require additional terms. A simple yield stress law ${\tau }_y=C+\mu (\rho gz-P_f)$ represents cohesion $C$, friction coefficient $\mu$ and pore pressure $P_f$, illustrating why hydration and interface weakening can change the outcome.

Vary the early-planet controls rather than assuming the answer. Explore plausible initial mantle potential temperature, radiogenic heating, crust thickness and composition, water inventory, grain-size evolution, surface temperature, and boundary strength. Hotter mantle lowers viscosity but can produce thicker buoyant basaltic crust; it does not automatically make sustained subduction easier. A cold strong plate can transmit slab pull yet resist bending and initiation. Water weakening of rocks, melt-induced weakening, inherited faults and mechanically weak sediments can change this balance. Test proposed triggers such as mantle plume impingement or gravitational instability without presupposing an established global plate-force system.

Useful diagnostics include the Rayleigh number,with thermal expansion coefficient $\alpha$, layer thickness $D$, thermal diffusivity $\kappa$ and temperature contrast $\Delta T$, together with surface mobility, the localization of deformation, plate longevity, melt production and material recycling. A large Rayleigh number establishes vigorous convection under suitable conditions, not necessarily a mobile lid. A convincing model should sustain one-sided subduction, connected boundary evolution and seafloor spreading after any artificial trigger is removed. Compare initiation thresholds with continuation thresholds: hysteresis can allow an established regime to persist under conditions in which it would not spontaneously start.

As an example of the type of experiment, three-dimensional models have tested mantle plume-induced subduction initiation, finding sensitivity to plate buoyancy, magmatic weakening and hydration of the interface. The scientific use of such results here is to identify a feasible mechanism and its predicted signatures, not to adopt their success as a direct measurement of an ancient onset date.^[ref]

Connect a model regime to a date through independent evidence. Couple the experiments to candidate planetary cooling histories, then predict observable pressure-temperature-time paths, magmatic compositions, deformation patterns and recycling rates for each regime. If independently dated rocks constrain a thermal or compositional state, ask which regimes can reproduce that state and the observed geological associations. Repeat over uncertain initial conditions and constitutive parameters, check numerical resolution, and compare two- and three-dimensional results where geometry matters. Models that prescribe a descending slab, fix plate velocity, or insert a weak boundary may be excellent forward models of a working subduction zone but do not demonstrate spontaneous initiation.

The result is a conditional statement: persistent plate tectonics is physically feasible under a range of conditions, and dated geological observations may favor that range at a particular time. Uncertain cooling histories, several feasible regimes and uncertain rheology prevent numerical feasibility alone from dating onset. The strongest constraint comes when the same parameter range explains multiple independently dated observations and alternative regimes fail those tests.

A long-term decline in mean interior temperature, distinct from a short transient, a local cold anomaly or a change in the pressure used for a potential temperature. In a mantle thermal budget, cooling requires heat loss to exceed internal generation plus heat entering from below. The rate depends on both the heat sources and the efficiency of transport.

Resistance of a stratified fluid to a small vertical parcel displacement. For dry homogeneous air, $N^2=(g/\theta )d\theta /dz$ in terms of potential temperature; $N^2>0$ gives a restoring buoyancy force. Moisture and composition gradients modify the parcel comparison. Across an ideal sharp interface, a denser lower fluid is stable to an infinitesimal interchange.

Authors: Claude Herzberg, Kent Condie, Jun Korenaga.

Earth and Planetary Science Letters 292, 79–88. Models primary liquids and ambient mantle potential temperature; estimates thick ancient oceanic crust and distinguishes non-arc basalts from komatiites.

A mountain does not by itself determine whether an atmospheric layer becomes thinner. In the fixed-lid model of mountain-crossing isentropic PV balance, raising the lower boundary by $h_m$ reduces layer depth from $H$ to $H-h_m$. Conserving potential vorticity then requires reduced absolute vertical vorticity; in the Northern Hemisphere this gives an anticyclonic relative tendency if other changes are neglected.

For an isentropic atmospheric layer, the relevant thickness is its isentropic mass density, not simply its geometric depth. Its upper potential temperature surface can rise, deform or exchange pressure loading as flow crosses terrain. Both boundaries and the thermodynamic structure must therefore be specified before inferring compression. The relation $f+{\zeta }_{\theta }=q_{\theta }{\sigma }_{\theta }$ gives the vorticity change once that mass-thickness change is known; it cannot determine the upper-boundary response on its own.