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53-467: The APMB is likely compositionally zoned with the lower 18–30 km containing andesitic melts and the upper 9–18 km containing dacitic melts. Estimates for the percentage of andesitic melt vary from 8 vol% on the low end and up to 30 vol% on the high end. These andesitic melts also have a high water content (~7-10 wt.% water) indicated by the high electrical conductivity measured in the APMB. Measurements for

106-574: A salt glacier . Differential loading causes salt deposits covered by overburden ( sediment ) to rise upward toward the surface and pierce the overburden, forming diapirs (including salt domes ), pillars, sheets, or other geological structures. In addition to Earth-based observations, diapirism is thought to occur on Neptune's moon Triton , Jupiter's moon Europa , Saturn's moon Enceladus , and Uranus's moon Miranda . Diapirs commonly intrude buoyantly upward along fractures or zones of structural weakness through denser overlying rocks. This process

159-570: A growth rate that is approximately constant in time. At this point, nonlinear terms in the equations of motion can no longer be ignored. The spikes and bubbles then begin to interact with one another in the third stage. Bubble merging takes place, where the nonlinear interaction of mode coupling acts to combine smaller spikes and bubbles to produce larger ones. Also, bubble competition takes places, where spikes and bubbles of smaller wavelength that have become saturated are enveloped by larger ones that have not yet saturated. This eventually develops into

212-472: A liquid jet. This instability, sometimes called the hosepipe (or firehose) instability, occurs due to surface tension, which acts to break a cylindrical jet into a stream of droplets having the same total volume but higher surface area. Many people have witnessed the RT instability by looking at a lava lamp , although some might claim this is more accurately described as an example of Rayleigh–Bénard convection due to

265-399: A minimum of 15% andesitic melt. Additionally, these resistivity values indicate that the melt has a water content up to 10 wt.% H 2 O, which makes up approximately 10% of the APMB. Diapir A diapir ( / ˈ d aɪ . ə p ɪər / ; from French diapir [djapiʁ] , from Ancient Greek διαπειραίνω ( diapeiraínō )  'to pierce through')

318-466: A region of turbulent mixing, which is the fourth and final stage in the evolution. It is generally assumed that the mixing region that finally develops is self-similar and turbulent, provided that the Reynolds number is sufficiently large. The inviscid two-dimensional Rayleigh–Taylor (RT) instability provides an excellent springboard into the mathematical study of stability because of the simple nature of

371-736: A significant risk when trying to drill through them. There is an analogy to a Galilean thermometer . Rock types such as evaporitic salt deposits, and gas charged muds are potential sources of diapirs. Diapirs also form in the Earth's mantle when a sufficient mass of hot, less dense magma assembles. Diapirism in the mantle is thought to be associated with the development of large igneous provinces and some mantle plumes . Explosive, hot volatile rich magma or volcanic eruptions are referred to generally as diatremes . Diatremes are not usually associated with diapirs, as they are small-volume magmas which ascend by volatile plumes, not by density contrast with

424-527: Is κ = ∇ 2 η = η x x . {\displaystyle \kappa =\nabla ^{2}\eta =\eta _{xx}.\,} Thus, p G ( z = η ) − p L ( z = η ) = σ η x x . {\displaystyle p_{G}\left(z=\eta \right)-p_{L}\left(z=\eta \right)=\sigma \eta _{xx}.\,} However, this condition refers to

477-439: Is a spatial wavenumber. Thus, the problem reduces to solving the equation ( D 2 − α 2 ) Ψ j = 0 ,   D = d d z ,   j = L , G . {\displaystyle \left(D^{2}-\alpha ^{2}\right)\Psi _{j}=0,\,\,\,\ D={\frac {d}{dz}},\,\,\,\ j=L,G.\,} The domain of

530-474: Is a type of intrusion in which a more mobile and ductilely deformable material is forced into brittle overlying rocks. Depending on the tectonic environment, diapirs can range from idealized mushroom-shaped Rayleigh–Taylor instability structures in regions with low tectonic stress such as in the Gulf of Mexico to narrow dikes of material that move along tectonically induced fractures in surrounding rock. The term

583-526: Is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. Examples include the behavior of water suspended above oil in the gravity of Earth , mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions , supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion. Water suspended atop oil

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636-476: Is an everyday example of Rayleigh–Taylor instability, and it may be modeled by two completely plane-parallel layers of immiscible fluid, the denser fluid on top of the less dense one and both subject to the Earth's gravity. The equilibrium here is unstable to any perturbations or disturbances of the interface: if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards,

689-445: Is described by the vector g = − g e z {\displaystyle \mathbf {g} =-g\,\mathbf {e} _{z}} . The velocity field and pressure field in this equilibrium state, denoted with an overbar, are given by where the reference location for the pressure is taken to be at z = 0 {\displaystyle z=0} . Let this interface be slightly perturbed, so that it assumes

742-402: Is equivalent to the situation when the fluids are accelerated , with the less dense fluid accelerating into the denser fluid. This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion. As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing "plumes" flowing upwards (in

795-418: Is known as diapirism . The resulting structures are also referred to as piercement structures . In the process, segments of the existing strata can be disconnected and pushed upwards. While moving higher, they retain many of their original properties, e.g. pressure; their pressure can be significantly different from the pressure of the shallower strata they get pushed into. Such overpressured "floaters" pose

848-417: Is necessary to fix conditions at the boundaries and interface. This determines the wave speed c , which in turn determines the stability properties of the system. The first of these conditions is provided by details at the boundary. The perturbation velocities w i ′ {\displaystyle w'_{i}\,} should satisfy a no-flux condition, so that fluid does not leak out at

901-454: Is simply ∂ η ∂ t = w ′ ( 0 ) , {\displaystyle {\frac {\partial \eta }{\partial t}}=w'\left(0\right),\,} where the velocity w ′ ( η ) {\displaystyle w'\left(\eta \right)\,} is linearized on to the surface z = 0 {\displaystyle z=0\,} . Using

954-439: Is the formation and growth of a large diapir arising from the APMB. Lower-density magma than the surrounding rocks is produced during partial melting in the APMB, causing a plume of buoyant magma to rise from the center of the magma body. This causes material to be removed from the APMB to feed the growing diapir, resulting in a region of subsidence surrounding the uplift zone. Data collected between 1992 and 2010 demonstrates that

1007-425: Is the growth rate of the perturbation. Then the linear stability analysis based on the inviscid governing equations shows that Thus, if ρ 2 < ρ 1 {\displaystyle \rho _{2}<\rho _{1}} , the base state is stable and while if ρ 2 > ρ 1 {\displaystyle \rho _{2}>\rho _{1}} , it

1060-613: Is the required interfacial condition. The free-surface condition: At the free surface z = η ( x , t ) {\displaystyle z=\eta \left(x,t\right)\,} , the kinematic condition holds: ∂ η ∂ t + u ′ ∂ η ∂ x = w ′ ( η ) . {\displaystyle {\frac {\partial \eta }{\partial t}}+u'{\frac {\partial \eta }{\partial x}}=w'\left(\eta \right).\,} Linearizing, this

1113-596: Is to say, surface tension stabilises large wavenumbers or small length scales. Then the maximum growth rate occurs at the wavenumber k m = k c / 3 {\displaystyle k_{m}=k_{c}/{\sqrt {3}}} and its value is The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, ( u ′ ( x , z , t ) , w ′ ( x , z , t ) ) . {\displaystyle (u'(x,z,t),w'(x,z,t)).\,} Because

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1166-574: Is unstable for all wavenumbers. If the interface has a surface tension γ {\displaystyle \gamma } , then the dispersion relation becomes which indicates that the instability occurs only for a range of wavenumbers 0 < k < k c {\displaystyle 0<k<k_{c}} where k c 2 = ( ρ 2 − ρ 1 ) g / γ {\displaystyle k_{c}^{2}=(\rho _{2}-\rho _{1})g/\gamma } ; that

1219-476: The Young–Laplace equation: p G ( z = η ) − p L ( z = η ) = σ κ , {\displaystyle p_{G}\left(z=\eta \right)-p_{L}\left(z=\eta \right)=\sigma \kappa ,\,} where σ is the surface tension and κ is the curvature of the interface, which in a linear approximation

1272-533: The supernova explosion 1000 years ago. The RT instability has also recently been discovered in the Sun's outer atmosphere, or solar corona , when a relatively dense solar prominence overlies a less dense plasma bubble. This latter case resembles magnetically modulated RT instabilities. Note that the RT instability is not to be confused with the Plateau–Rayleigh instability (also known as Rayleigh instability) of

1325-563: The Altiplano-Puna Volcanic Complex (APVC) in order to characterize the magmatic structures beneath the surface. These stations found a low velocity region approximately 10–20 km beneath the surface that was interpreted to be a sill-like magma body associated with the APVC. Seismic studies and modeling continues to take place in this area, further constraining the extent and characteristics of this magma body. A 3D density model of

1378-631: The Andes as well as determine characteristics of the APMB. Magnetotelluric stations were deployed across the Central Andes and resolved a highly conductive region beneath the Altiplano-Puna plateau, which appeared to coincide with the low velocity zone associated with the APMB. Further magnetotelluric studies showed that the region has low electrical resistivities of <3 Ωm. Resistivity values in this range are interpreted to only occur with magma that contains

1431-487: The Central Andes was developed based on modeling of Bouguer anomalies and it provided a more detailed view of the region's lithospheric structure and an estimation of the amount of partial melt present in the APMB (~9%). Continued investigation of Bouguer anomaly data led to the discovery of a column-like, low density structure extending from the top of the APMB with a diameter of approximately 15 km. Electromagnetic methods have also been used to investigate structures in

1484-464: The active heating of the fluid layer at the bottom of the lamp. The evolution of the RTI follows four main stages. In the first stage, the perturbation amplitudes are small when compared to their wavelengths, the equations of motion can be linearized, resulting in exponential instability growth. In the early portion of this stage, a sinusoidal initial perturbation retains its sinusoidal shape. However, after

1537-514: The base state. Consider a base state in which there is an interface, located at z = 0 {\displaystyle z=0} that separates fluid media with different densities, ρ 1 {\displaystyle \rho _{1}} for z < 0 {\displaystyle z<0} and ρ 2 {\displaystyle \rho _{2}} for z > 0 {\displaystyle z>0} . The gravitational acceleration

1590-494: The boundaries z = ± ∞ . {\displaystyle z=\pm \infty .\,} Thus, w L ′ = 0 {\displaystyle w_{L}'=0\,} on z = − ∞ {\displaystyle z=-\infty \,} , and w G ′ = 0 {\displaystyle w_{G}'=0\,} on z = ∞ {\displaystyle z=\infty \,} . In terms of

1643-416: The end of this first stage, when non-linear effects begin to appear, one observes the beginnings of the formation of the ubiquitous mushroom-shaped spikes (fluid structures of heavy fluid growing into light fluid) and bubbles (fluid structures of light fluid growing into heavy fluid). The growth of the mushroom structures continues in the second stage and can be modeled using buoyancy drag models, resulting in

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1696-461: The fluid is assumed incompressible, this velocity field has the streamfunction representation u ′ = ( u ′ ( x , z , t ) , w ′ ( x , z , t ) ) = ( ψ z , − ψ x ) , {\displaystyle {\textbf {u}}'=(u'(x,z,t),w'(x,z,t))=(\psi _{z},-\psi _{x}),\,} where

1749-451: The fluids determines the structure of the subsequent non-linear RT instability flows (assuming other variables such as surface tension and viscosity are negligible here). The difference in the fluid densities divided by their sum is defined as the Atwood number , A. For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; for A close to 1, the much lighter fluid "below"

1802-425: The gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, the fluid movement can be closely approximated by linear equations , and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, and non-linear equations are required to describe fluid motions. In general, the density disparity between

1855-506: The heavier fluid takes the form of larger bubble-like plumes. This process is evident not only in many terrestrial examples, from salt domes to weather inversions , but also in astrophysics and electrohydrodynamics . For example, RT instability structure is evident in the Crab Nebula , in which the expanding pulsar wind nebula powered by the Crab pulsar is sweeping up ejected material from

1908-791: The horizontal momentum equation of the linearised Euler equations for the perturbations, ∂ u i ′ ∂ t = − 1 ρ i ∂ p i ′ ∂ x {\displaystyle {\frac {\partial u_{i}'}{\partial t}}=-{\frac {1}{\rho _{i}}}{\frac {\partial p_{i}'}{\partial x}}\,} with i = L , G , {\displaystyle i=L,G,\,} to yield p i ′ = ρ i c D Ψ i , i = L , G . {\displaystyle p_{i}'=\rho _{i}cD\Psi _{i},\qquad i=L,G.\,} Putting this last equation and

1961-639: The jump condition on p G ′ − p L ′ {\displaystyle p'_{G}-p'_{L}} together, c ( ρ G D Ψ G − ρ L D Ψ L ) = g η ( ρ G − ρ L ) + σ η x x . {\displaystyle c\left(\rho _{G}D\Psi _{G}-\rho _{L}D\Psi _{L}\right)=g\eta \left(\rho _{G}-\rho _{L}\right)+\sigma \eta _{xx}.\,} Substituting

2014-406: The normal-mode and streamfunction representations, this condition is c η = Ψ {\displaystyle c\eta =\Psi \,} , the second interfacial condition. Pressure relation across the interface: For the case with surface tension , the pressure difference over the interface at z = η {\displaystyle z=\eta } is given by

2067-521: The partial melt percentage in the APMB also vary, with seismic imaging indicating that it is anywhere from 10-40% partial melt. For a magma body with ~20% partial melt, the viscosity is estimated to be <10 Pa s. The Altiplano-Puna region around the Uturuncu volcano is experiencing a type of deformation termed 'sombrero uplift,' which means a central zone of uplift surrounded by a region of subsidence. One potential explanation for this sombrero uplift pattern

2120-392: The position z = f ( x , t ) {\displaystyle z=f(x,t)} . Correspondingly, the base state is also slightly perturbed. In the linear theory, we can write where k {\displaystyle k} is the real wavenumber in the x {\displaystyle x} -direction and σ {\displaystyle \sigma }

2173-422: The potential energy of the configuration is lower than the initial state. Thus the disturbance will grow and lead to a further release of potential energy , as the denser material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards. This was the set-up as studied by Lord Rayleigh. The important insight by G. I. Taylor was his realisation that this situation

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2226-630: The preservation of preexisting hydrocarbons . There are many salt domes and salt glaciers in the Zagros mountains , formed by the collision of two tectonic plates , the Eurasian Plate and the Arabian Plate . There are underwater salt domes in the Gulf of Mexico. Rayleigh%E2%80%93Taylor instability The Rayleigh–Taylor instability , or RT instability (after Lord Rayleigh and G. I. Taylor ),

2279-402: The problem is the following: the fluid with label 'L' lives in the region − ∞ < z ≤ 0 {\displaystyle -\infty <z\leq 0\,} , while the fluid with the label 'G' lives in the upper half-plane 0 ≤ z < ∞ {\displaystyle 0\leq z<\infty \,} . To specify the solution fully, it

2332-463: The region is uplifting at ~10 mm/year and subsiding at a slower rate (only a few mm/year). More recent InSAR data, collected between September 2014 and December 2017, shows that the uplift rate over this period has decreased to 3–5 mm/year and may experience short-term velocity reversals. Additionally, there is evidence that the uplift and subsidence rates have balanced out over the past 16,000 years to create no net deformation. These aspects of

2385-602: The streamfunction, this is Ψ L ( − ∞ ) = 0 , Ψ G ( ∞ ) = 0. {\displaystyle \Psi _{L}\left(-\infty \right)=0,\qquad \Psi _{G}\left(\infty \right)=0.\,} The other three conditions are provided by details at the interface z = η ( x , t ) {\displaystyle z=\eta \left(x,t\right)\,} . Continuity of vertical velocity: At z = η {\displaystyle z=\eta } ,

2438-481: The subscripts indicate partial derivatives . Moreover, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays irrotational , hence ∇ × u ′ = 0 {\displaystyle \nabla \times {\textbf {u}}'=0\,} . In the streamfunction representation, ∇ 2 ψ = 0. {\displaystyle \nabla ^{2}\psi =0.\,} Next, because of

2491-863: The surface z=0 .) Using hydrostatic balance , in the form P L = − ρ L g z + p 0 , P G = − ρ G g z + p 0 , {\displaystyle P_{L}=-\rho _{L}gz+p_{0},\qquad P_{G}=-\rho _{G}gz+p_{0},\,} this becomes p G ′ − p L ′ = g η ( ρ G − ρ L ) + σ η x x , on  z = 0. {\displaystyle p'_{G}-p'_{L}=g\eta \left(\rho _{G}-\rho _{L}\right)+\sigma \eta _{xx},\qquad {\text{on }}z=0.\,} The perturbed pressures are evaluated in terms of streamfunctions, using

2544-440: The surrounding mantle. Diapirs or piercement structures are structures resulting from the penetration of overlaying material. By pushing upward and piercing overlying rock layers, diapirs can form anticlines (arch-like shape folds ), salt domes (mushroom/ dome-shaped diapirs), and other structures capable of trapping hydrocarbons such as petroleum and natural gas . Igneous intrusions themselves are typically too hot to allow

2597-583: The total pressure (base+perturbed), thus [ P G ( η ) + p G ′ ( 0 ) ] − [ P L ( η ) + p L ′ ( 0 ) ] = σ η x x . {\displaystyle \left[P_{G}\left(\eta \right)+p'_{G}\left(0\right)\right]-\left[P_{L}\left(\eta \right)+p'_{L}\left(0\right)\right]=\sigma \eta _{xx}.\,} (As usual, The perturbed quantities can be linearized onto

2650-458: The translational invariance of the system in the x -direction, it is possible to make the ansatz ψ ( x , z , t ) = e i α ( x − c t ) Ψ ( z ) , {\displaystyle \psi \left(x,z,t\right)=e^{i\alpha \left(x-ct\right)}\Psi \left(z\right),\,} where α {\displaystyle \alpha \,}

2703-523: The uplift and subsidence cannot be easily explained by the diapir model, so other possible mechanisms for driving the deformation are being investigated. One such mechanism that might explain the deformation is the movement of volatiles in a column connected to the APMB. Movement like this may explain the surface deformation rate that varies on monthly or yearly scales and appears to have resulted in no net deformation over longer periods. Between 1996 and 1997, several broadband seismic stations were deployed over

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2756-798: The vertical velocities match, w L ′ = w G ′ {\displaystyle w'_{L}=w'_{G}\,} . Using the stream function representation, this gives Ψ L ( η ) = Ψ G ( η ) . {\displaystyle \Psi _{L}\left(\eta \right)=\Psi _{G}\left(\eta \right).\,} Expanding about z = 0 {\displaystyle z=0\,} gives Ψ L ( 0 ) = Ψ G ( 0 ) + H.O.T. , {\displaystyle \Psi _{L}\left(0\right)=\Psi _{G}\left(0\right)+{\text{H.O.T.}},\,} where H.O.T. means 'higher-order terms'. This equation

2809-433: Was introduced by Romanian geologist Ludovic Mrazek , who was the first to understand the principle of salt tectonics and plasticity . The term diapir may be applied to igneous intrusions , but it is more commonly applied to non-igneous, relatively cold materials, such as salt domes and mud diapirs. If a salt diapir reaches the surface, it can flow because salt becomes ductile with a small amount of moisture, forming

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