Glossary: Difference between revisions

From Fluids Wiki
Jump to navigation Jump to search
m (Change notation for strain rate)
 
(129 intermediate revisions by 7 users not shown)
Line 7: Line 7:


'''Note''': The [http://glossary.ametsoc.org/wiki/Main_Page AMS Glossary] is a good source for definitions, should the definition that you seek not be available below.
'''Note''': The [http://glossary.ametsoc.org/wiki/Main_Page AMS Glossary] is a good source for definitions, should the definition that you seek not be available below.
See also [[Dimensionless Numbers]], and [[Fluids equations]].


----
----


== A-D ==
== A-B ==
<div id="Barotropic fluid"></div>
<div id="Added mass"></div>
; '''Added mass'''
: The inertia added to a system due to the fact that an accelerating or decelerating body must move some volume of surrounding fluid with it as it moves.
 
<div id="Available Potential Energy"></div>
; '''[[Available Potential Energy]]'''
: The potential energy available for conversion into other forms of energy. For intuitive purposes one may think of this as <math> APE = \int_\Omega \left(\rho-\rho^*\right)dV </math> where <math> \rho </math> is the density field, and <math> \rho^* </math> is the redistributed density field such that the potential energy is minimized.
 
<div id="Azimuth"></div>
; '''Azimuth'''
: The horizontal angle. In polar coordinates, the angular direction.
 
<div id="Barotropic fluid"></div>
; '''Barotropic fluid'''
; '''Barotropic fluid'''
: A fluid in which <math> \nabla p\times \nabla\rho = 0 </math>. This means that surfaces of constant pressure and constant density are parallel. Fluids in which <math> p = p(\rho) </math> or the density is constant (<math> \rho = \rho_0 </math>) are necessarily barotropic.  
: A fluid in which <math> \nabla p\times \nabla\rho = 0 </math>. This means that surfaces of constant pressure and constant density are parallel. Fluids in which <math> p = p(\rho) </math> or the density is constant (<math> \rho = \rho_0 </math>) are necessarily barotropic.  
Line 21: Line 35:
<div id="Boundary Layer"></div>
<div id="Boundary Layer"></div>
; '''Boundary Layer'''  
; '''Boundary Layer'''  
: Region near a surface in which friction becomes important.
: Region near a boundary in which viscosity becomes important.
 
<div id="Buoyancy Frequency"></div>
; '''[[Buoyancy Frequency]]'''
: The frequency at which an infinitesimally perturbed [[#Fluid Parcel|fluid parcel]] oscillates around its rest state in the absence of friction. Formally given as <math> N^2(z) = -\frac{g}{\rho_0}\frac{d\overline{\rho}(z)}{dz} </math>, where <math>\overline{\rho}(z)</math> is the background stratification.


<div id="beta plane"></div>
<div id="beta plane"></div>
;  '''<math>\beta</math> - plane'''  
;  '''<math>\beta</math> - plane'''  
: The <math>\beta </math> - plane approximation assumes that the Coriolis frequency varies linearly with latitude i.e. <math>f=f_0+\beta y</math>. <math>f_0=2\Omega\sin(\theta_0)</math> and <math>\beta=\frac{2\Omega}{a}\cos(\theta_0)</math> where <math>\Omega</math> is the period of Earth's rotation, <math>\theta_0</math> is the reference latitude, and <math>a</math> is the mean radius of the Earth. Wikipedia's entry on this is a good one. cf. [[#fplane|<math>f</math>-plane]]
: The <math>\beta </math> - plane approximation assumes that the Coriolis frequency varies linearly with latitude i.e. <math>f=f_0+\beta y</math>. <math>f_0=2\Omega\sin(\theta_0)</math> and <math>\beta=\frac{2\Omega}{a}\cos(\theta_0)</math> where <math>\Omega</math> is the period of Earth's rotation, <math>\theta_0</math> is the reference latitude, and <math>a</math> is the mean radius of the Earth. Wikipedia's entry on this is a good one. cf. [[#f-plane|<math>f</math>-plane]]
 
== C-D ==


<div id="Capillary Wave"></div>
<div id="Capillary Wave"></div>
Line 33: Line 53:
<div id="Chaotic Advection"></div>
<div id="Chaotic Advection"></div>
; '''[[Chaotic Advection]]'''  
; '''[[Chaotic Advection]]'''  
: The advection of particles under a chaotic flow map or dynamical system.  
: The advection of particles under the flow map of a chaotic [[#Dynamical System | dynamical system]].  


<div id="Characteristic Scale"></div>
<div id="Characteristic Scale"></div>
Line 42: Line 62:
; '''Circulation'''  
; '''Circulation'''  
: The circulation of a flow is the integral of the vorticity over a surface, <math>\Gamma=\int_{\Omega}\vec{\omega} dA</math>, and represents the vorticity flux through the surface <math>\Omega</math>. By Stokes' theorem, <math>\Gamma=\int_{\partial\Omega}\vec{u}\cdot ds</math>.  
: The circulation of a flow is the integral of the vorticity over a surface, <math>\Gamma=\int_{\Omega}\vec{\omega} dA</math>, and represents the vorticity flux through the surface <math>\Omega</math>. By Stokes' theorem, <math>\Gamma=\int_{\partial\Omega}\vec{u}\cdot ds</math>.  
<div id="Coriolis Frequency"></div>
;'''Coriolis Frequency'''
: <math> f = 2\Omega\sin\theta </math>, where <math> \Omega </math> is the rate of earth's rotation, and <math> \theta </math> is the latitude (with the northern hemisphere to be positive). See also [[#f-plane|<math>f</math>-plane]], [[#beta plane|<math>\beta</math>-plane]].


<div id="Correlation Time"></div>
<div id="Correlation Time"></div>
;'''[[Correlation Time]]'''  
;'''[[Correlation Time]]'''  
: The time it takes for the auto correlation function of a process to decrease by a given amount.
: The time it takes for the auto correlation function of a process to decrease by a given amount.
<div id="Diffusivity"></div>
;'''Diffusivity'''
: Rate of diffusion (unit: m<math>^2</math>/s). Mass diffusivity: the rate at which mass (molecular) of the substance diffuses through a unit surface in a unit time. Thermal diffusivity: <math> \kappa = \frac{k}{\rho C_p} </math>, where <math> k </math> is the thermal conductivity, <math> \rho </math> is the density, and <math> C_p </math> is the specific heat. Momentum diffusivity: see [[#Viscosity|kinematic viscosity]].


<div id="DNS"></div>
<div id="DNS"></div>
Line 53: Line 81:
<div id="Dispersion Relation"></div>
<div id="Dispersion Relation"></div>
;'''Dispersion Relation'''
;'''Dispersion Relation'''
: The relationship between wavenumber, <math> k </math>, and (angular) wave frequency, <math> \omega </math>; typically written as <math> \omega = \omega(k) </math>. See also [[#Phase Velocity|Phase Velocity]], [[#Group Velocity|Group Velocity]], and [[#Dispersive Wave|Dispersive Waves]].
: For a given wave, the dispersion relation is the relationship between wavenumber, <math> k </math>, and (angular) wave frequency, <math> \omega </math>; typically written as <math> \omega = \omega(k) </math>. See also [[#Phase Velocity|Phase Velocity]], [[#Group Velocity|Group Velocity]], and [[#Dispersive Wave|Dispersive Waves]].


<div id="Dispersive Wave"></div>
<div id="Dispersive Wave"></div>
;'''[[Dispersive Wave]]'''
;'''[[Dispersive Wave]]'''
: A wave whose [[#Dispersion Relation|dispersion relation]] is not of the form <math>\omega=ck</math>. This causes waves of different wavelengths to travel at different speeds, and so the wave disperses over time.
: A wave for which the [[#Group Velocity|group velocity]] is dependent on wavenumber, so energy in different wavelengths propagates at different velocities.


== E-J ==
<div id="DJL equations"></div>
;'''[[DJL equations|Dubreil-Jacotin-Long (DJL) equation]]'''
: A scalar equation equivalent to the steady, incompressible, stratified Euler equations.
 
<div id="Dynamical System"></div>
;'''Dynamical System'''
: A set of differential equations describing a classical mechanical system.  There are many generalizations and formalisms associated with this concept, but physically the most important point is that the solution to a dynamical system is a time evolution function for the system.  There are numerous examples of dynamical systems in classical mechanics, but simple examples include pendulums and predator prey models.  See also [[#Fixed Point|fixed point]], [[#Stable Manifold|stable manifold]], and [[#Unstable Manifold|unstable manifold]].
 
== E-G ==
<div id="Energy cascade"></div>
<div id="Energy cascade"></div>
;'''[[Energy Cascade]]'''  
;'''[[Energy Cascade]]'''  
: When the coherent structures of the continuum move to smaller and smaller scales until viscosity causes dissipation.
: When energy is supplied at large scales and then passed to smaller and smaller scales until viscosity causes dissipation.


<div id="Enstrophy"></div>
<div id="Enstrophy"></div>
;'''[[Enstrophy]]'''  
;'''[[Enstrophy]]'''  
: <math> \frac{1}{2}\omega^2 </math>, the norm squared of the vorticity.  
: <math> \frac{1}{2}\omega^2 </math>, the norm squared of the vorticity.
 
<div id="Euler Equations"></div>
;'''[[Euler equations]]'''
: The Euler equations are the [[Navier-Stokes equations]] with the viscous term neglected.


<div id="f-plane"></div>
<div id="Eulerian Measurement"></div>
;'''<math>f</math> - plane'''  
;'''[[Eulerian and Lagrangian measurements|Eulerian measurement]]'''  
: The <math>f </math> - plane approximation assumes that the Coriolis frequency is constant in latitude i.e. <math>f=f_0</math>. Where <math>f_0=2\Omega\sin(\theta_0)</math>, <math>\Omega</math> is the period of Earth's rotation, and <math>\theta_0</math> is the reference latitude. cf. [[#betaplane|<math>\beta</math>-plane]]  
: A measurement taken at a fixed position in space. See also [[#Lagrangian Measurement|Lagrangian measurement]].


<div id="Internal Solitary Wave"></div>
<div id="Fixed Point"></div>
;'''[[Internal Solitary Wave]]'''
;'''Fixed Point'''  
: A single, internal, travelling wave, often located at a [[#Pycnocline|pycnocline]].
: For a time-invariant [[#Dynamical System|dynamical system]] <math>\dot{x} = \vec{v}(\vec{x}), \vec{x}(0) = \vec{x_0}</math>, a fixed point is a point <math>\vec{x_f}</math>  in the domain such that <math>\vec{v}(\vec{x_f}) = \vec{0}</math>.  Fixed points are important in the analysis of time-invariant systems because they define both [[#Stable Manifold|stable]] and [[#Unstable Manifold|unstable manifolds]], which divide the domain into regions of different dynamics.


<div id="Internal Tide"></div>
<div id="Fluid Parcel"></div>
;'''Internal Tide'''
;'''Fluid Parcel'''  
: Internal Waves generated at a tidal frequency. See [[#internalwave|Internal Wave]].  
: A volume of fluid which (see chapter 1 of Kundu.):
<ul>
*deforms under outside forces
*is non-diffusive, such that molecules do not cross the boundary
*is large enough to be well defined by thermodynamic quantities
*is small enough to reach internal equilibrium much faster than the background flow
</ul>


<div id="Internal Wave"></div>
<div id="f-plane"></div>
; '''Internal Wave'''  
;'''<math>f</math> - plane'''  
: Waves in which the displaced quantity is an isopycnal. These can include [[#gravitywave|gravity waves]] and [[#Rossbywave|Rossby waves]].
: The <math>f </math> - plane approximation assumes that the Coriolis frequency is constant in latitude i.e. <math>f=f_0</math>. Where <math>f_0=2\Omega\sin(\theta_0)</math>, <math>\Omega</math> is the period of Earth's rotation, and <math>\theta_0</math> is the reference latitude. cf. [[#beta plane|<math>\beta</math>-plane]]  


<div id="Isentropic Surface"></div>
<div id="Geostrophic Balance"></div>
;'''Isentropic Surface'''  
;'''Geostrophic Balance'''  
: A surface of constant entropy.
: Geostrophic balance occurs when the Coriolis pseudo-force balances the pressure forces. Under geostrophic balance, flow is along lines of constant pressure.


<div id="Gravity Wave"></div>
<div id="Gravity Wave"></div>
; '''Gravity Wave'''  
; '''[[Gravity Wave]]'''  
: A wave in which the dominant restoring force is due to gravity acting to restore displaced mass.
: A wave in which the dominant restoring force is due to gravity acting to restore displaced mass.


<div id="Group Velocity"></div>
<div id="Group Velocity"></div>
;'''Group Velocity'''
;'''Group Velocity'''
: The velocity with which the energy of a wave packet propagates. Mathematically, <math>c_g=\frac{\partial \omega}{\partial \vec{k}}=\left(\partial_{k_x}\omega,\partial_{k_y}\omega,\partial_{k_z}\omega \right)</math>, where <math>c_g</math> is the group velocity and <math>\vec{k}=(k_x,k_y,k_z)</math> is the wavenumber. The group velocity is closely related to the velocity of a [[#Waveenvelope|wave envelope]] .
: The velocity with which the energy of a wave packet propagates. Mathematically, the group velocity  <math>\vec{c_g}= \nabla_{\vec{k}} \omega =\left(\frac{\partial \omega}{\partial k},\frac{\partial \omega}{\partial l},\frac{\partial \omega}{\partial m} \right)</math>, where <math>\vec{k}=(k,l,m)</math> is the wavenumber. The group velocity is closely related to the velocity of a [[#Wave Envelope|wave envelope]] .


<div id="Gyre"></div>
<div id="Gyre"></div>
;'''Gyre'''  
;'''Gyre'''  
: A vortex, a region dominated by a coherent rotating structure.
: A vortex, a region dominated by a coherent rotating structure.
== H-J ==


<div id="Halocline"></div>
<div id="Halocline"></div>
;'''Halocline'''  
;'''[[Stratification|Halocline]]'''  
: Region with a high gradient in salinity. See also [[#Pycnocline|pycnocline]] and [[#Thermocline|thermocline]] <div id="Halocline"></div>
: Region with a high gradient in salinity. See also [[#Pycnocline|pycnocline]] and [[#Thermocline|thermocline]] <div id="Halocline"></div>
<div id="Internal Solitary Wave"></div>
;'''[[Internal Solitary Wave]]'''
: A single, internal, travelling wave, often located at a [[#Pycnocline|pycnocline]].
<div id="Internal Tide"></div>
;'''Internal Tide'''
: [[#internalwave|Internal waves]] generated at a tidal frequency.
<div id="Internal Wave"></div>
; '''[[Internal Wave|Internal Wave]]'''
: A wave whose displaced quantity is interior to the fluid. These can include [[#gravitywave|gravity waves]] and [[#Rossbywave|Rossby waves]].
<div id="Isentropic"></div>
;'''[[Surfaces of Constant Property|Isentropic]]'''
: Of or having constant entropy. An isentropic surface is a surface of constant entropy.
<div id="Isobaric"></div>
;'''[[Surfaces of Constant Property|Isobaric]]'''
: Of or having constant pressure. An isobaric surface is a surface of constant pressure.
<div id="Isohaline"></div>
;'''[[Surfaces of Constant Property|Isohaline]]'''
: Of or having constant salinity. An isohaline surface is a surface of constant salinity.
<div id="Isopycnal"></div>
;'''[[Surfaces of Constant Property|Isopycnal]]'''
: Of or having constant density. An isopycnal surface is a surface of constant density.
<div id="Isothermal"></div>
;'''[[Surfaces of Constant Property|Isothermal]]'''
: Of or having constant temperature. An isothermal surface is a surface of constant temperature.


== K-Q ==
== K-Q ==
<div id="KdV equation"></div>
;'''[[KdV equation|Korteweg–de Vries (KdV) equation]]'''
: A nonlinear equation used to describe long [[Internal Wave|internal waves]] (among other things).
<div id="Kelvin Wave"></div>
;'''Kelvin Wave'''
: Gravity waves that are boundary trapped and are in [[#Geostrophic Balance|geostrophic balance]] in the direction orthogonal to the boundary. Kelvin waves require the presence of a boundary such as a coastline, channel wall, or the equator. Like [[#Poincare Wave|Poincaré waves]], Kelvin waves are rotating gravity waves. In the Northern (Southern) hemisphere, Kelvin waves propagate with the boundary on the right (left) with respect to the direction of propagation.
<div id="Lagrangian Coherent Structures"></div>
;'''Lagrangian Coherent Structures'''
: An invariant manifold which separates the flow of a time-varying system into regions of qualitatively distinct dynamics.  They will typically be time-dependent curves, and their definition requires a choice of integration time.  A full description of the construction of LCS is beyond the scope of this glossary.  The important point for an intuitive understanding is that LCS are time-dependent analogues of [[#Separatrix|separatrices]].
<div id="Lagrangian Measurement"></div>
;'''[[Eulerian and Lagrangian measurements|Lagrangian measurement]]'''
: A measurement taken while moving with a particle, i.e., while moving with the fluid. See also [[#Eulerian Measurement|Eulerian measurement]].
<div id="Large Scale Flow"></div>
<div id="Large Scale Flow"></div>
;'''Large Scale Flow'''  
;'''Large Scale Flow'''  
: In geophysical fluid dynamics this refers to the flow dominated by the Earth's rotation, so almost geostrophic flow.
: In geophysical fluid dynamics this refers to the flow dominated by the Earth's rotation, so almost geostrophic flow.


<div id="Large Eddy Simulation"></div>
<div id="LES"></div>
;'''Large Eddy Simulation'''  
;'''Large Eddy Simulation (LES)'''  
: A simulation in which a turbulence model has been included to approximate small scale motion and thus reduce the complexity of the problem.
: A simulation in which a turbulence model has been included to approximate small scale motion and thus reduce the complexity of the problem.
<div id="Material Derivative"></div>
;'''[[Material Derivative]]'''
: A [[#Lagrangian Measurement|Lagrangian quantity]], the material derivative describes the rate of change of a function from the perspective of a fluid particle moving with the flow.  If <math>f</math> is the function in question, the material derivative is defined as <math>\frac{Df}{Dt} = \frac{\partial f}{\partial t}+ \vec{u} \cdot \nabla f</math>.  This derivative has many names, including but not limited to advective derivative, hydrodynamic derivative, Lagrangian derivative, and total derivative.
<div id="Material Volume"></div>
;'''Material Volume'''
: A fixed piece of matter which moves with the flow. It is comprised of the same particles for all time.


<div id="Meridional"></div>
<div id="Meridional"></div>
;'''Meridional'''  
;'''Meridional'''  
: Along a north-south direction; or along a meridian.
: Along a north-south direction; or along a meridian.
<div id="Mixing"></div>
;'''[[Mixing]]'''
: This has the standard English language meaning, but there are many more connotations in fluid mechanics.
<div id="Navier-Stokes equations"></div>
;'''[[Navier-Stokes equations]]'''
: The equations of motion for a Newtonian fluid.


<div id="Nepheloid Layer"></div>
<div id="Nepheloid Layer"></div>
;'''Nepheloid Layer'''  
;'''[[Nepheloid Layer]]'''  
: A layer of water with a high concentration of suspended sediment which occurs near the deep ocean bottom, close to the base of the continental slope.
: A layer of water with a high concentration of suspended sediment.


<div id="Normal Mode"></div>
<div id="Normal Mode"></div>
Line 131: Line 243:
<div id="Phase Velocity"></div>
<div id="Phase Velocity"></div>
;'''Phase Velocity'''
;'''Phase Velocity'''
:The velocity at which a wave crest or trough propagates. Mathematically, <math>c=\frac{\omega}{|k|}\hat{k}</math> where <math>c</math> is the phase velocity, <math>|k|</math> is the magnitude of the wavenumber, and <math>\hat{k}</math> is the unit vector corresponding to <math>k</math>. cf. [[#Group Velocity|group velocity]].
:The velocity at which a wave crest or trough propagates. Mathematically, <math>\vec{c_p}=\frac{\omega}{|k|}\hat{k}</math> where <math>c</math> is the phase velocity, <math>|k|</math> is the magnitude of the wavenumber, and <math>\hat{k}</math> is the unit vector corresponding to <math>\vec{k}</math>. cf. [[#Group Velocity|group velocity]].  The magnitude of the phase velocity, the phase speed, is sometimes called the celerity of the wave.


<div id="Poincare Wave"></div>
<div id="Poincaré Map"></div>
;'''[[Poincaré Map]]'''
: If a [[#Dynamical System|dynamical system]] has a periodic orbit (in either time or space), the trajectories of the particles in that region can be studied using a Poincaré map.  Define a plane <math>S</math> which is normal to the flow, which we call a Poincaré section.  The Poincaré map is defined as a first return map: given <math>x \in S</math>, the flow will map <math>x</math> back onto <math>S</math> after one period, say to the point <math>x'</math>.  We then define the Poincaré map as <math>P(x) = x' </math>.  The Poincaré map, when it can be defined, allows us to study trajectories by studying sequences of points on a lower dimensional surface than the original system.
 
<div id="Poincaré Wave"></div>
;'''Poincaré Wave'''
;'''Poincaré Wave'''
: Gravity waves that are slow/large enough to feel the effects of rotation, but for which gravity remains the dominant restoring force.
: Gravity waves that are slow/large enough to feel the effects of rotation, but for which gravity remains the dominant restoring force. Under the linear [[Shallow water equations]] with an [[#f-plane|f-plane]], the dispersion relation for Poincaré waves is <math>\omega^2 = f_0^2 + c_0^2K^2</math>, where <math>f_0</math> is the Coriolis frequency, <math>c_0^2=gH</math> is the shallow-water gravity-wave speed, and <math>K=\frac{2\pi}{\lambda}</math> is the horizontal wavenumber.  


<div id="Pycnocline"></div>
<div id="Pycnocline"></div>
;'''Pycnocline'''  
;'''[[Stratification|Pycnocline]]'''  
: Region with a  high gradient in density.  See also [[#Halocline|halocline]] and [[#Thermocline|thermocline]].
: Region with a  high gradient in density.  See also [[#Halocline|halocline]] and [[#Thermocline|thermocline]].
<div id="Q and R"></div>
;'''[[Q and R]]'''
: Two invariants of the [[Velocity gradient tensor|velocity gradient tensor]] which are useful for defining the flow.


== R-Z ==
== R-Z ==
<div id="Reynolds Decomposition"></div>
;'''Reynolds Decomposition'''
:When studying turbulent flows, quantities can be decomposed into a mean part and a deviation from the mean (also called the turbulent fluctuations). For example, the horizontal velocity <math> u </math> is written as <math> u = \bar u + u^{\prime} </math>, where <math>\bar{u}</math> is the mean part and <math> u^{\prime}</math> is the fluctuation. The fluctuations have zero mean, i.e., <math> <u^{\prime}> = 0 </math>. Also, if <math> u </math> is stationary in time, then <math> \bar u </math> is the time average of <math> u </math>.
<div id="Rossby Wave"></div>
<div id="Rossby Wave"></div>
;'''Rossby Wave'''  
;'''Rossby Wave'''  
: Waves in which the dominant restoring force is to due the conservation of potential vorticity.
: Waves in which the dominant restoring force is to due the conservation of potential vorticity. These waves are a result of gradient in potential vorticity imparted by the latitudinal variation of the Coriolis frequency. Waves for which the gradient in potential vorticity is provided by topography are termed topographic Rossby waves.
 
<div id="Saltation"></div>
;'''Saltation'''
:One of the ways in which sediment in a bed-load layer may be transported. If the near-bed flow velocity exceeds some critical value, the sediment particles jump downstream and this is termed saltation. The particles stay close to the surface and move in smooth, approximately parabolic trajectories.
 
<div id="Separatrix"></div>
;'''[[Separatrix]]'''
: A curve in a time-invariant [[#Dynamical System|dynamical system]] which separates the phase space into regions of distinct dynamics.  Points which start on different sides of a separatrix will be separated as they follow their trajectories.  See also [[#Lagrangian Coherent Structures|Lagrangian coherent structures]].
 
<div id="Shallow water equations"></div>
;'''[[Shallow water equations]]'''
: The equations of motion for a fluid where horizontal scales are much larger than the depth.


<div id="Soliton"></div>
<div id="Soliton"></div>
;'''Soliton'''  
;'''Soliton'''  
:a single wave crest or trough that propagates at a constant speed, without changing its shape, even after interacting with other waves.
:A single wave crest or trough that propagates at a constant speed, without changing its shape, even after interacting with other waves.
 
<div id="Stable Manifold"></div>
;'''[[Stable and Unstable Manifolds|Stable Manifold]]'''
: The stable manifold of a [[#Fixed Point|fixed point]] is the set of all points which are attracted to the fixed point over time.  Mathematically, if <math>\vec{x_f}</math> is the fixed point, and <math>\phi_t</math> is the flow map for the [[#Dynamical System |dynamical system]], then if <math>\vec{x}</math> is a point on the stable manifold, then <math> \lim \limits_{t \to \infty} \phi_t(\vec{x})= \vec{x_f}</math>.  See also [[#Unstable Manifold|unstable manifold]].


<div id="Stratification"></div>
<div id="Stratification"></div>
;'''Stratification'''
;'''[[Stratification]]'''
: The way in which a fluids density varies with depth.  
: The way in which a fluids density varies with depth.  


Line 159: Line 299:


<div id="Thermocline"></div>
<div id="Thermocline"></div>
;'''Thermocline'''  
;'''[[Stratification|Thermocline]]'''  
: Region with a high gradient in temperature. See also [[#Halocline|halocline]] and [[#Pycnocline|pycnocline]]
: Region with a high gradient in temperature. See also [[#Halocline|halocline]] and [[#Pycnocline|pycnocline]]


Line 165: Line 305:
;'''Thermohaline Flow'''  
;'''Thermohaline Flow'''  
: Flow in the ocean due to density gradients which are caused by surface heat (thermal fluxes) and freshwater or saline fluxes.
: Flow in the ocean due to density gradients which are caused by surface heat (thermal fluxes) and freshwater or saline fluxes.
<div id="Tracer"></div>
;'''[[Tracer]]'''
: A quantity transported by the flow.
<div id="Turbulence Intensity"></div>
;'''Turbulence Intensity'''
: A quantity that gives a measure of the level or strength of turbulence. It is defined as the ratio of the root-mean-square of the turbulent velocity fluctuations, <math> u'_{rms} </math>, to the mean velocity, <math> \bar u </math>: Turbulence intensity  =  <math> u'_{rms}/\bar u </math>  = <math> \frac{\sqrt{\frac{1}{3}\left( {u'^2}_x + {u'^2}_y + {u'^2}_z \right)} }{\sqrt{ {{\overline{u}^2}_x} +  {{\overline{u}^2}_y} +  {{\overline{u}^2}_z}} }</math>
<div id="Unstable Manifold"></div>
;'''[[Stable and Unstable Manifolds|Unstable Manifold]]'''
: The unstable manifold of a [[#Fixed Point|fixed point]] is the set of all points which are repelled by the fixed point over time.  Mathematically, if <math>\vec{x_f}</math> is the fixed point, and <math>\phi_t</math> is the flow map for the [[#Dynamical System |dynamical system]], then if <math>\vec{x}</math> is a point on the unstable manifold, then <math> \lim \limits_{t \to -\infty} \phi_t(\vec{x})= \vec{x_f}</math>.  See also [[#Stable Manifold|stable manifold]].
<div id="Velocity gradient tensor"></div>
;'''Velocity gradient tensor'''
: A useful quantity in formulating fluid deformation and rotation rates. The strain rate tensor and rotation tensor are defined in terms of this quantity.
<div id="Viscosity"></div>
;'''Viscosity'''
: A quantity measuring the magnitude of internal friction. Dynamic viscosity: <math> \mu </math> (unit: <math>\text{kg}\cdot \text{m}^{-1} \text{s}^{-1}</math>). Kinematic viscosity: <math> \nu = \mu/\rho </math> (unit: <math>\text{m}^2/\text{s}</math>), also called [[#Diffusivity|momentum diffusivity]].
<div id="Viscous dissipation rate"></div>
;'''Viscous dissipation rate'''
: The irreversible rate of kinetic energy being converted into internal (or thermal) energy through viscosity. Often defined as <math>2\mu S_{ij}S_{ij}</math> where <math>S_{ij}</math> is the strain rate tensor.


<div id="Wave Envelope"></div>
<div id="Wave Envelope"></div>
;'''Wave Envelope'''
;'''[[Wave Envelope]]'''
: The curve which outlines the extremes of a Wave Packet. See [[#Wavepacket|Wave Packet]]  
: The curve which outlines the extremes of a [[#Wave Packet|wave packet]].
 
<div id="Wave equations"></div>
;'''[[Wave equations|Wave equation]]'''
: An equation to describe the vibrations of a medium.
 
<div id="Wavelet"></div>
;'''[[Wavelet]]'''
: A wavelet is a localized wave that are near zero outside of a specific region. Wavelets are used to decompose a given signal into frequency and time. Similar to a [[#Wave Packet|wave packet]].


<div id="Wave Packet"></div>
<div id="Wave Packet"></div>
;'''Wave Packet'''
;'''Wave Packet'''
: Composition of waves in a localized region acting in matrimony.
: A group of waves which travel together, often as the result of a burst of energy.  This leads to the concept of a [[#Wave Envelope|wave envelope]]. Also called a wave train.


<div id="Zonal"></div>
<div id="Zonal"></div>
;'''Zonal'''
;'''Zonal'''
: Along an east-west direction; or along a latitude circle.
: Along an east-west direction; or along a latitude circle.

Latest revision as of 10:26, 28 May 2018

Glossary of Terms for Fluid Dynamics

Add as you feel necessary. When needed, provide a link to a reference page or other terms.

Purpose: Many of the terms on this list have multiple definitions depending on context. The context for these definitions is geophysical and environmental fluid dynamics.

Disclaimer: this list is mostly the result of googling, and as such should not be referenced directly.

Note: The AMS Glossary is a good source for definitions, should the definition that you seek not be available below.

See also Dimensionless Numbers, and Fluids equations.


A-B

Added mass
The inertia added to a system due to the fact that an accelerating or decelerating body must move some volume of surrounding fluid with it as it moves.
Available Potential Energy
The potential energy available for conversion into other forms of energy. For intuitive purposes one may think of this as where is the density field, and is the redistributed density field such that the potential energy is minimized.
Azimuth
The horizontal angle. In polar coordinates, the angular direction.
Barotropic fluid
A fluid in which . This means that surfaces of constant pressure and constant density are parallel. Fluids in which or the density is constant () are necessarily barotropic.
Baroclinic motion
Motion caused by the misalignment of the surfaces of constant pressure and constant density (i.e. ).
Boundary Layer
Region near a boundary in which viscosity becomes important.
Buoyancy Frequency
The frequency at which an infinitesimally perturbed fluid parcel oscillates around its rest state in the absence of friction. Formally given as , where is the background stratification.
- plane
The - plane approximation assumes that the Coriolis frequency varies linearly with latitude i.e. . and where is the period of Earth's rotation, is the reference latitude, and is the mean radius of the Earth. Wikipedia's entry on this is a good one. cf. -plane

C-D

Capillary Wave
Waves in which the dominant restoring force is due to surface tension. Typical length scales are under 7cm (Kundu, 4th ed.).
Chaotic Advection
The advection of particles under the flow map of a chaotic dynamical system.
Characteristic Scale
This scale is context dependent. In an engineering situation like a jet out of a small hole one scale is given by the size of the hole, and another, less easily quantifiable scale will be the length over which the jet mixes with the ambient fluid.
Circulation
The circulation of a flow is the integral of the vorticity over a surface, , and represents the vorticity flux through the surface . By Stokes' theorem, .
Coriolis Frequency
, where is the rate of earth's rotation, and is the latitude (with the northern hemisphere to be positive). See also -plane, -plane.
Correlation Time
The time it takes for the auto correlation function of a process to decrease by a given amount.
Diffusivity
Rate of diffusion (unit: m/s). Mass diffusivity: the rate at which mass (molecular) of the substance diffuses through a unit surface in a unit time. Thermal diffusivity: , where is the thermal conductivity, is the density, and is the specific heat. Momentum diffusivity: see kinematic viscosity.
Direct Numerical Simulation (DNS)
Simulation in which you make no assumption on turbulence, and typically attempt to resolve as much as possible. cf. LES
Dispersion Relation
For a given wave, the dispersion relation is the relationship between wavenumber, , and (angular) wave frequency, ; typically written as . See also Phase Velocity, Group Velocity, and Dispersive Waves.
Dispersive Wave
A wave for which the group velocity is dependent on wavenumber, so energy in different wavelengths propagates at different velocities.
Dubreil-Jacotin-Long (DJL) equation
A scalar equation equivalent to the steady, incompressible, stratified Euler equations.
Dynamical System
A set of differential equations describing a classical mechanical system. There are many generalizations and formalisms associated with this concept, but physically the most important point is that the solution to a dynamical system is a time evolution function for the system. There are numerous examples of dynamical systems in classical mechanics, but simple examples include pendulums and predator prey models. See also fixed point, stable manifold, and unstable manifold.

E-G

Energy Cascade
When energy is supplied at large scales and then passed to smaller and smaller scales until viscosity causes dissipation.
Enstrophy
, the norm squared of the vorticity.
Euler equations
The Euler equations are the Navier-Stokes equations with the viscous term neglected.
Eulerian measurement
A measurement taken at a fixed position in space. See also Lagrangian measurement.
Fixed Point
For a time-invariant dynamical system , a fixed point is a point in the domain such that . Fixed points are important in the analysis of time-invariant systems because they define both stable and unstable manifolds, which divide the domain into regions of different dynamics.
Fluid Parcel
A volume of fluid which (see chapter 1 of Kundu.):
    • deforms under outside forces
    • is non-diffusive, such that molecules do not cross the boundary
    • is large enough to be well defined by thermodynamic quantities
    • is small enough to reach internal equilibrium much faster than the background flow
- plane
The - plane approximation assumes that the Coriolis frequency is constant in latitude i.e. . Where , is the period of Earth's rotation, and is the reference latitude. cf. -plane
Geostrophic Balance
Geostrophic balance occurs when the Coriolis pseudo-force balances the pressure forces. Under geostrophic balance, flow is along lines of constant pressure.
Gravity Wave
A wave in which the dominant restoring force is due to gravity acting to restore displaced mass.
Group Velocity
The velocity with which the energy of a wave packet propagates. Mathematically, the group velocity , where is the wavenumber. The group velocity is closely related to the velocity of a wave envelope .
Gyre
A vortex, a region dominated by a coherent rotating structure.

H-J

Halocline
Region with a high gradient in salinity. See also pycnocline and thermocline
Internal Solitary Wave
A single, internal, travelling wave, often located at a pycnocline.
Internal Tide
Internal waves generated at a tidal frequency.
Internal Wave
A wave whose displaced quantity is interior to the fluid. These can include gravity waves and Rossby waves.
Isentropic
Of or having constant entropy. An isentropic surface is a surface of constant entropy.
Isobaric
Of or having constant pressure. An isobaric surface is a surface of constant pressure.
Isohaline
Of or having constant salinity. An isohaline surface is a surface of constant salinity.
Isopycnal
Of or having constant density. An isopycnal surface is a surface of constant density.
Isothermal
Of or having constant temperature. An isothermal surface is a surface of constant temperature.

K-Q

Korteweg–de Vries (KdV) equation
A nonlinear equation used to describe long internal waves (among other things).
Kelvin Wave
Gravity waves that are boundary trapped and are in geostrophic balance in the direction orthogonal to the boundary. Kelvin waves require the presence of a boundary such as a coastline, channel wall, or the equator. Like Poincaré waves, Kelvin waves are rotating gravity waves. In the Northern (Southern) hemisphere, Kelvin waves propagate with the boundary on the right (left) with respect to the direction of propagation.
Lagrangian Coherent Structures
An invariant manifold which separates the flow of a time-varying system into regions of qualitatively distinct dynamics. They will typically be time-dependent curves, and their definition requires a choice of integration time. A full description of the construction of LCS is beyond the scope of this glossary. The important point for an intuitive understanding is that LCS are time-dependent analogues of separatrices.
Lagrangian measurement
A measurement taken while moving with a particle, i.e., while moving with the fluid. See also Eulerian measurement.
Large Scale Flow
In geophysical fluid dynamics this refers to the flow dominated by the Earth's rotation, so almost geostrophic flow.
Large Eddy Simulation (LES)
A simulation in which a turbulence model has been included to approximate small scale motion and thus reduce the complexity of the problem.
Material Derivative
A Lagrangian quantity, the material derivative describes the rate of change of a function from the perspective of a fluid particle moving with the flow. If is the function in question, the material derivative is defined as . This derivative has many names, including but not limited to advective derivative, hydrodynamic derivative, Lagrangian derivative, and total derivative.
Material Volume
A fixed piece of matter which moves with the flow. It is comprised of the same particles for all time.
Meridional
Along a north-south direction; or along a meridian.
Mixing
This has the standard English language meaning, but there are many more connotations in fluid mechanics.
Navier-Stokes equations
The equations of motion for a Newtonian fluid.
Nepheloid Layer
A layer of water with a high concentration of suspended sediment.
Normal Mode
For a linear PDE, the normal modes are the functions which describe the spatial structure of the standing waves that solve that PDE. We can then approximate any wave that solves the PDE, including non-standing waves, by using the normal modes as a basis.
Particle Image Velocimetry (PIV)
A non-invasive method used to measure whole velocity fields by taking two images shortly after each other, and calculating the distance individual tracer particles have travelled within the time interval. From the known time interval and the measured displacement, the velocity is calculated.
Phase Velocity
The velocity at which a wave crest or trough propagates. Mathematically, where is the phase velocity, is the magnitude of the wavenumber, and is the unit vector corresponding to . cf. group velocity. The magnitude of the phase velocity, the phase speed, is sometimes called the celerity of the wave.
Poincaré Map
If a dynamical system has a periodic orbit (in either time or space), the trajectories of the particles in that region can be studied using a Poincaré map. Define a plane which is normal to the flow, which we call a Poincaré section. The Poincaré map is defined as a first return map: given , the flow will map back onto after one period, say to the point . We then define the Poincaré map as . The Poincaré map, when it can be defined, allows us to study trajectories by studying sequences of points on a lower dimensional surface than the original system.
Poincaré Wave
Gravity waves that are slow/large enough to feel the effects of rotation, but for which gravity remains the dominant restoring force. Under the linear Shallow water equations with an f-plane, the dispersion relation for Poincaré waves is , where is the Coriolis frequency, is the shallow-water gravity-wave speed, and is the horizontal wavenumber.
Pycnocline
Region with a high gradient in density. See also halocline and thermocline.
Q and R
Two invariants of the velocity gradient tensor which are useful for defining the flow.

R-Z

Reynolds Decomposition
When studying turbulent flows, quantities can be decomposed into a mean part and a deviation from the mean (also called the turbulent fluctuations). For example, the horizontal velocity is written as , where is the mean part and is the fluctuation. The fluctuations have zero mean, i.e., . Also, if is stationary in time, then is the time average of .
Rossby Wave
Waves in which the dominant restoring force is to due the conservation of potential vorticity. These waves are a result of gradient in potential vorticity imparted by the latitudinal variation of the Coriolis frequency. Waves for which the gradient in potential vorticity is provided by topography are termed topographic Rossby waves.
Saltation
One of the ways in which sediment in a bed-load layer may be transported. If the near-bed flow velocity exceeds some critical value, the sediment particles jump downstream and this is termed saltation. The particles stay close to the surface and move in smooth, approximately parabolic trajectories.
Separatrix
A curve in a time-invariant dynamical system which separates the phase space into regions of distinct dynamics. Points which start on different sides of a separatrix will be separated as they follow their trajectories. See also Lagrangian coherent structures.
Shallow water equations
The equations of motion for a fluid where horizontal scales are much larger than the depth.
Soliton
A single wave crest or trough that propagates at a constant speed, without changing its shape, even after interacting with other waves.
Stable Manifold
The stable manifold of a fixed point is the set of all points which are attracted to the fixed point over time. Mathematically, if is the fixed point, and is the flow map for the dynamical system, then if is a point on the stable manifold, then . See also unstable manifold.
Stratification
The way in which a fluids density varies with depth.
Surface Wave
Waves in which the displaced quantity is a water-air interface. These can include gravity waves, Rossby waves, and capillary waves.
Thermocline
Region with a high gradient in temperature. See also halocline and pycnocline
Thermohaline Flow
Flow in the ocean due to density gradients which are caused by surface heat (thermal fluxes) and freshwater or saline fluxes.
Tracer
A quantity transported by the flow.
Turbulence Intensity
A quantity that gives a measure of the level or strength of turbulence. It is defined as the ratio of the root-mean-square of the turbulent velocity fluctuations, , to the mean velocity, : Turbulence intensity = =
Unstable Manifold
The unstable manifold of a fixed point is the set of all points which are repelled by the fixed point over time. Mathematically, if is the fixed point, and is the flow map for the dynamical system, then if is a point on the unstable manifold, then . See also stable manifold.
Velocity gradient tensor
A useful quantity in formulating fluid deformation and rotation rates. The strain rate tensor and rotation tensor are defined in terms of this quantity.
Viscosity
A quantity measuring the magnitude of internal friction. Dynamic viscosity: (unit: ). Kinematic viscosity: (unit: ), also called momentum diffusivity.
Viscous dissipation rate
The irreversible rate of kinetic energy being converted into internal (or thermal) energy through viscosity. Often defined as where is the strain rate tensor.
Wave Envelope
The curve which outlines the extremes of a wave packet.
Wave equation
An equation to describe the vibrations of a medium.
Wavelet
A wavelet is a localized wave that are near zero outside of a specific region. Wavelets are used to decompose a given signal into frequency and time. Similar to a wave packet.
Wave Packet
A group of waves which travel together, often as the result of a burst of energy. This leads to the concept of a wave envelope. Also called a wave train.
Zonal
Along an east-west direction; or along a latitude circle.