In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
, the baroclinity (often called baroclinicity) of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid. In
meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...
a baroclinic flow is one in which the density depends on both temperature and pressure (the fully general case). A simpler case,
barotropic
In fluid dynamics, a barotropic fluid is a fluid whose density is a function of pressure only. The barotropic fluid is a useful model of fluid behavior in a wide variety of scientific fields, from meteorology to astrophysics.
The density of most ...
flow, allows for density dependence only on pressure, so that the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
of the
pressure-gradient force
In fluid mechanics, the pressure-gradient force is the force that results when there is a difference in pressure across a surface. In general, a pressure is a force per unit area, across a surface. A difference in pressure across a surface the ...
vanishes.
Baroclinity is proportional to:
:
which is proportional to the sine of the angle between surfaces of constant
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
and surfaces of constant
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
. Thus, in a ''barotropic'' fluid (which is defined by zero baroclinity), these surfaces are parallel.
In Earth's atmosphere, barotropic flow is a better approximation in the
tropics
The tropics are the regions of Earth surrounding the Equator. They are defined in latitude by the Tropic of Cancer in the Northern Hemisphere at N and the Tropic of Capricorn in
the Southern Hemisphere at S. The tropics are also referred to ...
, where density surfaces and pressure surfaces are both nearly level, whereas in higher latitudes the flow is more baroclinic. These midlatitude belts of high atmospheric baroclinity are characterized by the frequent formation of
synoptic-scale
cyclone
In meteorology, a cyclone () is a large air mass that rotates around a strong center of low atmospheric pressure, counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere as viewed from above (opposite to an anti ...
s, although these are not really dependent on the baroclinity term ''per se'': for instance, they are commonly studied on
pressure coordinate iso-surfaces where that term has no contribution to
vorticity production.
Baroclinic instability
Baroclinic instability is a fluid dynamical instability of fundamental importance in the
atmosphere
An atmosphere () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A s ...
and in the
oceans
The ocean (also the sea or the world ocean) is the body of salt water that covers approximately 70.8% of the surface of Earth and contains 97% of Earth's water. An ocean can also refer to any of the large bodies of water into which the worl ...
. In the atmosphere it is the dominant mechanism shaping the
cyclone
In meteorology, a cyclone () is a large air mass that rotates around a strong center of low atmospheric pressure, counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere as viewed from above (opposite to an anti ...
s and
anticyclone
An anticyclone is a weather phenomenon defined as a large-scale circulation of winds around a central region of high atmospheric pressure, clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere as viewed from abov ...
s that dominate
weather
Weather is the state of the atmosphere, describing for example the degree to which it is hot or cold, wet or dry, calm or stormy, clear or cloudy. On Earth, most weather phenomena occur in the lowest layer of the planet's atmosphere, the ...
in mid-latitudes. In the ocean it generates a field of
mesoscale eddies
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid b ...
(100 km or smaller) that play various roles in oceanic dynamics and the transport of
tracer
Tracer may refer to:
Science
* Flow tracer, any fluid property used to track fluid motion
* Fluorescent tracer, a substance such as 2-NBDG containing a fluorophore that is used for tracking purposes
* Histochemical tracer, a substance used for tr ...
s.
Whether a fluid counts as ''rapidly rotating'' is determined in this context by the
Rossby number, which is a measure of how close the flow is to solid body rotation. More precisely, a flow in solid body rotation has
vorticity that is proportional to its
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
. The Rossby number is a measure of the departure of the vorticity from that of solid body rotation. The Rossby number must be small for the concept of baroclinic instability to be relevant. When the Rossby number is large, other kinds of instabilities, often referred to as inertial, become more relevant.
The simplest example of a stably stratified flow is an incompressible flow with density decreasing with height.
In a compressible gas such as the atmosphere, the relevant measure is the vertical gradient of the
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, which must increase with height for the flow to be stably stratified.
The strength of the stratification is measured by asking how large the vertical shear of the horizontal winds has to be in order to destabilize the flow and produce the classic
Kelvin–Helmholtz instability. This measure is called the ''
Richardson number The Richardson number (Ri) is named after Lewis Fry Richardson (1881–1953). It is the dimensionless number that expresses the ratio of the buoyancy term to the flow shear term:
:
\mathrm = \frac = \frac \frac
where g is gravity, \rho is de ...
''. When the Richardson number is large, the stratification is strong enough to prevent this shear instability.
Before the classic work of
Jule Charney
Jule Gregory Charney (January 1, 1917 – June 16, 1981) was an American meteorologist who played an important role in developing numerical weather prediction and increasing understanding of the general circulation of the atmosphere by devisin ...
and
Eric Eady
Eric Thomas Eady (5 September 1915 – 26 March 1966) was a British meteorology researcher and author of the Eady Model of baroclinic instability, modelling baroclinic generation of weather systems.
Eady was born in Ealing and attended Eal ...
on baroclinic instability in the late 1940s, most theories trying to explain the structure of mid-latitude eddies took as their starting points the high Rossby number or small Richardson number instabilities familiar to fluid dynamicists at that time. The most important feature of baroclinic instability is that it exists even in the situation of rapid rotation (small Rossby number) and strong stable stratification (large Richardson's number) typically observed in the atmosphere.
The energy source for baroclinic instability is the
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
in the environmental flow. As the instability grows, the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of the fluid is lowered.
In growing waves in the atmosphere, cold air moving downwards and equatorwards displaces the warmer air moving polewards and upwards.
Baroclinic instability can be investigated in the laboratory using a rotating, fluid filled
annulus
Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to:
Human anatomy
* ''Anulus fibrosus disci intervertebralis'', spinal structure
* Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
. The annulus is heated at the outer wall and cooled at the inner wall, and the resulting fluid flows give rise to baroclinically unstable waves.
The term "baroclinic" refers to the mechanism by which
vorticity is generated. Vorticity is the curl of the velocity field. In general, the evolution of vorticity can be broken into contributions from advection (as vortex tubes move with the flow),
stretching and twisting (as vortex tubes are pulled or twisted by the flow) and baroclinic vorticity generation, which occurs whenever there is a density gradient along surfaces of constant pressure. Baroclinic flows can be contrasted with
barotropic
In fluid dynamics, a barotropic fluid is a fluid whose density is a function of pressure only. The barotropic fluid is a useful model of fluid behavior in a wide variety of scientific fields, from meteorology to astrophysics.
The density of most ...
flows in which density and pressure surfaces coincide and there is no baroclinic generation of vorticity.
The study of the evolution of these baroclinic instabilities as they grow and then decay is a crucial part of developing theories for the fundamental characteristics of midlatitude weather.
Baroclinic vector
Beginning with the equation of motion for a frictionless fluid (the
Euler equations) and taking the curl, one arrives at the
equation of motion for the curl of the fluid velocity, that is to say, the
vorticity.
In a fluid that is not all of the same density, a source term appears in the
vorticity equation whenever surfaces of constant density (
isopycnic
An isopycnic surface is a surface of constant density inside a fluid. Isopycnic surfaces contrast with isobaric or isothermal surfaces, which describe surfaces of constant pressure and constant temperature respectively. Isopycnic surfaces are so ...
surfaces) and surfaces
of constant pressure (
isobaric surfaces) are not aligned. The
material derivative of the local vorticity is given by:
:
(where
is the velocity and
is the
vorticity,
is the pressure, and
is the density). The baroclinic contribution is the vector:
:
This vector, sometimes called the solenoidal vector, is of interest both in compressible fluids and in incompressible (but inhomogeneous) fluids. Internal
gravity wave
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
s as well as unstable Rayleigh–Taylor modes can be analyzed from the perspective of the baroclinic vector. It is also of interest in the creation of vorticity by the passage of shocks through inhomogeneous media, such as in the
Richtmyer–Meshkov instability The Richtmyer–Meshkov instability (RMI) occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which ...
.
Experienced divers are familiar with the very slow waves that can be excited at a
thermocline or a
halocline
In oceanography, a halocline (from Greek ''hals'', ''halos'' 'salt' and ''klinein'' 'to slope') is a Cline (hydrology), cline, a subtype of chemocline caused by a strong, vertical salinity gradient within a body of water. Because salinity (in c ...
, which are known as ''internal waves''. Similar waves can be generated between a layer of water and a layer of oil. When the interface between these two surfaces is not horizontal and the system is close to hydrostatic equilibrium, the gradient of the pressure is vertical but the gradient of the density is not. Therefore the baroclinic vector is nonzero, and the sense of the baroclinic vector is to create vorticity to make the interface level out. In the process, the interface overshoots, and the result is an oscillation which is an internal gravity wave. Unlike surface gravity waves, internal gravity waves do not require a sharp interface. For example, in bodies of water, a gradual gradient in temperature or salinity is sufficient to support internal gravity waves driven by the baroclinic vector.
References
Bibliography
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External links
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Fluid dynamics
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