Baroclinity

In fluid dynamics

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, the baroclinity (sometimes 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

Meteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...

 a baroclinic atmosphere is one for which the density depends on both the temperature and the pressure; contrast this with barotropic

Barotropic

In meteorology, a barotropic atmosphere is one in which the pressure depends only on the density and vice versa, so that isobaric surfaces are also isopycnic surfaces . The isobaric surfaces will also be isothermal surfaces, hence the geostrophic wind is independent of height...

 atmosphere, for which the density depends only on the pressure. In atmospheric terms, the barotropic zones of the Earth are generally found in the central latitudes, or tropics

Tropics

The tropics is a region of the Earth surrounding the Equator. It is limited in latitude by the Tropic of Cancer in the northern hemisphere at approximately  N and the Tropic of Capricorn in the southern hemisphere at  S; these latitudes correspond to the axial tilt of the Earth...

, whereas the baroclinic areas are generally found in the mid-latitude/polar regions.

Baroclinity is proportional to


which is proportional to sine of the angle between surfaces of constant pressure

Pressure

Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

 and surfaces of constant density

Density

The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

. Thus, in a barotropic fluid (which is defined by zero baroclinity), these surfaces are parallel.

Areas of high atmospheric baroclinity are characterized by the frequent formation of cyclone

Cyclone

In meteorology, a cyclone is an area of closed, circular fluid motion rotating in the same direction as the Earth. This is usually characterized by inward spiraling winds that rotate anticlockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere of the Earth. Most large-scale...

s.

Baroclinic instability

Baroclinic instability is a fluid dynamical instability of fundamental importance in the atmosphere

Atmosphere

An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...

 and in the oceans. In the atmosphere it is the dominant mechanism shaping the cyclone

Cyclone

In meteorology, a cyclone is an area of closed, circular fluid motion rotating in the same direction as the Earth. This is usually characterized by inward spiraling winds that rotate anticlockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere of the Earth. Most large-scale...

s and anticyclone

Anticyclone

An anticyclone is a weather phenomenon defined by the United States' National Weather Service's glossary as "[a] large-scale circulation of winds around a central region of high atmospheric pressure, clockwise in the Northern Hemisphere, counterclockwise in the Southern Hemisphere"...

s that dominate weather

Weather

Weather is the state of the atmosphere, to the degree that it is hot or cold, wet or dry, calm or stormy, clear or cloudy. Most weather phenomena occur in the troposphere, just below the stratosphere. Weather refers, generally, to day-to-day temperature and precipitation activity, whereas climate...

 in mid-latitudes. In the ocean it generates a field of mesoscale

Mesoscale meteorology

Mesoscale meteorology is the study of weather systems smaller than synoptic scale systems but larger than microscale and storm-scale cumulus systems. Horizontal dimensions generally range from around 5 kilometers to several hundred kilometers...

 (100 km or smaller) eddies

Eddy (fluid dynamics)

In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid flows past an obstacle. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object...

 that play various roles in oceanic dynamics and the transport of tracer

Tracer

Tracer may refer to:* Histochemical tracer, a substance used for tracing purposes in histochemistry, the study of the composition of cells and tissues...

s. Baroclinic instability is a concept relevant to rapidly rotating, strongly stratified fluids.

Whether a fluid counts as rapidly rotating is determined in this context by the Rossby number

Rossby number

The Rossby number, named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing fluid flow. The Rossby number is the ratio of inertial to Coriolis force, terms v\cdot\nabla v\sim U^2 / L and \Omega\times v\sim U\Omega in the Navier–Stokes equations, respectively...

, 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

Angular velocity

In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

. 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

Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

, which must increase with height for the flow to be stably stratified.
One measures the strength of the stratification 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

Kelvin-Helmholtz instability

The Kelvin–Helmholtz instability, after Lord Kelvin and Hermann von Helmholtz, can occur when velocity shear is present within a continuous fluid, or when there is sufficient velocity difference across the interface between two fluids. One example is wind blowing over a water surface, where the...

. This measure is the Richardson number. When the Richardson number is large, the stratification is strong enough to prevent this shear instability.

Before the classic work of Jule Charney and Eric Eady

Eric Eady

Eric Eady was a British meteorology researcher and author of the Eady Model of baroclinic instability, which give rise to weather systems. Eady received a BSc in mathematics and in 1937 he become a weather forecaster in the UK Meteorological Office. In 1946 he started a PhD in mathematics at the...

 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

Potential energy

In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

 in the environmental flow. As the instability grows, the center of mass

Center of mass

In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

 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 (mathematics)

In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...

. 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

Vortex stretching

In fluid dynamics, vortex stretching is the lengthening of vortices in three-dimensional fluid flow, associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum....

 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

Barotropic

In meteorology, a barotropic atmosphere is one in which the pressure depends only on the density and vice versa, so that isobaric surfaces are also isopycnic surfaces . The isobaric surfaces will also be isothermal surfaces, hence the geostrophic wind is independent of height...

 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 fluid (say, the Euler equations or the Navier-Stokes equations

Navier-Stokes equations

In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...

) and taking the curl, one arrives at the equation of motion for the curl of the fluid velocity

Vorticity equation

The vorticity equation is an important prognostic equation in the atmospheric sciences. Vorticity is a vector, therefore, there are three components...

, 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

Vorticity equation

The vorticity equation is an important prognostic equation in the atmospheric sciences. Vorticity is a vector, therefore, there are three components...

 whenever surfaces of constant density (isopycnic

Isopycnic

Isopycnic means "of the same density." In particular, an isopycnic surface is a surface of constant density. This term is a bit more obscure than the similar terms isobaric or isothermal surfaces, which describe surfaces of constant pressure and constant temperature respectively...

 surfaces) and surfaces
of constant pressure (isobaric

Isobaric

Isobaric may refer to:*in thermodynamics, an isobaric process, i.e. one that is carried out at constant pressure;...

 surfaces) are not aligned. The material derivative of the local vorticity is given by


where is the velocity and is the vorticity, is pressure, and is density). The baroclinic contribution is the vector



This vector is of interest both in compressible fluids and in incompressible (but inhomogenous) fluids. Internal gravity wave

Gravity wave

In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media which has the restoring force of gravity or buoyancy....

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 inhomogenous media, such as in the Richtmeyer-Meshkov instability.

Divers may be familiar with the very slow waves that can be excited at a thermocline or a halocline; these are 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.