This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Baroclinity" – news ·newspapers ·books ·scholar ·JSTOR(September 2009) (Learn how and when to remove this message)

Influid dynamics, thebaroclinity (often calledbaroclinicity) of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid.^[1][2] Inmeteorology, a baroclinic flow is one in which the density depends on both temperature and pressure (the fully general case). A simpler case,barotropic flow, allows for density dependence only on pressure, so that thecurl of thepressure-gradient force vanishes.

Baroclinity is proportional to:

${\displaystyle \mathrm{\nabla }p\times \mathrm{\nabla }\rho }$

which is proportional to the sine of the angle between surfaces of constantpressure and surfaces of constantdensity. Thus, in abarotropic fluid (which is defined by zero baroclinity), these surfaces are parallel.^[3][4][5]

In Earth's atmosphere, barotropic flow is a better approximation in thetropics, where density surfaces and pressure surfaces are both nearly level, whereas in higher latitudes the flow is more baroclinic.^[6] These midlatitude belts of high atmospheric baroclinity are characterized by the frequent formation ofsynoptic-scalecyclones,^[7] although these are not really dependent on the baroclinity termper se: for instance, they are commonly studied onpressure coordinate iso-surfaces where that term has no contribution tovorticity production.

Before the classic work ofJule Charney andEric Eady on baroclinic instability in the late 1940s.^[8][9]

Baroclinic instability can be investigated in the laboratory using a rotating, fluid filled annulus. 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.^[10][11]

Beginning with the equation of motion for a frictionless fluid (theEuler equations) and taking the curl, one arrives at theequation of motion for the curl of the fluid velocity, that is to say, thevorticity.^{[citation needed]}

In a fluid that is not all of the same density, a source term appears in thevorticity equation whenever surfaces of constant density (isopycnic surfaces) and surfacesof constant pressure (isobaric surfaces) are not aligned. Thematerial derivative of the local vorticity is given by:^{[citation needed]}

${\displaystyle \frac{D\overrightarrow{\omega }}{Dt}\equiv \frac{\mathrm{\partial }\overrightarrow{\omega }}{\mathrm{\partial }t}+\left(\overrightarrow{u}\cdot \overrightarrow{\mathrm{\nabla }}\right)\overrightarrow{\omega }=\left(\overrightarrow{\omega }\cdot \overrightarrow{\mathrm{\nabla }}\right)\overrightarrow{u}-\overrightarrow{\omega }\left(\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{u}\right)+\underset{\text{baroclinic contribution}}{\underbrace{\frac{1}{{\rho }^2}\overrightarrow{\mathrm{\nabla }}\rho \times \overrightarrow{\mathrm{\nabla }}p}}}$

(where${\displaystyle \overrightarrow{u}}$ is the velocity and${\displaystyle \overrightarrow{\omega }=\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{u}}$ is thevorticity,^[12]${\displaystyle p}$ is the pressure, and${\displaystyle \rho }$ is the density). The baroclinic contribution is the vector:^[13]

${\displaystyle \frac{1}{{\rho }^2}\overrightarrow{\mathrm{\nabla }}\rho \times \overrightarrow{\mathrm{\nabla }}p}$

This vector, sometimes called the solenoidal vector,^[14] is of interest both in compressible fluids and in incompressible (but inhomogeneous) fluids. Internalgravity waves 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,^[15][16] such as in theRichtmyer–Meshkov instability.^{[17][citation needed]}

Experienced divers are familiar with the very slow waves that can be excited at athermocline or ahalocline, which are known asinternal 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.^{[citation needed]}

• Holton, James R. (2004). Dmowska, Renata; Holton, James R.; Rossby, H. Thomas (eds.).An Introduction to Dynamic Meteorology. International Geophysics Series. Vol. 88 (4th ed.). Burlington, MA:Elsevier Academic Press.ISBN 978-0-12-354015-7.
• Gill, Adrian E. (1982). Donn, William L. (ed.).Atmosphere-Ocean Dynamics. International Geophysical Series. Vol. 30. San Diego, CA:Academic Press.ISBN 978-0-12-283522-3.
• Pedlosky, Joseph (1987) [1979].Geophysical Fluid Dynamics (2nd ed.). New York:Springer-Verlag.ISBN 978-0-387-96387-7.
• Tritton, D.J. (1988) [1977].Physical Fluid Dynamics (2nd ed.). New York, NJ:Oxford University Press.ISBN 978-0-19-854493-7.
• Vallis, Geoffrey K. (2007) [2006]. "Vorticity and Potential Vorticity".Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge:Cambridge University Press.ISBN 978-0-521-84969-2.

• Fluid dynamics
• Atmospheric dynamics

• Articles with short description
• Short description is different from Wikidata
• Articles needing additional references from September 2009
• All articles needing additional references
• All articles with unsourced statements
• Articles with unsourced statements from May 2018