Frontogenesis, in meteorology, refers to the formation or strengthening of an atmospheric front

Weather front

A weather front is a boundary separating two masses of air of different densities, and is the principal cause of meteorological phenomena. In surface weather analyses, fronts are depicted using various colored lines and symbols, depending on the type of front...

. During frontogenesis, the temperature gradient tightens and as a result the thermal wind

Thermal wind

The thermal wind is a vertical shear in the geostrophic wind caused by a horizontal temperature gradient. Its name is a misnomer, because the thermal wind is not actually a wind, but rather a wind shear.- Physical Intuition :...

 becomes imbalanced. To maintain balance, the geostrophic wind

Geostrophic wind

The geostrophic wind is the theoretical wind that would result from an exact balance between the Coriolis effect and the pressure gradient force. This condition is called geostrophic balance. The geostrophic wind is directed parallel to isobars . This balance seldom holds exactly in nature...

 aloft and below adjust, such that regions of divergence/convergence form. Mass continuity

Continuity equation

A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...

 would require a vertical transport of air along the cold front where there is divergence (lowered 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 :...

). Although this circulation is described by a series of processes, they are actually occurring at the same time, observable along the front as a thermally direct circulation. There are several factors that influence the final shape and tilt of the circulation around the front, ultimately determining the type and location of clouds and precipitation.

3-Dimensional Equation

The three-dimensional form of the frontogenesis equation is


where each dimension begins with a diabatic

Diabatic

A diabatic process is one in which heat transfer takes place, which is the opposite of an adiabatic process. In quantum chemistry, the potential energy surfaces are obtained within the adiabatic or Born-Oppenheimer approximation...

 term; in the direction



in the direction



and in the direction

.

The equation also includes horizontal and vertical deformation

Deformation (meteorology)

Deformation is the rate of change of shape of fluid bodies. Meteorologically, this quantity is very important in the formation of atmospheric fronts, in the explanation of cloud shapes, and in the diffusion of materials and properties.-Equations:...

 terms; in the direction



and in the direction



and in the vertical direction

.

The final terms are the tilting term and the vertical divergence

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

 term; the tilting term is represented in the three-dimensional frontogenesis equation in the and directions





and the vertical divergence

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

term is present as