The
vorticity equation is an important
prognostic equation in the
atmospheric sciencesAtmospheric sciences is an umbrella term for the study of the atmosphere, its processes, the effects other systems have on the atmosphere, and the effects of the atmosphere on these other systems. Meteorology includes atmospheric chemistry and atmospheric physics with a major focus on weather...
. Vorticity is a vector, therefore, there are three components. The equation of vorticity (three components in the canonical form) describes the material derivative (that is, the local change due to local change with time and
advectionAdvection, in chemistry and engineering, is a transport mechanism of a substance or a conserved property with a fluid in motion. The fluid motion in advection is described mathematically as a vector field, and the material transported is typically described as a scalar concentration of substance,...
) of
vorticityVorticity is a concept used in fluid dynamics. In the simplest sense, vorticity is the tendency for elements of the fluid to "spin."More formally, vorticity can be related to the amount of "circulation" or "rotation" in a fluid.The average vorticity in a small region of fluid flow is equal to the...
, and thus can be stated in either
relative or
absolute form.
The more compact version is that for
absolute vorticity, component , using the pressure system:
Here, is
densityThe density of a material is defined as its mass per unit volume. The symbol of density is ρ .- Formula :Mathematically:where: is the density, is the mass, is the volume....
,
u,
v, and are the components of wind
velocityIn physics, velocity is the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
, and is the 2-dimensional (i.e.
The
vorticity equation is an important
prognostic equation in the
atmospheric sciencesAtmospheric sciences is an umbrella term for the study of the atmosphere, its processes, the effects other systems have on the atmosphere, and the effects of the atmosphere on these other systems. Meteorology includes atmospheric chemistry and atmospheric physics with a major focus on weather...
. Vorticity is a vector, therefore, there are three components. The equation of vorticity (three components in the canonical form) describes the material derivative (that is, the local change due to local change with time and
advectionAdvection, in chemistry and engineering, is a transport mechanism of a substance or a conserved property with a fluid in motion. The fluid motion in advection is described mathematically as a vector field, and the material transported is typically described as a scalar concentration of substance,...
) of
vorticityVorticity is a concept used in fluid dynamics. In the simplest sense, vorticity is the tendency for elements of the fluid to "spin."More formally, vorticity can be related to the amount of "circulation" or "rotation" in a fluid.The average vorticity in a small region of fluid flow is equal to the...
, and thus can be stated in either
relative or
absolute form.
The more compact version is that for
absolute vorticity, component , using the pressure system:
Here, is
densityThe density of a material is defined as its mass per unit volume. The symbol of density is ρ .- Formula :Mathematically:where: is the density, is the mass, is the volume....
,
u,
v, and are the components of wind
velocityIn physics, velocity is the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
, and is the 2-dimensional (i.e. horizontal-component-only)
delIn vector calculus, del is a vector differential operator represented by the nabla symbol: .Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember...
.
The terms on the RHS denote the positive or negative generation of
absolute vorticity by
divergenceIn vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the...
of air, twisting of the axis of rotation, and
baroclinityIn fluid dynamics, the baroclinity is a measure of the stratification in a fluid. In meteorology a baroclinic atmosphere is one for which the density depends on both the temperature and the pressure; contrast this with barotropic atmosphere, for which the density depends only on the pressure...
, respectively.
Fluid dynamics
The vorticity equation describes the evolution of the vorticity of a fluid element as it moves around. The vorticity equation can be derived from the conservation of momentum equation. In its general vector form it may be expressed as follows,
where, is the velocity vector, is the density, is th pressure, is the viscous stress tensor and is the body force term.
Equivalently in tensor notation,
where, we have used the
Einstein summation conventionIn mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulas...
, and is the
Levi-Civita symbolThe Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus. It is named after the Italian mathematician and physicist Tullio Levi-Civita.-Definition:In three dimensions, the Levi-Civita...
.
Physical Interpretation
- The term is the material derivative of the vorticity vector . It describes the rate of change of vorticity of a fluid particle (or in other words the angular acceleration of the fluid particle). This can change due to the unsteadiness in the flow captured by (the unsteady term) or due to the motion of the fluid particle as it moves from one point to another, (the convection term).
- The first term on the RHS of the vorticity equation, , describes the stretching or tilting of vorticity due to the velocity gradients. Note that this is a tensor with nine terms.
- The next term, , describes stretching of vorticity
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....
due to flow compressibility. Sometimes the negative sign is included in the term.
- The third term, is the baroclinic term. It accounts for the changes in the vorticity due to the intersection of density and temperature surfaces.
- , accounts for the diffusion of vorticity due to the viscous effects.
- provides for changes due to body forces.
Simplifications
- In case of conservative body forces
A conservative force is defined as a force with the following property: when a particle moves in any closed loop, the force acting along the path multiplied by the distance travelled always sums to zero....
, .
- For a barotropic fluid
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...
, . This is also true for a constant density fluid where .
- For inviscid fluids, .
Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to,
Alternately, in case of incompressible, inviscid fluid with conservative body forces,