FLUID ROTATION Circulation and Vorticity

Arbitrary blob of fluid rotating in a horizontal plane Circulation: A measure of the rotation within a finite element of a fluid

In meteorology, changes in circulation are associated with changes in the intensity of weather systems. We can calculate changes in circulation by taking the time derivative of the circulation: Circulation is a macroscopic measure of rotation of a fluid and is a seldom used quantity in synoptic meteorology and atmospheric dynamics.

Calculate the circulation within a small fluid element with area

The relative vorticity is the microscopic equivalent of macroscopic circulation

Consider an arbitrary large fluid element, and divide it into small squares. Sum circulations: common side cancels Make infinitesimal boxes: each is a point measure of vorticity and all common sides cancel

Consider an arbitrary large fluid element, and divide it into small squares. Fill area with infinitesimal boxes: each is a point measure of vorticity and all common sides cancel so that: The circulation within the area is the area integral of the vorticity

Understanding vorticity: A natural coordinate viewpoint Natural coordinates: s direction is parallel to flow, positive in direction of flow n direction is perpendicular to flow, positive to left of flow

Note that only the curved sides of this box will contribute to the circulation, since the wind velocity is zero on the sides in the n direction Denote the distance along the top leg as  s Denote the distance along the bottom leg as  s + d(  s) Denote the velocity along the bottom leg as V Use Taylor series expansion and denote velocity along the top leg as (negative because we are integrating counterclockwise) CALCULATE CIRCULATION Note that d (  s) =   n

CALCULATE VORTICITY

Shear

Flow curvature

Vorticity due to the earth’s rotation Consider a still atmosphere: Earth’s rotation R no motion along this direction

after some algebra and trigonometry……

3D relative vorticity vector Cartesian expression for vorticity Vertical component of vorticity vector (rotation in a horizontal plane Absolute vorticity (flow + earth’s vorticity) Absolute vorticity

The vorticity equation in height coordinates (1)(2) Expand total derivative

Rate of change of relative vorticity Following parcel Divergence acting on Absolute vorticity (twirling skater effect) Tilting of vertically sheared flow Gradients in force Of friction Pressure/density solenoids

Rate of change of relative vorticity Following parcel Divergence acting on Absolute vorticity (twirling skater effect) Tilting of vertically sheared flow Gradients in force Of friction Pressure/density solenoids

Rate of change of relative vorticity Following parcel Divergence acting on Absolute vorticity (twirling skater effect) Tilting of vertically sheared flow Gradients in force Of friction Pressure/density solenoids geostrophic wind Cold advection pattern m (or  ) large acceleration small m (or  ) small acceleration large Solenoid: field loop that converts potential energy to kinetic energy

Rate of change of relative vorticity Following parcel Divergence acting on Absolute vorticity (twirling skater effect) Tilting of vertically sheared flow Gradients in force Of friction Pressure/density solenoids Geostrophic wind = constant N-S wind component due to friction

The vorticity equation in pressure coordinates (1)(2) Expand total derivative

Local rate of change of relative vorticity Horizontal advection of absolute vorticity on a pressure surface Vertical advection of relative vorticity Divergence acting on Absolute vorticity (twirling skater effect) Tilting of vertically sheared flow Gradients in force Of friction The vorticity equation In English: Horizontal relative vorticity is increased at a point if 1) positive vorticity is advected to the point along the pressure surface, 2) or advected vertically to the point, 3) if air rotating about the point undergoes convergence (like a skater twirling up), 4) if vertically sheared wind is tilted into the horizontal due a gradient in vertical motion 5) if the force of friction varies in the horizontal. Solenoid terms disappear in pressure coordinates: we will work in P coordinate from now on