Circulation and Vorticity

Circulation refers to the tendency for a chain of air parcels in the atmosphere to rotate cyclonically (counterclockwise in the northern hemisphere). If an area of air is of interest, you would use circulation to address the problem.

Vorticity referes to the tendency for the wind shear around a point to produce cyclonic rotation. If a point in the atmosphere is of interest, you would use vorticity to address the problem.

Circulation Formally defined as the path integral of the tangential velocity around an air parcel (or closed chain of fluid elements):

v v v =•≡ cosα dlVldVC (1) ∫ t ∫ t (a) (b) (c) (d)(e) where: a) Vt is the tangential velocity at any point along the path b) dl is an increment along the curve, or part of the distance around it c) |Vt| is the magnitude of the tangential velocity d) α is the angle between the tangential velocity vector and the dl vector e) |dl| is the magnitude of the dl vector

Circulation Theorem

The same forces that cause horizontal and vertical accelerations (PGF, Coriolis force, gravitational accelerations, friction) also cause changes in circulation. They are related by the circulation theorem:

DC −= ρ −1dp (2) Dt ∫ The term on the left hand side is the total change of circulation (C) with time. The term on the right hand side is called the solenoidal term. Essentially, it represents the effect that the pressure gradient force has on the circulation. If the atmosphere is barotropic (density is a function of pressure only), then the solenoidal term = 0.

A solenoid can be thought of as a parallelogram formed by intersecting lines of constant density and pressure:

This situation arises when, for example, a sea breeze forms due to temperature (and thus density) differences between land and water areas. The warm land creates warmer, and less dense air, in the column above.

Vorticity

Vorticity is more commonly used in meteorology. It is the microscopic measure of rotation in a fluid around a point. Mathematically, vorticity is defined as the curl of the velocity vector:

v ωa ×∇≡ Va (3)

The ‘a’ subscripts in the equation above (3) refer to the “absolute” reference frame, or absolute vorticity. The absolute vorticity includes the spin due to the Earth’s rotation. Relative vorticity does not include the spin of the Earth, and is calculated as:

v ω ×∇≡ V (4)

In Cartesian coordinates, the total vorticity is:

⎛ ∂w ∂v ∂u ∂w ∂v ∂u ⎞ ω = ⎜ − − ,, − ⎟ (5) ⎝ ∂y ∂z ∂z ∂x ∂x ∂v ⎠

Although the horizontal component of vorticity can be important in a lot of meteorological applications (e.g. tornado formation, boundary layer rolls), the vertical component of vorticity is most commonly applied in large-scale dynamics. The vertical component of absolute vorticity (η) and relative vorticity (ς) are thus defined as:

v v v v η ( ×∇•= Vk a ), ζ ( ×∇•= Vk ) (6)

This is the same expression that you derived on previous homework and exam problems. For hereafter, any mention of absolute and relative vorticities are considered to be the vertical component unless otherwise specified.

Interpretation of Vorticity Signs

Positive relative vorticity is associated with cyclonic (counterclockwise) circulation in the Northern Hemisphere.

Negative relative vorticity is associate with anti-cyclonic (clockwise) circulation in the Northern Hemisphere.

Consider the following contour map of relative vorticity:

The regions of positive vorticity (centered over northeast Nebraska and the Hudson Bay) represent the centers of mid-latitude cyclones. The centers of negative relative vorticity (Colorado and the Gulf of Mexico) represent High pressure centers. This can be confirmed by the surface map below from the same time period:

Planetary Vorticity

Planetary vorticity is the difference between absolute and relative vorticity. It is the local vertical component of the vorticity of the earth due to its rotation:

v v Vk e sin2 φ ≡Ω=×∇• f (7)

Thus, the relative vorticity in Cartesian coordinates is:

∂v ∂u ζ = − (8) ∂x ∂y

and the absolute vorticity in Cartesian coordinates is:

∂v ∂u η = − + f (9) ∂x ∂y

As seen from the figure below, the absolute vorticity is dominated by the planetary rotation in the mid-latitudes:

Vorticity in Natural Coordinates

A physical interpretation of relative vorticity is most easily obtained by looking at vorticity in natural coordinates. Recall the unit vectors of natural coordinates:

t - Oriented parallel to the horizontal velocity at each point n - Oriented perpendicular to the horizontal velocity and pointing positively to the left of the flow direction k - Directed vertically upward

v If we have a wind vector V and a radius of curvature Rs, the relative vorticity can be written as:

v ∂V V ζ −= + (10) ∂n Rs (a) (b)

The terms on the RHS can be interpreted as follows: a) Rate of change of the wind speed normal to the direction of the flow. This is called the shear vorticity. Thus, vorticity can be created even for straight line flow. Consider the case of a jet stream maximum in the upper troposphere:

A cyclonic (positive relative vorticity) circulation develops north of the jet maximum, while and anti-cyclonic (negative relative vorticity) circulation develops south of the jet.

b) The turning of the wind along a streamline. This is called the curvature vorticity. Note that curved flow could have a net zero relative vorticity if the shear vorticity and curvature vorticity are equal and opposite of one another.