Understanding Atmospheric Divergence and Vorticity Concepts

Atmospheric Science 4320 / 7320 Anthony R. Lupo

Day one • Let’s talk about fundamental Kinematic Concepts • In lab, we talked about divergence, which is a scalar quantity:

Day one • We can prove that divergence is the fractional change with time of some horizontal area A.

Day one/two • Then • finally:

Day two • We could extend the concept to 3-D and get:

Day two • Horizontal divergence in terms of a line integral, invoking Green’s theorem (2-D) and Gauss’ (3-D) theorem • Define area A (with tangential wind vectors)

Day two • We must also assume Green’s theorem holds defining a line integral:

Day two • Green’s theorem (where P = u and Q = v) • S is the oriented surface, or the position vector on the curve is R • thus ds = dr • F is a vector field in the normal direction on S, in our case V (which is tangent to the curve) where we consider:

Day two • So F is: The normal component is:

Day two • And we invoke Green’s theorem; • Now recall vector identity: Ax(BxC) = (AdotC)B - C(AdotB)

Day two • And see that: • And what is the second term on the RHS equal to????

Day two • In 3-D we invoke Gauss’s theorem: • Stoke’s Theorem

Day two • Recall Green’s theorem: • And,

Day two • Then

Day two • And;

Day two • And if you don’t understand this: • You might have a ….little …trouble…

Day two • Horizontal divergence in Natural Coordinates: • (s,n,z,t) • The Velocity in Natural Components: • and,

Day two • so, the horizontal divergence is: • aha, product rule! (which terms will drop out as 0?

Day two • Recall: • And;

Day two • Sooo, • A B

Day two • In the Above relationship • Term A refers to the speed divergence: • Term B is the directional divergence (Diffluence)

Day two • Speed Div

Day two • Diffluence (directional div)

Day two • In a typical synoptic situation, these terms tend to act opposite each other: • Confluence and speed increasing: • Diffluence and speed decreasing:

Day two • Alternative derivation of horizontal divergence in natural coordinates: • Then take derivative (product rule again:

Day two/three • If we “rotate” i and j (x and y) to coincide with s and n (s and n) then: • Thus,

Day two/three • and, • Then, the celebrated result!

Day three • We can also perform our simple-minded area analysis along the same lines:

Day three • Then • finally:

Day three • Divergence in the Large-Scale Meteorological Coordinate System: • The divergence refers to the cartesian coordinate which is an invariant coordinate. • On large-scales we need to take into account the curvature of the earth’s surface.

Day three • However, the earth is curved, thus all else being equal, if we move a large airmass (approximated as 2 - D)northward (southward), the area gets smaller (larger) implying convergence (divergence). • We can consider the parcel moving upward or downward also (the area or volume):

Day three • These discrepencies arise from the fact that the Earth is a sphere, and thus we cannot hold i, j, and k constant. • Recall we re-worked the Navier - Stokes equations to be valid on a spherical surface:

Day three • The equation

Day three • We can also define divergence: • Thus, for example (can you do the rest?)

Day three • then; • Recall, we defined expressions for:

Day three • thus, we get, for the divergence:

Day three • How important are these “correction terms” for each scale?

Day three • And; Then (this approximation is fine too!),

Day three • Orders of magnitude of Horizontal divergence and vertical motions:

Day three/four • Vorticity and Circulation/ unit area • Vorticity is a vector whose magnitude is directly proportional to the circulation/unit area of a plane normal to the vorticity vector. • Vorticity = Curl(V), ROT(V), or

Day four • We are primarily interested in the vertical component of vorticity due to circulations in the horizontal plane: • (xi) (eta) (zeta)

Day four • The vertical component: • This is called relative vorticity:

Day four • Circulation (Kelvin’s Theorem): • consider a closed curve S, and by definition: • or • R = the position vector • dr = change in the position vector

Day four • from Green’s Theorem: • then:

Day four • Thus, circulation per unit area: • We need to show that: • Recall, line integral definition:

Day four • Then,

Day four • And theeen,

Day four • So,

Day four • Thus, we get the vertical component of Vorticity: • Let’s Examine the vorticity on a sphere: • Vorticity:

Day four • Look at using ‘foil’, u-comp only:

Understanding Atmospheric Divergence and Vorticity Concepts