Chapter 7: Circulation and Vorticity

Transcription

1 Chapter 7: Circulation and Vorticity Circulation C = u ds Integration is performed in a counterclockwise direction C is positive for counterclockwise flow!!! Kelvin s Circulation Theorem The rate of change of circulation can be expressed as: DC = Du ds = 1 ρ p s ds + Φ s ds + friction What is the magnitude of each term in this equation? For a barotropic fluid (density is a function only of pressure): 1 ρ p ( ) p s ds = 1 ρ p & 1 = ( ' ρ p 2 ( ) dp ( ) 1 ( ) ρ p 1 = 0 around a closed circuit ) + *

2 Geopotential term: Φ ds = dφ s = [ Φ 2 Φ 1 ] = 0 around a closed circuit In a frictionless (inviscid) flow the friction term is also zero. In a barotropic, inviscid fluid the circulation is constant!!! Bjerknes Circulation theorem DC = 0 Changes in circulation can arise due to friction or baroclinicity. The sea breeze circulation How does the temperature over the land and over the ocean vary during the course of a day? What impact does this have on the thickness of an atmospheric column? How does the atmosphere respond to this horizontal variation in thickness? Example: Calculating the circulation associated with a sea breeze

3 Relative circulation Circulation in the atmosphere arises due to our rotating frame of reference. C absolute = C earth + C relative where, C earth = u earth ds RΩ 2πRsin φ C earth 2πΩR 2 sin φ If C absolute is conserved what does this imply about changes in C relative for meridional flow? Vorticity For solid body rotation: ζ = C A ζ = δc δa = 2πδrV πδr 2 = 2V δr = 2ω The earth undergoes solid body rotation with an angular velocity of ω = Ωsin φ, so: ζ earth = 2Ωsin φ = f What is the sign of ζ earth in the Northern and Southern hemispheres?

4 The circulation around ABCD can be calculated as: C = u ds = ( udx + vdy) δc = C AB + C BC + C CD + C DA $ = uδx + & v + v % x δx ' $ ) δy & u + u ( % y δy ' ) δx vδy ( = v u δxδy x y δyδx $ = v x u ' & ) δa % y( Then the relative vorticity, ζ, is given by: ζ = δc δa ζ = v x u y What is the sign of ζ for clockwise and counterclockwise flow? What does this imply about the sign of ζ for flow around low and high pressure centers? Example: Calculation of relative vorticity from a weather map

5 Conservation of Potential Vorticity What conditions were required for constant circulation according to Kelvin s circulation theorem? On a constant potential temperature, θ, surface: ρ = p RT = p $ & Rθ % p 0 p R ' d c p $ R ) = p c v c p p d c p 0 ( & % Rθ ' ) ( This is analogous to a barotropic fluid. Therefore, on a constant potential temperature surface the pressure gradient term is zero and Kelvin s circulation theorem is satisfied. This implies that: DC absolute C absolute = A θ = 0 ( ζ θ + f )da C ζ θ + f = lim absolute δa 0 δa δa( ζ θ + f ) = constant Consider an air parcel that is confined between two potential temperature surfaces, θ and θ + δθ, separated by pressure interval δp. The motion of this air parcel will be adiabatic. The mass of the parcel is given by: δm = ρδzδa = ( δp g)δa and must be conserved following the motion.

6 This gives: δa = δmg δp δmg δp δθ δθ δa = δmg δθ ' δθ * ), ( δp+ δa = constant g δθ δp Combining this result with δa( ζ θ + f ) = constant gives an expression for Rossby-Ertel potential vorticity, P: & constant g δθ ) ( + ζ θ + f ' δp* What does the δθ δp term represent? ( ) = constant P = g δθ δp ζ θ + f ( ) = constant Potential vorticity depends on the depth of the fluid and the absolute vorticity. Example: Conservation of potential vorticity and flow over the Rocky Mountains

7 Air column depth Change in δθ δp Increase δp Decrease δp Increase δp Decrease (return to original value) Decrease Increase Decrease Increase (return to original value) Change in ζ + f Increase Decrease Increase Decrease Sign of ζ Positive Negative Positive Negative Resulting northward southward lee side trough southward motion Change in f Increase Decrease Increase Decrease For westerly flow across a mountain range a lee wave will form downstream of the mountain. Stretching a column of the atmosphere results in generation of cyclonic vorticity. Shrinking of a column of the atmosphere results in generation of anticylonic vorticity.

8 The Vorticity Equation Using Navier-Stokes equations scaled for mid-latitude weather systems we can derive an equation for the time rate of change of vorticity. D h u h = 1 % p d ' ρ & x D h u y (1) D h x (2) D h (2)- (1) - D h i + p d y ( j * fk ) u h = 1 p d ρ x + fv (1) D h v = 1 p d ρ y % u( ' * = 1 2 p d ρ x y + f v y + v f y % v( ' * = 1 2 p d & x) ρ x y f u x u f x (1) - (2)- % v x u ( % u ' * = f x + v ( ' * u f x v f y D hζ + u f x + v f y = f % u x + v ( ' * ζ t + u ζ x + v ζ y + u f x + v f y = f % u x + v ( ' * fu (2) What causes the relative vorticity, at a fixed location, to change in time? How does vorticity change for a non-divergent flow? For quasi-geostrophic flow: D g % u ( ζ g + f ) = f a x + v ( a ' * & y ) D g ( ζ g + f ) = f w z What does this equation tell us about changes in relative vorticity in a quasi-geostrophic framework?