Circulation and Vorticity Example: Rotation in the atmosphere – water vapor satellite animation Circulation – a macroscopic measure of rotation for a finite area of a fluid Vorticity – a microscopic measure of rotation at any point in a fluid Circulation is a scalar quantity while vorticity is a vector quantity. Circulation Theorem Circulation is defined as the line integral around a closed contour in a fluid of the velocity component tangent to the contour C ≡ U ⋅ dl = U cosαdl ∫ ∫ € By convention C is evaluated for counterclockwise integration around the contour. When will C be positive (negative)? Consider a fluid in solid Body rotation with angular velocity W. For a circular ring of fluid with radius R: V = WR dl = Rdl Then the circulation is: C = ∫ ΩRdl = 2πΩR2 What is the relationship Between€ C and angular momentum in this case? What is the relationship Between C and angular velocity in this case? C can be used to describe the rotation in cases where it is difficult to define an axis of rotation and thus angular velocity. How will C change over time? Take total derivative of C in an inertial reference frame: DaCa Da DaU a Dadl = ∫ U a ⋅ dl = ∫ ⋅ dl + ∫ U a ⋅ Dt Dt Dt Dt D U But: ∫ a a ⋅ dl is just the line integral of the acceleration of wind: Dt € D U 1 a a = − ∇p − ∇Φ, Dt ρ € where we have neglected the friction force, do not need to consider the Coriolis force for an inertial reference frame, and have expressed the € gravitational force as a gradient of the geopotential. dl Dadl Noting that: =U a then = DaU a and: Dt Dt Dadl 1 ∫ U a ⋅ = ∫ U a ⋅ DaU a = ∫ Da (U a ⋅U a ) = 0 Dt 2 € € Using this the change in aBsolute circulation following the motion is: € D C 1 a a = ∫ − ∇p ⋅ dl − ∫ ∇Φ⋅ dl Dt ρ The line integral of the gravitational force is given by: € − ∇Φ ⋅ dl = −gk ⋅ dl = −gdz = −dΦ = 0 ∫ ∫ ∫ ∫ Then the change in circulation is given by: € D C 1 1 a a = ∫ − ∇p ⋅ dl = ∫ − dp Dt ρ ρ What is the physical interpretation of each term in this equation? € The sea breeze circulation What are the physical mechanisms that lead to sea Breeze formation? Example: Calculate the change in circulation associated with a sea Breeze 1 For a barotropic fluid r = r(p) and − dp = 0 so: ∫ ρ D C a a = 0 Dt € and absolute circulation is conserved following the motion. € This is known as Kelvin’s circulation theorem. What is the relationship Between aBsolute and relative circulation? Ca – absolute circulation Ce – circulation due to the rotation of the Earth C – relative circulation Ca = Ce + C The circulation due to the rotation of the Earth is given by: C = U ⋅ dl , where e ∫ e U e = Ω ×r is the velocity due to the rotation of the Earth € Stokes’ theorem: € ∫ F ⋅ ds = ∫∫ (∇ × F ) ⋅ n dA, A where n is the unit vector normal to the area A and the direction of n is defined by the right-hand rule for counterclockwise integration of the line € integral. € € Using Stokes’ theorem we can rewrite our equation for Ce as: Ce = ∫ U e ⋅ dl = ∫∫ (∇ ×U e ) ⋅ n dA A € Using the vector identity: ∇ ×U = ∇ × Ω ×r = ∇ × Ω × R = Ω∇ ⋅ R = 2Ω ( e ) ( ) ( ) For a horizontal plane n is directed in the vertical direction and € ∇ ×U ⋅ n = 2Ω ⋅ n = 2Ωsin φ = f , ( e ) € where we have used Ω = Ωcosφj + Ωsin φk € Then: € , Ce = ∫∫ (∇ ×Ue )⋅ n dA = 2ΩsinφA = 2ΩAe A where Ae is the area projected onto the equatorial plane and: Ae = Asinf Then: C = Ca – Ce = Ca – 2WAsinf = Ca – 2WAe The relative circulation (C) is the difference Between the aBsolute circulation (Ca) and the circulation due to the rotation of the Earth (Ce). Taking D Dt of this equation gives: DC DC DA = a − 2Ω e Dt Dt Dt € DC dp DA = − − 2Ω e Dt ∫ ρ Dt € For a barotropic fluid this reduces to: DC DA = −2Ω e Dt Dt Integrating from an initial state (1) to a final state (2) gives: € C2 − C1 = −2Ω(Ae2 − Ae1) = −2Ω(A2 sin φ2 − A1 sin φ1) What is the physical interpretation of this result? € Vorticity Vorticity – a microscopic measure of rotation at any point in a fluid Vorticity is defined as the curl of the velocity ( ∇ ×V ) Absolute vorticity: ω a = ∇ ×U a € Relative vorticity: ω = ∇ ×U € i j k ∂ ∂ ∂ ω = ∇ ×U€ = ∂x ∂y ∂z u v w ' ∂w ∂v* ' ∂u ∂w* ' ∂v ∂u* = ) − ,i + ) − , j + ) − , k ( ∂y ∂z+ ( ∂z ∂x + ( ∂x ∂y+ It is typical to consider only the vertical component of the vorticity vector: € Vertical component of aBsolute vorticity: η ≡ k ⋅ ∇ ×U ( a ) ∂v ∂u Vertical component of relative vorticity: ζ ≡ k ⋅ (∇ ×U ) = − ∂x ∂y € Under what conditions will z be positive? € What is the relationship Between aBsolute and relative voriticity? Vertical component of planetary vorticity: k ⋅ ∇ ×U = 2Ωsin φ = f ( e ) Then: η = ζ + f € The absolute vorticitiy is equal to the sum of the relative and planetary vorticity. € What is the relationship Between vorticity and circulation? Consider the circulation for flow around a rectangular area dxdy: $ ∂v ' $ ∂u ' δC = uδx + & v + δx)δ y −& u + δy)δ x − vδy % ∂x ( % ∂y ( $ ∂v ∂u' δC = & − )δ xδy % ∂x ∂y( δC = ζδA δC ζ = δA € We can also relate circulation and vorticity using Stokes’ theorem and the definition of vorticity: C = ∫ U ⋅ dl C = ∫∫ (∇ ×U ) ⋅ n dA A C = ∫∫ ζdA A C = ζ A C ζ = A This indicates that the average vorticity is equal to the circulation divided by the area enclosed by the contour used in calculating the circulation. € For solid body rotation: C = 2ΩπR2 and € ζ = 2Ω In this case the vorticity (z) is twice the angular velocity (W) € Vorticity in Natural Coordinates From this figure the circulation is: % ∂V ( δC = V[δs + d(δs)] −'V + δn*δ s & n ) ∂ ∂V δC = Vd(δs) − δnδs ∂n € From the geometry shown in this figure: d(δs) = δβδn, where db is the angular change in wind direction This then gives: € ∂V δC = Vδβδn − δnδs ∂n & δβ ∂V ) δC = (V − +δ nδs ' δs ∂n * δC δβ ∂V = V − δnδs δs ∂n In the limit as δn,δs → 0: € δC ∂β ∂V = ζ = V − δnδs ∂s ∂n € ∂β 1 Noting that = then ∂s R € s V ∂V ζ = − R ∂n € s What is the physical interpretation of this equation? € Can purely straigthline flow have non-zero vorticity? Where would this occur in the real world? Can curved flow have zero vorticity? Potential Vorticity Kelvin’s circulation theorem states that circulation is constant in a barotropic fluid. Can we expand this result to apply for less restrictive conditions? Density can Be expressed as a function of p and q using the ideal gas law and the definition of q. R cp # & R cp p # ps & p ρ = and θ = T% ( ⇒ T = θ% ( RT $ p ' $ ps ' € € Then: R cp (1−R cp ) R cp cv cp R cp pps p ps p ps ρ = R c = = Rθp p Rθ Rθ On an isentropic surface r is only a function of p and the solenoidal term is given by: € dp (1−cv cp ) ∝ dp = 0 ∫ ρ ∫ Then for adiabatic motion a closed chain of fluid parcels on an isentropic surface satisfies Kelvin’s circulation theorem: € DC D(C + 2ΩδAsin φ) a = = 0 Dt Dt For an approximately horizontal isentropic surface: € C ≈ ζθ δA, where zq is the relative vorticity evaluated on an isentropic surface. € Kelvin’s circulation theorem is then given by: D(C + 2ΩδAsin φ) D[δA(ζθ + 2Ωsin φ)] D[δA(ζθ + f )] = = = 0 Dt Dt Dt and € δA(ζθ + f ) = Const. following the air parcel motion € Consider an air parcel of mass dM confined between two q surfaces: δp From the hydrostatic equation = −ρg and δM = ρδV = ρδzδA: δz δp δM = − δA g € € δp For dM conserved following the air parcel motion δM = − δA = Const. g € Then: δMg % δθ(% δMg( % δθ€( δA = − = ' − *' * = (Const.)g' − * δp & δp)& δθ ) & δp) δM where Const. = δθ € Using this expression to replace dA in δA(ζθ + f ) = Const. gives: € % δθ( (Const.)g' − *( ζθ + f ) = Const. p & δ ) € € Ertel’s Potential Vorticity (P) is defined as: ' δθ* P ≡ (ζθ + f )) −g , ( δp+ Ertel’s potential vorticity has units of K kg-1 m2 s-1 and is typically given as a potential vorticity unit (PVU = 10-6 K kg-1 m2 s-1) € What is the sign of P in the Northern hemisphere? For adiabatic, frictionless flow P is conserved following the motion. The term potential vorticity is used for several similar quantities, but in all cases it refers to a quantity which is the ratio of the absolute vorticity to the vortex depth. δp For Ertel’s potential vorticity the vortex depth is given by − δθ How will ζθ + f change if the vortex depth increases (decreases)? € Potential vorticity for an incompressiBle fluid €For an incompressible fluid r is constant and the mass, M, is given by: M = rhdA, where h is the depth of the fluid being considered M Rearranging gives: δA = ρh ζ + f And the potential vorticity is given By: h € As aBove, the potential voriticity is constant for frictionless flow following the air parcel motion. € For a constant depth fluid: ζ + f = η = Const. (aBsolute vorticity is conserved following the motion) This puts a strong constraint on the types of motion that are possible. € Consider a zonal flow that has z0 = 0 initially so h0 = f0 Since ζ + f = η = Const., then at some later time z + f = h0 = f0 For a flow that turns to the north f > f0 and z = f0 – f < 0 € For a flow that turns to south f < f0 and z = f0 – f > 0 Can these conditions Be satisfied for Both easterly and westerly flow? For a fluid in which the depth can vary potential vorticity, rather than absolute voriticity, is conserved.