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Potential Vorticity Equation

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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.
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    EOSC 512 2019 6 Fundamental Theorems: Vorticity and Circulation In GFD, and especially the study of the large-scale motions of the atmosphere and ocean, we are particularly interested in the rotation of the fluid. As a consequence, again assuming the motions are of large enough scale to feel the e↵ects of (in particular, the di↵erential) rotation of the outer shell of rotating, spherical planets: The e↵ects of rotation play a central role in the general dynamics of the fluid flow. This means that vorticity (rotation or spin of fluid elements) and circulation (related to a conserved quantity...) play an important role in governing the behaviour of large-scale atmospheric and oceanic motions. This can give us important insight into fluid behaviour that is deeper than what is derived from solving the equations of motion (which is challenging enough in the first place). In this section, we will develop two such theorems and principles related to the conservation of vorticity and circulation that are particularly useful in gaining insight into ocean and atmospheric flows. 6.1 Review: What is vorticity? Vorticity was previously defined as: @uk !i = ✏ijk (6.1) @xj Or, in vector notation: ~! = ~u r⇥ ˆi ˆj kˆ @ @ @ = (6.2) @x @y @z uvw @w @v @u @w @v @u = ˆi + ˆj + ˆi @y − @z @z − @x @x − @y ✓ ◆ ✓ ◆ ✓ ◆ Physically, the vorticity is two times the local rate of rotation (or “spin”) of a fluid element. Here, it is important to distinguish between circular motion and the rotation of the element. Here, the fluid element moving from A to B on the circular path has no vorticity, while the fluid element moving from C to D has non-zero vorticity.
  • Chapter 7 Several Forms of the Equations of Motion

    Chapter 7 Several Forms of the Equations of Motion

    CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium, the equations of motion for compress- ible flow become the famous Navier-Stokes equations. In Cartesian coordinates, ,l , ------ +0------- ()lUi = ,t ,xi ,lUi , ------------- + -------- ()lUiU j + Pbij – lGi – ,t ,x j , .-------- ()2µSij – ()()23 µµ– v bijSkk = 0 . (7.1) ,x j ,l()ek+ , ------------------------ + ------- ()lUiht–g,()T ,xi – lGiUi – ,t ,xi , ------- ()2µU S – ()()23 µµ– b U S = 0 j ij v ij j kk ,xi The Navier-Stokes equations are the foundation of the science of fluid mechanics. With the inclusion of an equation of state, virtually all flow solving revolves around finding solutions of the Navier-Stokes equations. Most exceptions involve fluids where the relation between stress and rate-of-strain is nonlinear such as polymers, or where the equation of state is not very well understood (for example supersonic flow in water) or rarefied flows where the Boltzmann equation must be used to explicitly account for particle collisions. bjc 7.1 4/1/13 The momentum equation expressed in terms of vorticity The equations can take on many forms depending on what approximations or assumptions may be appropriate to a given flow. In addition, transforming the equations to different forms may enable one to gain insight into the nature of the solutions. It is essential to learn the many different forms of the equations and to become practiced in the manipulations used to transform them. 7.1.1 INCOMPRESSIBLE NAVIER-STOKES EQUATIONS If there are no body forces and the flow is incompressible, ¢ U = 0 the Navier-Stokes equations reduce to what is probably their most familiar form.
  • FLUID VORTICES FLUID MECHANICS and ITS Applicanons Volume 30

    FLUID VORTICES FLUID MECHANICS and ITS Applicanons Volume 30

    FLUID VORTICES FLUID MECHANICS AND ITS APPLICAnONS Volume 30 Series Editor: R. MOREAU MADYLAM Ecole Nationale Superieure d'Hydraulique de Grenoble BOlte Postale 95 38402 Saint Martin d'Heres Cedex, France Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advance­ ment. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particulary open to cross fertilisation with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological ~nd ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For a list of related mechanics titles, see final pages. Fluid Vortices Edited by SHELDON 1. GREEN Department ofMechanical Engineering, University ofBritish Colwnbia, Vancouver, Canada. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
  • A Potential Vorticity Theory for the Formation of Elongate Channels in River Deltas and Lakes

    A Potential Vorticity Theory for the Formation of Elongate Channels in River Deltas and Lakes

    University of Pennsylvania ScholarlyCommons Department of Earth and Environmental Departmental Papers (EES) Science 12-2010 A Potential Vorticity Theory for the Formation of Elongate Channels in River Deltas and Lakes Federico Falcini Douglas J. Jerolmack University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/ees_papers Part of the Environmental Sciences Commons, Geomorphology Commons, and the Sedimentology Commons Recommended Citation Falcini, F., & Jerolmack, D. J. (2010). A Potential Vorticity Theory for the Formation of Elongate Channels in River Deltas and Lakes. Journal of Geophysical Research: Earth Surface, 115 (F4), F04038-. http://dx.doi.org/10.1029/2010JF001802 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ees_papers/76 For more information, please contact [email protected]. A Potential Vorticity Theory for the Formation of Elongate Channels in River Deltas and Lakes Abstract Rivers empty into oceans and lakes as turbulent sediment-laden jets, which can be characterized by a Gaussian horizontal velocity profile that spreads and decays downstream because of shearing and lateral mixing at the jet margins. Recent experiments demonstrate that this velocity field controls river-mouth sedimentation patterns. In nature, diffuse jets are associated with mouth bar deposition forming bifurcating distributary networks, while focused jets are associated with levee deposition and the growth of elongate channels that do not bifurcate. River outflows from elongate channels are similar in structure to cold filaments observed in ocean currents, where high potential vorticity helps to preserve coherent structure over large distances. Motivated by these observations, we propose a hydrodynamic theory that seeks to predict the conditions under which elongate channels form.