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6 Fundamental Theorems: Vorticity and Circulation

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6 Fundamental Theorems: Vorticity and Circulation 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. In a similar vein, it is important to keep in mind the distinction between vorciticity (local rotation of fluid elements) and the curvature of streamlines (motion of the flow in circular-like orbits). Page 58 EOSC 512 2019 Vorticity in a rotating fluid: In a rotating fluid, recall: (~u ) =(~u ) + ⌦~ ~r (6.3) inertial rotating ⇥ The vorticity associated with the velocity in an inertial frame is related to the vorticity in a rotating frame by: ~ (~! )inertial =(~! )rotating + ⌦ ~r r⇥ ⇥ (6.4) ~ ⇣ ⌘ =(~! )rotating +2⌦ Thus, the vorticity in the inertial frame is equal to vorticity seen in the rotating frame (relative vorticity)plus the vorticity of the velocity due to the frame’s rotation. 6.2 Circulation Circulation is defined for any vector field J~ around some closed curve C as Γ = J~ d~x J~ · I C (6.5) = Ji dxi CI Where d~x is the di↵erential line element vector along C. By convention, the contour is taken in the counter-clockwise sense. J~ d~x implies that the circulation involves the component of J tangent to the curve. · If J~ is taken to be the velocity vector, we call this the circulation and denote Γ~u =Γ. Γ= ~u d~x (6.6) · CI The flow’s circulation is closely related to its vorticity in an integral sense. To see this, we can use Stoke’s theorem to rewrite this line integral in the form of an area integral involving the curl of the vector field dotted with the area’s normal vector Γ= ~u d~x Applying Stoke’s theorem · CI = ( ~u ) nˆ dA r⇥ · (6.7) VZ = ~! nˆ dA · AZ ~! nˆ dA is sometimes referred to as the vortex strength of the so-called vortex tube with cross-sectional area A, A · Rshown on the following page. Page 59 EOSC 512 2019 The vortex tube is a cylindrical tube in space whose surface elements are composed of vortex lines passing through the same closed curve C. Similarly to a streamline, a vortex line is a line in the fluid that, is everywhere tangent to the vorticity vector. It can be important to note that the strength of the vector vorticity is not constant along a vortex line, in the same way that velocity is not (necessarily) constant along a streamline. But, the strength of a vortex tube is constant along the tube, in the same way that the transport between two streamlines is constant in compressible flow. Then, considering the following picture, To make the strength constant, we must enforce, il ~! nˆ dA = il ~! nˆ dA (6.8) A1 1 · 1 1 A2 2 · 2 2 6.3 Kelvin’s Circulation Theorem This is a statement of the conservation of circulation under certain conditions. Consider a closed contour C drawn in the fluid that moves with the fluid such that the motion of the fluid elements on the contour determine its subsequent location and shape. Page 60 EOSC 512 2019 The velocity vector indicates a fluid element on the contour moving with the fluid velocity at that point. One can think of the contour as being composed of a “pearl necklace” of fluid elements (pearls) that moves and deforms in a way defined by the individual motion of the pearls. Consider the time rate of change of the circulation on the contour. DΓ D = ~u d~x Dt Dt · CI D~u Dd~x = d~x + ~u (6.9) Dt · · Dt CI CI 1 2 | {z } | {z } Where 1 accounts for change in time of ~u along the contour, and 2 accounts for change due to changing contour position and shape. First, we will consider term 2 . If the line element d~x is moving with the fluid, it stretches and rotates depending on the velocity di↵erence between its endpoints. That is, Dd~x = ~u. As this distance d~x 0, this can be written as: Dt ! Dd~x ~u = ~u d~u · Dt · CI CI 1 = d ~u 2 (6.10) 2 | | CI =0 This integral vanishes, as it is the integral of a perfect di↵erential around a closed path! Thus, our expression for DΓ Dt reduces to: DΓ D~u = d~x (6.11) Dt Dt · CI Note we have not yet specified which velocity we are using to define the circulation. It is convenient for our purposes to follow the absolute velocity seen in the inertial frame, D(Γ) D(~u ) inertial = inertial d~x (6.12) Dt Dt · CI D(~u )inertial We can substitute for Dt using the momentum equation. Here, for simplicity we will take the viscosity coefficients to be constants, but you could continue to carry through the terms that depend on the spatial gradients Page 61 EOSC 512 2019 of µ and λ if you wanted or needed it to. We will further assume an incompressible flow. Under these assumptions, D(Γ) ⇢ inertial = P + ⇢F~ + µ 2~u (6.13) Dt r r inertial Further, we will assume that the body force per unit mass F can be derived from a scalar potential, as in the case of gravity, F~ = Φ .Then: r g F~ d~x = Φ d~x · − r g · CI CI = dΦ d~x (6.14) − g CI 0 The integral vanishes because this is again the integral of a perfect di↵erential around a closed path. Then: D(Γ) P inertial = r d~x + ⌫ 2~u d~x (6.15) Dt − ⇢ · r inertial · CI CI This is a statement of Kelvin’s circulation theorem! It is convenient for physical interpretation to write: • P d~x =dP – an incremental change in pressure along the contour element dvecx. r · • ~u = ( ~u ) ~! = ~! if assuming incompressible flow. r r r· r⇥ r ⇥ So we can write Kelvin’s circulation theorem as DΓ dP a = ⌫ ( ~! ) d~x (6.16) Dt − ⇢ − r⇥ · CI CI This says that for any closed contour C moving with the fluid, circulation can be produced or destroyed in two ways. The first is the baroclinic production term that arises when surfaces of density and pressure do not coincide (more coming on this shortly!). The second is through frictional e↵ects that cause vorticity to di↵use through the walls of the vortex tube. Before discussing these processing in more detail, we will first consider the conditions under which circulation is conserved. • If ⇢ = ⇢(P ) only, the the surfaces of constant density and pressure coincide. Such a fluid is called barotropic. dP The simplest example being a fluid of constant density. In this case, ⇢(P ) is a perfection di↵erential, so the contour integral vanishes. • and if ⌫ 0 (friction can be neglected – Note that this only needs to be true on the contour!) ⇡ The circulation is conserved. As we will see, this is a powerful constraint that governs certain types of large-scale flows in the ocean and atmosphere. Page 62 EOSC 512 2019 6.3.1 Baroclinic Production dP We can called the ⇢ term the baroclinic term, and will be non-zero whenever the surfaces do not coincide. − C Mathematically, thisH term can be written in a few di↵erent ways: dP P ⇢ P = r d~x = r ⇥r nˆ dA (6.17) − ⇢ − ⇢ · ⇢2 · CI CI AZ Physically, it represents the generation of vorticity and circulation due to a tendency of density surfaces to “slump” in the presence of a pressure gradient when these two surfaces are not aligned. Consider a pressure gradient whose direction is being imposed by gravity. The pressure gradient will be purely downward. We then add a density gradient that involves a component arising from density increasing to the left, The gradients of the pressure and density fields produces a tendency for the heavy fluid on the left to sink and the light fluids on the right to rise. This induces a positive circulation of the fluid, (counter-clockwise) indicated in red. Another way to look at this is to exmaine a fluid parcel whose centre of gravity is displace to the left by the presence of a density gradient.
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