CSU ATS601 Fall 2015 8 Rotational Flow In the previous section we learned that Earth?s rotation tends to make the (large-scale) flow quasi-two dimensional, dominated by geostrophic balance. Geostrophic flow is non-divergent at leading order. This type of (large-scale) flow is therefore almost entirely determined by its rotational component and this is intuitively clear, for example, from the dominant cyclonic and anticyclonic flow patterns in mid-latitudes. However, even small-scale flows can be predominantly rotational in character (think of a tornado). It will turn out in this section that fundamental conceptual insight into the behavior of geophysical flow can be gained from a rotational perspective, that is, by studying vorticity and its governing equation. After introducing descriptors of rotational flow, such as vorticity and circulation, the section will discuss the vorticity equation and circulation theorems and lead to a “discovery” of Rossby waves. Potential vorticity conservation and its implications will be introduced later-on from the perspective of the shallow water equations. 8.1 Vorticity and circulation The vorticity ! is defined as the curl of the velocity: ! v =(@yw - @zv, @zu - @xw, @xv - @yu) (8.1) ⌘r⇥ For fluids on a rotating planet (like the atmosphere and ocean), this is actually only the relative vorticity, since there is also vorticity associated with the rotation of the planet itself. We call Earth’s vorticity due to rotation the planetary vorticity. From the section on rotation, we saw that the intertial velocity vi = vr + ⌦ r, and thus, the absolute vorticity (sum of planetary and relative vorticity) is ⇥ !a (v + ⌦ r)= v + 2⌦ = ! + 2⌦ (8.2) ⌘r⇥ ⇥ r⇥ where we have used the fact that (⌦ r)=2⌦. This is not immediately obvious, but we will delay r⇥ ⇥ demonstrating this equality until we discuss circulation. In the previous section, we saw that geostrophic flow is two-dimensional and at leading order non- divergent (i.e. incompressible by continuity). Such flow can be described by a streamfunction defined previously as @ @ u =- and v = (8.3) @y @x or alternatively u = kˆ (8.4) ⇥r E. A. Barnes 63 updated 11:10 on Friday 16th October, 2015 CSU ATS601 Fall 2015 which in Cartesian coordinates is @ @ @ @ @ u = kˆ =(0, 0, 1) ( , , )=(- , ,0) (8.5) ⇥r ⇥ @x @y @z @y @x If the horizontal velocity can be defined by a streamfunction , then by definition, the flow is non- divergent. This can be seen easily using : @u @v u = + =-@xy + @yx = 0 (8.6) r· @x @y That the flow is non-divergent and can be written in terms of a streamfunction implies that the stream- function and the vorticity ! must be related. Indeed they are! Specifically, the vertical component of the vorticity (!z = ⇣) is the Laplacian of the streamfunction: ! = ⇣ = 2 (8.7) z r Vorticity represents a local (i.e. microscopic) measure of the rotational component of the flow. The value of vorticity at a point does not depend on the particular choice of an axis of rotation - it is a measure of the local “spin” of a fluid element. “If you put a paddle wheel in the flow, the paddle wheel will rotate if the vorticity is non-zero.” The circulation C is also a measure of the rotational component of the flow, but at a more global (macro- scopic) level. Circulation is defined as the integral of the velocity vector around a closed fluid loop: C v dr (8.8) ⌘ · I Now, it would make sense that vorticity and circulation are related, as the circulation is just a macro- scopic measure of the rotational component of the flow, while vorticity is a microscopic measure. We will now demonstrate that our intuition is correct! Consider a case where the line integral for C lies in the x-y plane, and assume that the circuit is a small rectangle with sides δx, δy as shown below. ∂u (u + δy)δx − ∂y ∂v (v)δy (v + δx)δy − ∂x (u)δx Figure: Example of the circulation around a rectangular circuit. E. A. Barnes 64 updated 11:10 on Friday 16th October, 2015 CSU ATS601 Fall 2015 Since the velocity components vary in space, the meridional components between the left and right, for example, differ by @xvδx, and likewise, the components between the top and bottom differ by @yuδy. Thus, the circulation is given by the sum of the flow along the sides of the rectangular circuit: C = v dr (8.9) · I @v @u = uδx + v + δx δy - u + δy δx - vδy (8.10) @x @y ✓ ◆ ✓ ◆ @v @u = δxδy - δyδx (8.11) @x @y @v @u = - δxδy (8.12) @x @y ✓ ◆ That is, the circulation in this case corresponds to the (vertical component of) vorticity inside the circuit (i.e. @xv - @yu) times the surface area of the domain enclosed by the circuit! This can be even more clearly seen by invoking Kelvin-Stokes Theorem (sometimes referred to as Green’s Theorem for two dimensions) . Kelvin-Stokes Theorem F dS = F dr S r⇥ · δr · This states that the surface integral ofR the curl of a vectorH field over a surface S is related to the line integral of the vector field over the boundary of S (called δr). Applying this theorem to our problem at hand, we see that C = v dr = ( v) dS = ! dS = ! nˆdS (8.13) · r⇥ · · · I SZ SZ SZ where nˆ is the unit vector normal to the surface S. Thus, we see that the circulation is an integral measure of the vorticity of the flow. Put another way: vorticity at a point is the circulation per unit area. ⇣>0 for anticlockwise rotation. • circulation around a path is the integral of the normal component of vorticity over any surface • bounded by the path. C > 0 for anticlockwise rotation. Figure: Example of how circulation around a fluid loop ecompassing a surface S is the integral of the vorticity over the surface. E. A. Barnes 65 updated 11:10 on Friday 16th October, 2015 CSU ATS601 Fall 2015 8.1.1 Solid (rigid) body rotation Recall that above we made the statement that (⌦ r)=2⌦ (8.14) r⇥ ⇥ The relationship between circulation and vorticity can now be used to show why this is the case. If we consider an circular fluid loop of radius r in solid body (rigid body) rotation ⌦, the loop will have tangential velocity V = ⌦r. In this case, the circulation is C = 2⇡r ⌦r = 2⇡r2⌦ (8.15) · Since the vorticity is just the circulation around an infitesimal surface boundary divided by the area of the surface, then C 1 ! nˆ = = v dr (8.16) · S S · I and plugging in our equation for C and using the fact that the area S = ⇡r2 we obtain 2⇡r2⌦ ! nˆ = = 2⌦ (8.17) · ⇡r2 Alternatively, one can also come to this conclusion using the definition of vorticity itself. That is, using cylindrical coordinates, imagine that the solid body is only rotating in a single direction and thus uz = ur = 0 and u✓ = ⌦r. Then the vorticity is non-zero only in the kˆ direction ! = v = !zkˆ (8.18) r⇥ where the kˆ component of the curl in cylindrical coordinates is given by 1 @ kˆ v = (ru✓) (8.19) ·r⇥ r @r Thus, 1 @ 1 @ ! = (ru )= (r2⌦)=2⌦ (8.20) z r @r ✓ r @r 8.1.2 “Irrotational vortex” (Vr vortex) This vortex is called the “Vr” vortex because the tangential velocity (v in this context) is such that the product vr is constant. Using our notation for cylindrical coordinates, that is, K u = (8.21) ✓ r E. A. Barnes 66 updated 11:10 on Friday 16th October, 2015 CSU ATS601 Fall 2015 where K is a constant that determines the vortex strength. Note that the angular momentum associated with this vortex is constant, since angular momentum is ru✓ = rK/r = K. Let’s calculate the vorticity associated with this vortex! Using the method above, where we realize that the only non-zero component of vorticity is in the kˆ direction 1 @ 1 @ K 1 @ K 1 @ kˆ v = (ru✓)= (r )= (r )= (K)=0 (8.22) ·r⇥ r @r r @r r r @r r r @r That is, at all points r = 0, the vorticity of the “Vr” vortex, or irrotational vortex, is zero! 6 Show examples of solid body and Vr vortex. 8.2 Circulation theorems 8.2.1 Kelvin’s Circulation Theorem Kelvin’s Circulation Theorem states that under certain circumstances, the circulation around a material fluid parcel is conserved. That is, the circulation is conserved along a circuit encompassing a material fluid parcel - no matter if the parcel changes shape, etc. In order for this theorem to apply we must assume the following the forces acting on the flow are conservative (e.g. no energy loss via friction or viscosity) • the fluid is barotropic (i.e. p = p(⇢)) • To prove this rather remarkable conservation property of circulation, we write the inviscid momentum equations (ignoring an explicit representation of rotation): Dv 1 =- p - Φ (8.23) Dt ⇢r r Then, we calculate the material derivative of the circulation DC D Dv D(dr) = v dr = dr + v (8.24) Dt Dt · Dt · · Dt I I ✓ ◆ Dv = dr + v dv (8.25) Dt · · I ✓ ◆ p = -r - Φ dr + v dv (8.26) ⇢ r · · I ✓ ◆ p = -r dr (8.27) ⇢ · I ✓ ◆ ⇢ p = r ⇥r nˆdS (8.28) ⇢2 · SZ E. A. Barnes 67 updated 11:10 on Friday 16th October, 2015 CSU ATS601 Fall 2015 where the Φ vanishes because it is an exact differentials integrated around a closed loop. The line integral r of v dv vanishes because v dv = dv2/2, and thus is also an exact differential integrated around a closed · · loop (and thus zero).