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 x, y as shown below.

u (u + y)x y

v (v)y (v + x)y x

(u)x

Figure: Example of the circulation around a rectangular circuit.

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Since the velocity components vary in space, the meridional components between the left and right, for example, differ by @xvx, and likewise, the components between the top and bottom differ by @yuy. 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 = ux + v + x y - u + y x - vy (8.10) @x @y ✓ ◆ ✓ ◆ @v @u = xy - yx (8.11) @x @y @v @u = - xy (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⇥ · r · 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 r). 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.

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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 - (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 - 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 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). The final step of the derivation uses Kelvin-Stoke’s Theorem, and the identity A ( A)g + A g = r⇥ ⇥r (8.29) r⇥ g g2 For a barotropic fluid, p = p(⇢), and so the gradient of p is always parallel to the gradient of ⇢. Thus, we obtain...

Kelvin’s Circulation Theorem: for barotropic, conservative flows

DC D = v dr = 0 (8.30) Dt Dt · I For a baroclinic fluid, we cannot assume that ⇢ is parallel to p and thus we have r r DC D ⇢ p = v dr = r ⇥r nˆdS (8.31) Dt Dt · ⇢2 · I SZ where the term inside the integral is called the solenoidal term (who’s meaning we will discuss in the next section). Remember, for this term to matter, the pressure gradient and density gradients cannot be parallel to one another. Now, we haven’t said a word about rotation - but fear not! Interesting note: For adiabatic flows (D✓/Dt = 0), interesting fundamental insight comes from carrying out the surface integral along an isentropic surface, i.e. n~ = ✓/| ✓|. In this case, the solenoidal term vanishes: ( ⇢ r r r ⇥ p) ✓ = 0 (this is because ✓ = ✓(p, ⇢), hence ✓ is perpendicular to ⇢ p). Hence, D v r ·r r r ⇥r Dt · dr = 0 for adiabatic flow subject only to conservative forces (related to the concept of potential vorticityH conservation, see later sections). Note that assuming D✓/Dt = 0 is crucial as it ensures the parcels to stay on a given isentropic surface. This is also the reason why choosing a density or pressure surface does not ensure conservation of circulation, despite the fact that ( ⇢ p) ⇢ = 0 and ( ⇢ p) p = 0 – parcels r ⇥r ·r r ⇥r ·r do not generally stay on either a density or pressure surface, even for adiabatic motion. However, for oceanic applications, choosing density surfaces is appropriate, since D⇢/Dt 0 (fluid is nearly incompressible). ⇡

8.2.2 Helmholtz’s Circulation Theorem

Helmholtz’s Circulation Theorem states that for an inviscid, barotropic fluid under the influence of con- servative forces, vortex lines are material lines, i.e. vortex lines contain the same material elements through- out time. This is the frozen-in property of vorticity, where

E. A. Barnes 68 updated 11:10 on Friday 16th October, 2015 CSU ATS601 Fall 2015

vortex lines: a line that is everywhere parallel to the local vorticity vector •

vortex tube: a tube formed by all of the vortex lines that pass through a closed surface S (or C in • panel b of the Figure 8.3 below).

For more information on Helmholtz’s Theorem and the “frozen-in” property, see Vallis 4.3.

With Kevlin’s circulation theorem, we can now better understand the behaviour of Taylor columns!

8.3 Vorticity equation (vertical component)

It turns out that the vertical component of the relative vorticity (⇣) is the dynamically most important (although not the largest) component because it contains information about the horizontal flow. The vertical component of relative vorticity is defined as:

Cartesian Coordinates: • @v @u ⇣ = - (8.32) @x @y

Spherical Coordinates: • 1 @v 1 @ ⇣ = - (u cos ✓) (8.33) a cos ✓ @ a cos ✓ @✓

In order to obtain prognostic (evolution) equation for ⇣, we first rewrite the horizontal advection terms in the horizontal momentum equations for Cartesian coordinates as:

1 2 2 1 2 2 u@xu + v@yu =-v⇣ + @x(u + v ) and u@xv + v@yv = u⇣ + @y(u + v ) (8.34) 2 2

Plugging this (seemingly unnecessarily messy expression) into the horizontal momentum equation in Carte- sian coordinates leads to

@xp 1 2 2 @tu - v(⇣ + f)+w@zu =- - @x(u + v )+X (8.35) ⇢ 2 @yp 1 2 2 @tv + u(⇣ + f)+w@zv =- - @y(u + v )+Y (8.36) ⇢ 2 where X and Y are additional, non-conservative forces (e.g. friction) that are not included explicitly here. Note how both the vertical components of the relative and planetary vorticity terms combine together to give an absolute vorticity term.

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Taking @x of the second equation minus @y of the first equation (and applying some algebra) yields D (⇣ + f)=-(⇣ + f)(@ u + @ v) (8.37) Dt x y

+ @zu@yw - @zv@xw (8.38)

-2 + ⇢ (@x⇢@yp - @y⇢@xp) (8.39)

+(@xY - @yX) (8.40)

This is an evolution equation for absolute vorticity ⇣ + f. The individual terms on the right-hand-side are:

1. vortex stretching: stretching by the horizontal divergence

2. vortex tilting: tilting by the vertical shear of the horizontal wind and/or horizontal shear of the vertical wind

3. solenoidal term: also called the baroclinic term (the vertical component of -( ⇢ p)/⇢2) r ⇥r 4. non-conservative terms: e.g. friction, molecular viscosity

Figure 6: Illustrations of (a) a vortex filament, (b) a vortex tube, (c) vortex stretching and (d) vortex tilting.

stretching and tilting terms give E. A. Barnes 70 updated 11:10 on Friday 16th October, 2015 D⃗ω a = ⃗ω .∇⃗v − ⃗ω ∇.⃗v Dt ∂ ∂u ∂v ∂w = ω uˆi + vˆj + wkˆ − ωkˆ + + (24) ∂z ! " #∂x ∂y ∂z $ ∂u ∂v ∂u ∂v = ˆiω + ˆjω − kˆω + ∂z ∂z #∂x ∂y $ Consider first the example depicted in Fig. 6 (c) where a horizontal divergent velocity field is present. This gives Dω ∂u ∂v kˆ = −ωkˆ + (25) Dt #∂x ∂y $ i.e. the divergent velocity field would act to decrease the vertical component of vorticity. This makes sense if the curve C consists of fluid elements moving with the velocity field

8 CSU ATS601 Fall 2015

Figure: Taken from Isla Simpson’s notes from Columbia University. http://www.columbia.edu/⇠irs2113/3 Circulation Vorticity PV.pdf

In the special case of two-dimensional, conservative, incompressible flow:

D (⇣ + f)=0 (8.41) Dt

that is, absolute vorticity is conserved!! In this instance, absolute vorticity conservation stabilizes eastward (westerly) flow, but destabilizes westward (easterly) flow! (see Holton Figure 4.8)

8.3.1 The beta effect

Note that D D⇣ Df D⇣ D⇣ (⇣ + f)= + = + v@ f = + v (8.42) Dt Dt Dt Dt y Dt Thus, the left-hand-side of the evolution equation of vorticity can be written as D @⇣ (⇣ + f)= + v ⇣ + v (8.43) Dt @t ·r The final term v appears because the Coriolis parameter varies with latitude. Thus, any meridional motion will change the planetary contribution of vorticity, and thus, will modify the absolute vorticity. Another way to see this is to return to the circulation. Define the relative circulation over some material loop as

Cr = vr dr (8.44) · I where vr = va - 2⌦ r. Then, we have ⇥

Cr = Ca - 2⌦ nˆdS = Ca - 2⌦A (8.45) · ? Z where A is the area enclosed by the projection of the material circuit onto the plane normal to ⌦ (i.e. ? equatorial plane). Thus, if the solenoidal term is zero, then the circulation theorem tells us that D (Cr + 2⌦A )=0 (8.46) Dt ? The relative circulation around the circuit will change if the area of the projection of the circuit onto the equatorial plane changes. This occurs when a parcel changes latitude!

In this case A = A sin ✓ and Cr = ⇣rA where A is the area of the fluid surface. So, if the fluid surface ? moves latitude, its relative vorticity must also change to keep the total circulation constant. That is,

D⇣r 2⌦ DA D 2⌦ cos ✓ 2⌦ =- ? =-2⌦ sin ✓ =-v =-v where = cos ✓ (8.47) Dt A Dt Dt a a

E. A. Barnes 71 updated 11:10 on Friday 16th October, 2015 CSU ATS601 Fall 2015 where we have used the fact that D/Dt(sin ✓)=cos ✓D/Dt✓ and v = aD/Dt✓. What this tells us is that the vertical component of the relative vorticity of a parcel changes due to its meridional movement!

8.4 Vector form of the vorticity equation (Vallis 4.2; Schubert 2.2; Holton 4.4)

We will now generalize the above to obtain an evolution equation for the full, three-dimensional, vortic- ity. The first step, as above, is to rewrite the advetive term:

1 1 (v )v = v2 - v ( v)= v2 - v ! ·r 2r ⇥ r⇥ 2r ⇥ which is a special case of the general vector identity

(a b)=(a )b +(b )a + a ( b)+b ( a) (8.48) r · ·r ·r ⇥ r⇥ ⇥ r⇥ with a = b = v. This allows us to rewrite the momentum equations as:

p 1 2 @tv + ! v + 2! v =-r - v - gk + X (8.49) ⇥ ⇥ ⇢ 2r where X is included to represent any non-conservative forces such as friction. The second and third terms on the LHS can be combined to !a v, where !a ! + 2! is the absolute vorticity. ⇥ ⌘ In order to form the vorticity equation we take the curl of this equation (and note @t! = 0):

p ⇢ @t!a + (!a v)=-r ⇥r + X . r⇥ ⇥ ⇢2 r⇥

The first term on the RHS results from ( a)= a + a with = ⇢-1 and a = p. r⇥ r⇥ r ⇥ r This is the solenoidal term which vanishes for a barotropic fluid. The second term on the RHS represents non-conservative forces such as friction, but note it is only the rotational component of these forces that contributes to the vorticity equation. What happened to the other terms on the RHS of the momentum equation above?

In order to evaluate (!a v) we use the vetor identity r⇥ ⇥

(a b)=a( b)-b( a)+(b )a -(a )b (8.50) r⇥ ⇥ r· r· ·r ·r with a = !a and b = v, i.e.:

(!a v)=!a( v)+(v )!a -(!a )v (8.51) r⇥ ⇥ r· ·r ·r

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(note, the term involving ! vanishes), hence: r· a

D!a p ⇢ + !a( v)=(!a )v - r ⇥r + X (8.52) Dt r· ·r ⇢2 r⇥

The LHS can alternatively be written compactly, using the continuity equation ( v =-⇢-1D⇢/Dt) r· and the reverse product rule, as:

D !a p ⇢ ⇢ =(!a )v - r ⇥r + X . Dt ⇢ ·r ⇢2 r⇥ ✓ ◆

The first term on the RHS, (!a )v (for a barotropic fluid under conservative forces the only forcing ·r term) represents the effects of vortex tilting and stretching. To see this imagine an initially vertically aligned vorticity vetor, i.e. (!a )v =(⇣ + f)@zv. Vertical shear of the horizontal wind (@zu or @zv) will force a ·r horizontal component of vorticity, resulting in vortex tilting. On the other hand, vertical shear of the vertical wind (@zw) will force a change in the vertical component of vorticity, resulting in stretching or contraction.

E. A. Barnes 73 updated 11:10 on Friday 16th October, 2015