1 Oct 2015EATS 3040-2013 Notes 4

HH = Holton and Hakim. An Introduction to Dynamic Meteorology, 5th Edition.

4. Circulation, Vorticity and Potential Vorticity

4.1 The Circulation Theorem

Definition C=∮U.dl=∮∣U∣cos(a)dl

In a simple circular vortex flow, integrating from 0 to 2π

C=∮U.dl = ∫ ΩR2dλ = 2πΩR2 = 2πRV

Circulation can be absolute (related to an inertial frame) or relative (in a frame rotating on Earth)

Newtons law applied to a closed chain of fluid elements around C gives,

∮ DaU aDt . dl=−∮grad pro

. dl−∮ g s k.dl (4.1)

where gs is true gravity (excluding centrifugal force)

DaU aDt

.dl=DaDt

(U a . dl )−U a .DaDt

(d l )

? are D/Dt and Da/Dt the same? HH give a footnote, OK for a scalar, not for a vector.

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IF the contour is a streamline, Ua = Dal/Dt and the last term becomes Ua . DaUa = Da(Ua2/2), a perfect differential and the line integral is zero. the last term in 4.1 is also zero so we have,

DCaDt

= DDt

(U a . dl )=−∮ ro−1dp (4.3)

The integral of (1/ρ)dp is called the solenoidal term. It is zero for barotropic situations - Kelvin's circulation theorem DCa/Dt = 0, but can be a circulation source in baroclinic situations (see sea breeze example of circulation in a vertical plane later).

For large scale motions we focus on circulation mostly in horizontal planes.

Part of the absolute circulation is due to Earth rotation, Ce = ∮U e .dl = 2AΩ sin φ whereUe = Ωxr and A is the area enclosed by the contour around which the circulation is computed or Ce = 2AeΩ if Ae is the projection of A onto the equatorial plane. See HH for details using Stoke's theorem.

If we consider relative circulation, C = Ca - Ce = Ca - 2ΩAe we can use (4.3) to obtain Bjerknes circulation theorem,

DCDt

=−∮ ro−1 dp−2OMEGA DAeDt(4.5)

For a barotropic situation (HH refer to a barotropic fluid with ρ = ρ(p) but air is not normally constrained in this manner) the first term on the RHS is zero and, as the chain of fluid elements move,

Ca = C + 2ΩAsinφ = constant

which is Kelvin's circulation theorem again. HH state "A negative absolute circulation in the Northern Hemisphere can develop only if a closed chain of fluid particles is advected across the equator from the Southern Hemisphere". I thyink that this relates to barotropic situations only. Note Ce is -ve in SH.

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Sea Breeze example.

Note that here the contour around which circulation is considered is in a vertical plane - more usual to consider horizontal surfaces. Definitely a baroclinic situation.

Dashed lines are lines of constant density. Solid boundary line is the loop around which the circulation is to be computed. If the plane is vertical, Ce is zero (Ue and dl are perpendicular) and C = Ca. Circulation theorem is

DCDt

=−∮ ro−1 dp=−∮ RT d ( lnp)

"Horizontal" lines are isobaric so no contribution, vertical lines give contribution

DCDt

=R ln (p0p1

)(T̄ 2−T̄ 1)>0

If v is a mean tangential velocity around the circuit, C = 2v (h + L). Suppose a 10° temperature difference. With p0 = 1000 hPa, p1 = 900 hPa, h = 1000m, L = 20 km we get ∂v/∂t ≈ 7 x 10-3 ms-2. After 3600s this would give v = 25 m/s - too strong for the average sea breeze.

Although surface temperature differences may be 10° that would be too high over 1 km, friction will slow the flow etc. There are more detailed numerical models of sea and lake breezes but earlymodels (perhaps Pearce, 1955?) used circulation models.

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4.2 Vorticity

Various places to start but one could start with the vertical component of vorticity,

ζ = lim (∮U.dl /A) as A → 0

where the contour is in the horizontal plane.

Or ω = curl U using either absoluteor relative velocity. For large scale atmospheric dyamics focus on the vertical components,

η = k.curlUa : ς = k.curlU

η = ∂va/∂x - ∂ua/∂y : ζ = ∂v/∂x - ∂u/∂y

Circulation limit

More generally, using Stoke's theorem,

∮U.dl=∫∫ curlU . n dA

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Vorticity in Natural Coordinates

δC = V[δs + d(δs)] - (V + δn ∂V/∂n) δs

where d(δs) = δn δβ and so

ζ = Lim [δC/(δsδn)] as δn,δs → 0

= -∂V/∂n + V/Rs

where Rs is the radius of curvature of the streamlines.

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Linear shear flow, with vorticty,]shear vorticity.

Flow around a corner, may have no vorticity

4.3 The Vorticity Equation

Cartesian Coordinates - fixed relative to Earth, with standard approximations and no friction, but fmay vary.

Du/Dt = -(1/ρ)∂p/∂x + fv Dv/Dt = -(1/ρ)∂p/∂y - fu

Consider - ∂/∂y of the u equation + ∂/∂x of the second noting ζ = ∂v/∂x - ∂u/∂y, we get

D(ζ + f)/Dt = - (ζ + f)(∂u/∂x + ∂v/∂y) - (∂w/∂x ∂v/∂z - ∂w/∂y ∂u/∂z)Stretching Tilting

+ (1/ρ2)(∂p/∂y ∂ρ/∂x - ∂p/∂x ∂ρ/∂y)solenoidal term

The tilting term; x component vorticity (∂v/∂z) tilted by -ve ∂w/∂x to produce +ve z component vorticity (ζ). With (1/ρ) replaced by α the solenoidal term can be writtenas -(∂p/∂y ∂α/∂x - ∂p/∂x ∂α/∂y) = - ( α x p).kSolenoid: In electicity. A current-carrying coil of wire that acts like a magnet when a current passes through it.

A coil of wire usually in cylindrical form that when carrying a current acts like a magnet so that a movable core is drawn into the coil when a current flows and that is used especially as a switch or control for a mechanical device (as a valve).

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4.3.2 Vorticity equation in pressure coordinates

Use k.curl of the momentum equation (with on pressure surfaces) and V the horizontal velocity

∂V/∂t + (V. ) V+ f k x V = - Φ (3.2)but first noting that

(V. ) V = (V.V)/2 + ζkxV (? HH 4.14 true for horizontal cpts only)as a variation of

(V. ) V = (V.V)/2 + (curl V)xV (this is true)and

x (a x b) = ( . b) a - (a . ) b - ( . a) b + (b . ) aso that we have

∂ζ /∂t = - (V. )(ζ + f) - ω ∂ζ/∂p - (ζ + f)( .V) + k.(∂V/∂p x ω)

rate of change advection stretching tilting

No solenoidal term - partial derivatives on constant pressure surfaces. But the term is usually small in Cartesian coordinates as well.

Scale Analysis of the vorticity equation Which are the significant terms?

Assume we are interested in motions with scales

u,v - 10 m/sw - 0.01 m/sL - 106 m - length scale, 1000 kmH - 104m - depth scale, 10 kmρ - 1 kg/m3 - typical air densityδρ/ρ - 10-2 - fractional density fluctuationδp - 103 Pa - (10hPa - typical pressure difference over 1000 kmT - 105 s - Time scale (L/U) about 30 hoursf - 10-4 s-1β - 10-11 m-1s-1 - rate of change of f with y

First note that ζ ( = ∂v/∂x - ∂u/∂y) is of order 10-5 s-1 but can be larger, maybe 10-4 s-1. in intense storms.

An analysis from the OQ-Net.

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For synoptic scale motions many terms can be assumed small and we end up with

Dh(ζ + f)/Dt = - f (∂u/∂x + ∂v/∂y)

while for intense storms,

Dh(ζ + f)/Dt = - (ζ + f)(∂u/∂x + ∂v/∂y)

For a low pressure centre (ζ > 0, f > 0 in NH) we expect convergence so (ζ + f) will increase and the vorticity will increase. For an anticyclone (ζ < 0, f > 0) in NH but we expect (ζ + f) > 0. Then divergence could lead to a lowering of (ζ + f) but at a slower rate and, when (ζ + f) = 0, it would cease lowering with ζ = -f.

One site that shows sfc divergence and vorticity.http://www.spc.noaa.gov/exper/mesoanalysis/new/viewsector.php?sector=16Select Kinematics, divergence and vorticity, no underlay Potential Vorticity:Isentropic surfaces - surface of constant potential temperature, on which ρ is a function of p. So if we consider a line integral on an isentropic surface there will be no solenoidal contribution tochanges in circulation around such a line.

Using Stokes Theorem Ca=∮U a . dl = ∫∫ ωa.n dA where n is normal to the isentropic surface in which the line integral is evaluated.

Mass in the cylinder is dm = ρdAdh and dA = (dm/ρ)(| θ|/dθ). If dA is small and the vorticity is essentially uniform over dA we have, for the circulation around this small circuit, since there is nosolenoidal term, via Kelvin's circulation theorem,

DCa/Dt = D[ωa.n dA]/Dt = 0 and we can write n = θ/| θ| and then get, after some manipulation

Ertel's Potential Vorticity Theorem, DΠ/Dt = 0 where Π = (ωa. θ)/ρ is the PV, which is thus conserved following the motion.

The figure above provides an illustration.

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dA

dh

Suppose no horizontal vorticity components, then

Π = (∂v/∂x - ∂u/∂y + f)(∂θ/∂z)/ρ

So Π is a product of absolute vertical vorticity and vertical potential temperature gradient. if ∂θ/∂z decreases, as in figure, then ∂v/∂x - ∂u/∂y + f must increase and if f is unchanged, ∂v/∂x - ∂u/∂y must increase - vortex stretching.

Ratio of vertical to horizontal contributions to PV scales as 1 + Ro-1, so if Ro = 0.1, ratio = 11. Also the f term dominates so Π ≈ f(∂θ/∂z)/ρ

Typical values (∂θ/∂z) ≈ 5 K / km and Π ≈ 0.5 x 10-6 Km2s-1kg-1 or 0.5 PVU

Use of PV surface to identify a dynamic tropopause. At the tropopause ρ is typicallyof order 0.4 - 0.5 kg m-3. Below tropopause, generall ∂θ/∂z < 5 Kkm-1 (0 for DALR, 4 for SALR) so PV < 1 PVU. Just above the tropopause ∂T/∂z is typically 0 Kkm-1 so ∂θ/∂z might be of order 10 Kkm-1 and PV > 2. Surfaces with PV = 1.5 or 2 PVU can be used to define the height of the dynamic tropopause,

Examples at http://www.pa.op.dlr.de/arctic/ecmwf.php These show θ, others show p or z,

Can either plot PV on an isentropic surface or potential temperature on a PV surface.

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Figure 4.9 from HH. Jan 12, 2012. a) is 500 hPa Geopotential height, b) is pressure c) is potential temperature and d) is wind speed, all on the dynamic tropopause. Ridge over western NA, trough in central NA. Trough has low tropopause, below 500hPa at some points.

See also:Quart. J. R. Met. Soc. (1985), 111, pp. 877-946On the use and significance of isentropic potential vorticity mapsBy B. J. HOSKINS, M. E. McINTYRE and A. W. ROBERTSON

SUMMARYThe two main principles underlying the use of isentropic maps of potential vorticity to represent dynamical processes in the atmosphere are reviewed, including the extension of those principles totake the lower boundary condition into account. The first is the familiar Lagrangian conservation principle, for potential vorticity (PV) and potential temperature, which holds approximately when advective processes dominate frictional and diabatic ones. The second is the principle of ‘invertibility’ of the PV distribution, which holds whether or not diabatic and frictional processes are important. The invertibility principle states that if the total mass underneath each isentropic surface is specified, then a knowledge of the global distribution of PV on each isentropic surface and of potential temperature at the lower boundary (which within certain limitations can be considered to be part of the PV distribution) is sufficient to deduce, diagnostically, all the other dynamical fields, such as winds, temperatures, geopotential heights, static stabilities, and vertical velocities, under a suitable balance condition. The statement that vertical velocities can be deduced is related to the well-known omega equation principle, and depends on having sufficient information about diabatic and frictional processes. Quasi-geostrophic, semigeostrophic, and ‘nonlinear normal mode initialization’ realizations of the balance condition are discussed. An important constraint on the mass-weighted integral of PV over a material volume and on its possible diabatic and frictional change is noted.

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Shallow Water Equations

Provides an idealization of atmospheric flow, illustrating various features, especially flow over mountain ranges.

Shallow implies horizontal scales >> vertical scales. Also implies in the oceans for tides, tsumanis, storm surges.

Assume U,V independent of z, pressure hydrostatic, constant density ρ0. Surface of fluid is assumed to be at z = h (x,y) and p(x,y,h) = constant p(h). Assume z = 0 is a geopotential surface.

Hydrostatic pressure gives, p(z) = p(h) + ρ0g(h-z) and (1/ρ) p = g h. The equation of motion,for the horizontal wind vector, V, becomes,

DhV/Dt + f kxV = - (1/ρ) hp = - g hh (4.33)

In this incompressible fluid (u,v,w) = 0 so, if w = 0 on z = 0, w(h) = -h hV, since V is independent of z.

Assuming w = 0 on z = 0 does not seem consistent with Figure 4.11 (below) where the lower boundary is not flat. Can we take account of that?

HH claim that Dp/Dt = 0 – doesn't make sense to me (consider a fluid element at the bottom of theshallow water layer if the layer depth changes), and not needed to assert that,

Dhh/Dt = w(h) = -h hV (4.36, 4.37)

Now take partial derivatives, ∂/∂x of the y-cpt of 4.33 - ∂/∂y of the x-cpt to obtain, as before,

Dh(ζ + f)/Dt = - (ζ + f)(∂u/∂x + ∂v/∂y)

We can also use 4.37 to see that

Dh(ζ + f)/Dt - (ζ + f)(1/h) Dhh/Dt = 0

Dividing again by h gives the perfect differential,

Dh{(ζ + f)/h}/Dt = 0

and shows that shallow water PV, (ζ + f)/h, is conserved. Assume ∂/∂y changes are smaller than ∂/∂x, except for f.

Recall that ζ = ∂v/∂x - ∂u/∂y

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Shallow Water Equation – Corrections to HH, pages115,116

Holton and Hakim's treatment of vorticity using the shallow water equations is full of errors – the major one being the assertion that Dp/Dt = 0 – which is wrong and unnecessary. Part of the problem is the use of h to represent the depth of fluid abovez = 0 and then to apply it relative to Fig 4.11 as if h is the thickness of a layer, not bounded below by z = 0.

IF we let h be the layer thickness we can consider a layer of fluid between zlb and zubwith

zub – zlb = h

The hydrostatic pressure assumption, with constant density ρ0 has ∂p/∂z = -ρ0 g (notemissing “-” in HH 4.31) and leads to

p(z) = ρ0 g(zub-z) + p(zub)

If we assume p(zub) = constant the pressure gradient term in the momentum equations,

(1/ρ0) hp = g h(zub)

Integration of the continuity equation .U = 0 (where U is 3D velocity and V is 2D horizontal velocity) in shallow water theory gives

Dhh/Dt = - h h .V

Partial derivatives, ∂/∂x of the y-cpt of HH 4.33 (with h replaced by zub) - ∂/∂y of thex-cpt are used to obtain, as before,

Dh(ζ + f)/Dt = - (ζ + f)(∂u/∂x + ∂v/∂y) = - (ζ + f) h .V

Multiplying by h and using the result above gives

h Dh(ζ + f)/Dt = (ζ + f) Dh h/Dt

Division by h2 then gives,

(1/h) Dh(ζ + f)/Dt - (ζ + f)(1/h)2Dh h/Dt = 0or

Dh[(ζ + f)/h]/Dt = 0 (HH 4.39)

So the result is correct, with h as the layer thickness, but HH make several errors in reaching it.

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Flow over topographic barriers

Figure 4.11 Westerly flow over a topographic barrier

Figure 4.12 Easterly flow over a topographic barrier

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A Rossby Wave

Barotropic Case, h and z constant

So Dh(ζ + f)/Dt = 0

Implications are that, if absolute vorticity is conserved, westerly zonal flow should remain zonal while easterly flow can curve N or S.

Ertel PV in Isentropic Coords. Not covered in detail

Polar Jet ( from http://www.nc-climate.ncsu.edu/secc_edu/images/jetstream3.jpg)

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