1 Oct 2015
EATS 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,
∮
Da U a
grad p
. dl =−∮
. dl −∮ g s k.dl
Dt
ro
(4.1)
where gs is true gravity (excluding centrifugal force)
Da U a
D
D
. dl = a (U a . dl )−U a . a (d l )
Dt
Dt
Dt
? 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,
DC a D
= (U a . dl )=−∮ ro−1 dp
Dt
Dt
(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 where
Ue = 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,
DA e
DC
=−∮ ro−1 dp−2 OMEGA
Dt
Dt
(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
DC
=−∮ ro−1 dp=−∮ RT d (lnp)
Dt
"Horizontal" lines are isobaric so no contribution, vertical lines give contribution
p
DC
=R ln ( 0 )( T̄2− T̄1 )>0
Dt
p1
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 early
models (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 =∫∫ curl U . 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.
Linear shear flow, with vorticty,]
shear vorticity.
Flow around a corner, may
have no vorticity
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4.3 The Vorticity Equation
Cartesian Coordinates - fixed relative to Earth, with standard approximations and no friction, but f
may 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 written
as -(∂p/∂y ∂α/∂x - ∂p/∂x ∂α/∂y) = - ( α x p).k
Solenoid: 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
velocity
on pressure surfaces) and V the horizontal
∂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 .
)a
so that we have
∂ζ /∂t = - (V. )(ζ + f) - ω ∂ζ/∂p - (ζ + f)(
rate of change
advection
.V) + k.(∂V/∂p x
stretching
ω)
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 w L H ρ δρ/ρ δp T f β -
10 m/s
0.01 m/s
106 m - length scale, 1000 km
104m - depth scale, 10 km
1 kg/m3 - typical air density
10-2 - fractional density fluctuation
103 Pa - (10hPa - typical pressure difference over 1000 km
105 s - Time scale (L/U) about 30 hours
10-4 s-1
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=16
Select 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 to
changes in circulation around such a line.
∮
Using Stokes Theorem C a = U a . dl = ∫∫ ωa.n dA where n is normal to the isentropic
surface in which the line integral is evaluated.
dA
dh
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 no
solenoidal 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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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 typically
of 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. 877-946
On the use and significance of isentropic potential vorticity maps
By B. J. HOSKINS, M. E. McINTYRE and A. W. ROBERTSON
SUMMARY
The 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 to
take 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
In this incompressible fluid
independent of z.
(4.33)
(u,v,w) = 0 so, if w = 0 on z = 0, w(h) = -h
hV,
since V is
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 the
shallow water layer if the layer depth changes), and not needed to assert that,
Dhh/Dt = w(h) = -h
h
V
(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 above
z = 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 zub
with
zub – zlb = h
The hydrostatic pressure assumption, with constant density ρ0 has ∂p/∂z = -ρ0 g (note
missing “-” 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 the
x-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 = 0
or
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
A Rossby
Wave
Figure 4.11 Westerly flow over a topographic barrier
Figure 4.12 Easterly flow over a topographic barrier
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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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