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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
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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
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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 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
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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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