14 Aug 2015
EATS 3040-2015 Notes 1
Some course material will be online at http://www.yorku.ca/pat/ESSE3040/
HH = Holton and Hakim. An Introduction to Dynamic Meteorology, 5th Edition. Most of
the images will be from http://booksite.elsevier.com/9780123848666/index.php
1. Introduction
Basics of atmospheric structure:
troposphere, tropopause, stratosphere
DALR, saturated lapse rate, typical soundings
http://weather.uwyo.edu/upperair/sounding.html
SkewT - log p plot of T and dew point temperature. Shows dry adiabats and saturated adiabats.
Tephigrams are similar. Also note various thermodynamic indices and wind data. Listings are
also available at the U of Wyoming site.
Why does T decrease in the troposphere and increase in the stratosphere? What is the height of
the tropopause?
Estimate dT/dz ≈ (-10-14)/3500 ≈ 6.9 K/km, between 1521m and 5010m. Today?
Tropopause height, 200hPa, 12.2 km
Basic Dynamical Variables:
We assume that air is a continuum- no need to worry about individual molecules so quantities, e.g.
temperature, velocity, are continuous functions of position (r).
SI base units (m, s, kg, K, and radians). Derived units (Hz - cycles/s, N, Pa, J, W)
N- Newton, unit for Force = mass x acceleration, kgms-2
Pa-Pascal, unit for stress, including pressure, force per unit area, Nm-2
J - Joule, unit for work or energy, Nm
W - Watt, Unit for power, rate of doing work, Js-1
Basic variables in Dynamical Meteorology
V=
(u,v) - Horizontal wind vector. U and V are normally zonal (W to E) and meridional (S to
N) components. Wind Direction usually given as the direction that the wind is coming
from. So if V=(10,0), wind is from ...... and if V=(-10,-10), wind is ......
U=
(u,v,w) - full 3 dimensional wind vector in Cartesian coordinates, U = V + wk
Also use pressure coordinates and, in place of vertical velocity, ω = Dp/Dt, D represents
differentiation following the fluid. So positiveω is negative w. (ω ≈ -ρgw)
Frequently assume w << u,v. Winds are approximately horizontal most of the time.
p
Pressure, mostly hydrostatic but dynamic pressures can be important at small scales.
T
Temperature, in degrees Kelvin
ρ
Density in kgm-3.
g
apparent gravitational acceleration (includes centrifugal force) - approximately 9.81 ms-2.
Ω
Earth's angular velocity, 7.292 x 10-5 s-1. Not exactly 2π/(24x60x60) = 7.272 x 10-5 s-1
(x 366/365) . Why?
Forces involved (see HH, 1.2, 1.3)
Conservation of mass and momentum, Newton's Laws, inertial frame of reference.
Normal stress (pressure) and pressure gradient force, -grad p
Gravitational force/unit mass, g
The apparent forces, centrifugal and Coriolis.
FIGURE 1.5 Relationship between the true gravitation vector g* and gravity g. For an idealized
homogeneous spherical Earth, g* would be directed toward the center of Earth. In reality, g* does
not point exactly to the center except at the equator and the poles. Gravity, g, is the vector sum of
g* and the centrifugal force and is perpendicular to the level surface of Earth, which approximates
an oblate spheroid.
FIGURE 1.8 Components of the Coriolis force due to relative motion along a latitude circle.
Viscous shear stress and forces - Navier-Stokes Equations
Ideal gas law and Hydrostatic pressures, Bernoulli Equation and dynamic pressures.
Pressure as a vertical coordinate.
1.4.1 The Hydrostatic Equation
FIGURE 1.9 Balance of forces for hydrostatic equilibrium. Small arrows show the upward and
downward forces exerted by air pressure on the air mass in the shaded block. The downward force
exerted by gravity on the air in the block is given by ρgdz, whereas the net pressure force given by
the difference between the upward force across the lower surface and the downward force across
the upper surface is –dp. Note that dp is negative, as pressure decreases with height. (After
Wallace and Hobbs, 2006.)
Hydrostatic pressure assumption, dp/dz = -ρg , so
∞
p ( z )=∫ ρ g dz
z
Geopotential changes in a column, dΦ = gdz = -αdp = -(RT/p)dp = - RTd(lnp)
So,
P1
Φ( z 2 )−Φ( z 1)=g 0 (Z 2−Z 1 )=R∫ T dlnp
p2
called the Hypsometric equation, which you will need. Z=Φ(z)/g0 is called the geopotential height
where g0 is a reference, global averags of g at mean sea level. We are often interested in layer
thickness,
P
1
Z T =( Z 2 −Z 1)=(R /g 0)∫ T dlnp
p2
In an isothermal atmosphere and with Z1 = 0 we would have
Z = H ln(p0/p) where H =RT/g0 is a scale height. And p(Z)=p0e-Z/H.
1.4.2 Pressure as a vertical coordinate
Look at analysis maps, http://weather.gc.ca/analysis/index_e.html
Gepotential heights at 850, 700, 500, 250 hPa. MSL pressures at surface.
C
A
B
FIGURE 1.10 Slope of pressure surfaces in the x, z plane.
We need to relate (∂p/∂x)z to (∂Z/∂x)p or (∂Φ/∂x)p
Note in the diagram δp = ρgδz but we need to be careful with signs.
Between A and C there is no pressure change.
Between A and B there is a pressure change (∂p/∂x)z δx
Between B and C there is a pressure change (∂p/∂z)x δz = - ρgδz
So, - ρgδz + (∂p/∂x)z δx = 0 and (1/ρ)(∂p/∂x)z = gδz/δx = (∂Φ/∂x)p in the limit as δx → 0.
Similarly (1/ρ)(∂p/∂y)z = gδz/δy = (∂Φ/∂y)p in the limit as δy → 0.
So in isobaric coordinates the pressure gradient is replaced by the gradient of the geopotential or
geopotential height on the isobaric surfaces. (note now now density dependence).
Other vertical coordinates, σ coordinates, isentropic coordinates, .....
Kinematics and Scale Analysis (HH 1.5) horizontal divergence, δ, and vertical component of
vorticity (curl U), ς. Pure deformation, d1, and d2. (see Fig 1.12)
FIGURE 1.12 Velocity fields associated with pure vorticity (a), pure divergence (b), pure
deformation (c), and a mixture of vorticity and convergence (d).
Scale analysis and dimensional analysis (HH 1.6).
Buckingham's pi theorem. (from Wikipedia)
In mathematical terms, if we have a physically meaningful equation such as
f(q1,q2,....,qn)=0, where the qi are the n physical variables, and they are expressed
in terms of k independent physical units, then the above equation can be restated
as F(π1,π2, ....., πn)=0 where the πi are dimensionless parameters constructed
from the qi by p = n − k dimensionless equations —the so-called Pi groups— of
the form πi = q1α1q2α2......qnαn, where the exponents ai are rational numbers (they
can always be taken to be integers: just raise it to a power to clear denominators).
The use of the πi as the dimensionless parameters was introduced by Edgar
Buckingham in his original 1914 paper on the subject from which the theorem
draws its name.
Examples: Period of a pendulum, Shallow and deep water wave propagation speeds.
Surface pressure map, from http://weather.gc.ca/
Geostrophic wind. Typical grad p ~ Delta p/L – but how to estimate L?
Earth radius ≈ 6371 km so 1 degrees latitude ≈ 6371π/180 ≈ 111 km.
Or use http://www.nhc.noaa.gov/gccalc.shtml from lat/long differences.
Also f? Earth rotation rate, Ω ≈ 2π/(24x3600) radians per second ≈ 7.27x10-5 s-1.
So f = 2Ω sinφ, where φ is latitude, ≈ 45 degrees, so f ≈ 1.03 x 10-4 s-1.
Near Toronto on this map |grad p| ≈ 400Pa/(3 x 111 km) = 1.2x10-3 Pa/m.
Map was 00Z on 14 Aug 2014
Coriolis parameter f ≈ 10-4 s-1. Air density ρa ≈ 1.2 kgm-3, |Ug| ≈ |grad p|/(f ρa),
CHECK dimensions or units.
|Ug| ≈ 1.2x10-3/(1.2 x 10-4) = 10 ms-1 in this case.
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