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Basic Dynamics-IJohn Thuburn
Basic Dynamics - I
Monday 2 June, 2008
Page 1
Basic Dynamics-IJohn Thuburn
Bibliography
• An Introduction to Dynamic Meteorology, J.R. Holton, Elsevier /
Academic Press
• Atmosphere Ocean Dynamics, A.E. Gill, Academic Press
• Geophysical Fluid Dynamics, J. Pedlosky, Springer
• Lectures on Geophysical Fluid Dynamics, R. Salmon, OUP
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Basic Dynamics-IJohn Thuburn
Outline
• The multiscale nature of atmospheric dynamics
• Dynamics in a rotating frame
• Governing equations
• Acoustic waves
• Inertio-gravity waves
• Phase velocity and group velocity
• Some key conservation properties
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Basic Dynamics-IJohn Thuburn
The multiscale nature of atmospheric dynamics
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Basic Dynamics-IJohn Thuburn
Dynamics in a rotating frame
Let
A = AxI + AyJ + AzK
= axi + ayj + azk
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Basic Dynamics-IJohn Thuburn
Time rate of change of an arbitrary vector
(DA
Dt
)
IF
= IDAx
Dt+ J
DAy
Dt+ K
DAz
Dt
= iDax
Dt+ j
Day
Dt+ k
Daz
Dt
+ ax
(Di
Dt
)
IF
+ ay
(Dj
Dt
)
IF
+ az
(Dk
Dt
)
IF
=
(DA
Dt
)
RF
+ Ω × A
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Basic Dynamics-IJohn Thuburn
Apply to position vector
uIF = uRF + Ω × x
Apply to velocity vector
aIF = aRF + 2Ω × uRF + Ω× (Ω× x)
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Basic Dynamics-IJohn Thuburn
Governing equations
Mass
∂ρ
∂t+ ∇.(ρu) = 0
Thermodynamics
Dθ
Dt= Q
Momentum
Du
Dt+ 2Ω × u = −
1
ρ∇p −∇Φ + F
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Basic Dynamics-IJohn Thuburn
along with
p = RTρ
and
T =
(p
p00
)κ
θ = Π(p)θ
where κ = R/Cp.
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Basic Dynamics-IJohn Thuburn
Some common approximations
• Spherical geoid: Φ = Φ(r)
• Quasi-hydrostatic: neglect Dw/Dt
• Anelastic: ∇(ρ0u) = 0
• Shallow atmosphere:
neglect Coriolis terms involving horizontal component of Ω;
replace 1/r by 1/a;
neglect some metric terms.
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Basic Dynamics-IJohn Thuburn
Quasigeostrophicequations
Planetarygeostrophic
Sphericalgeoid
CompressibleEulerequations
Quasi−Shallowatmosphere
Hydrostaticshallowatmosphere
Anelastic
Boussinesq
waterequations
Quasigeostrophicshallow waterequations
Barotropicvorticityequation
Semi−geostrophic
Shallow
hydrostatic
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Basic Dynamics-IJohn Thuburn
The importance of waves
• Acoustic waves: very fast, energetically very weak
• Inertio-gravity waves: fast, energetically weak
• Rossby waves and balanced vortical motion: energetically dominant
But fast waves are crucial for adjustment towards balance.
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Basic Dynamics-IJohn Thuburn
Acoustic waves
Neglect Coriolis and gravity
∂ρ
∂t+ ∇.(ρu) = 0
Du
Dt= −
1
ρ∇p
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Basic Dynamics-IJohn Thuburn
Linearize about an isothermal state of rest
∂ρ
∂t+ ρ0∇.(u) = 0
∂u
∂t= −
1
ρ0∇p = −
c2
ρ0∇ρ
where c2 = ∂p/∂ρ|θ = RT0/(1 − κ)
∂2ρ
∂t2− c2∇2ρ = 0
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Basic Dynamics-IJohn Thuburn
Seek wavelike solutions ∝ expik.x− ωt
to obtain the dispersion relation
ω2 = c2|k|2
Note u||k: waves are longitudinal
Also, acoustic waves are non-dispersive
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Basic Dynamics-IJohn Thuburn
Inertio-gravity waves
Make the Boussinesq approximation: assume the fluid to be
incompresible ∇.u = 0, and neglect variations in density except where
they appear in a buoyancy term, i.e. multiplied by g.
Also neglect Coriolis terms involving the horizontal component of Ω.
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Basic Dynamics-IJohn Thuburn
Du
Dt− fv = −
1
ρ0
∂p
∂x,
Dv
Dt+ fu = −
1
ρ0
∂p
∂y,
Dw
Dt= −
1
ρ0
∂p
∂z+ b,
∂u
∂x+
∂v
∂y+
∂w
∂z= 0,
Db
Dt+ wN2 = 0,
where
b = −gρ′
ρ0and N2 = −
g
ρ0
∂ρ0
∂z
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Basic Dynamics-IJohn Thuburn
Linearize about a hydrostatic state of rest
∂u
∂t− fv = −
1
ρ0
∂p
∂x,
∂v
∂t+ fu = −
1
ρ0
∂p
∂y,
∂w
∂t= −
1
ρ0
∂p
∂z+ b,
∂u
∂x+
∂v
∂y+
∂w
∂z= 0,
∂b
∂t+ wN2 = 0.
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Basic Dynamics-IJohn Thuburn
Seek wavelike solutions ∝ ei(kx+ly+mz−ωt) to obtain the dispersion
relation
ω2 =(k2 + l2)N2 + m2f2
k2 + l2 + m2
Note u ⊥ k: waves are transverse
For very deep waves, m2/(k2 + l2) ≪ 1, ω2 ≈ N2
For very shallow waves, (k2 + l2)/m2 ≪ 1, ω2 ≈ f2
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Basic Dynamics-IJohn Thuburn
Phase velocity
If a wavelike disturbance is ∝ expiφ(x, t)
wave crests and troughs are surfaces of constant phase φ.
For a plane wave
φ = k.x− ωt
= k.(x− cpt)
where cp = kω/|k|2 is the phase velocity. Crests and troughs move
at velocity cp.
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Basic Dynamics-IJohn Thuburn
Group velocity
How does a packet of waves propagate?
Consider a superposition of two 1D waves with similar wavenumber
and frequency, satisfying ω = ω(k)
q =1
2
(ei(k+δk)x−(ω+δω)t + ei(k−δk)x−(ω−δω)t
)
= cos(δk x − δω t)eikx−ωt (1)
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Basic Dynamics-IJohn Thuburn
Individual crests and troughs propagate at the phase velocity
cp = ω/k
but wave packets propagate at the group velocity
cg = δω/δk → ∂ω/∂k
In 3D
cg = ∇kω =
(∂ω
∂k,∂ω
∂l,
∂ω
∂m
)
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Basic Dynamics-IJohn Thuburn
Matlab demo of phase and group velocity
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Basic Dynamics-IJohn Thuburn
Some key conservation properties
Some conservation properties can be expressed as
∂A
∂t+ ∇.(F) = 0
Mass
A = ρ and F = ρu
Angular momentum
A = ρz. [r × (u + Ω × r)] and F = uA + pz× r
Energy
A = ρ(u2/2 + CvT + Φ) and F = u(A + p)
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Basic Dynamics-IJohn Thuburn
Potential temperatureDθ
Dt= 0
Potential vorticityDQ
Dt= 0
where Q = ζ.∇θ/ρ
Potential enstrophy
A = ρQ2/2 and F = uA
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