Modelling large-scale atmospheric
circulations using semi-geostrophic
theory
Mike Cullen and Keith Ngan Met Office
Colin Cotter and Abeed Visram (Imperial College), Bob Beare (Exeter University)
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Contents
This presentation covers the following areas
• Background
• Semi-geostrophic scaling
• Properties of the large-scale regime
• Application to validating numerical models
• Application to boundary layer-free atmosphere
interaction
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Met Office
Background
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Governing equations
On all relevant scales, the atmosphere is
governed by the compressible Navier-Stokes
equations, the laws of thermodynamics, phase
changes and source terms
The solutions of these equations are very
complicated, reflecting the complex nature of
observed flows
The accurate solution of these equations would
require computers 1030 times faster than now
available
Therefore cannot guarantee that numerical
model solutions will be useful
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Uses of reduced models
Show why the large scales can be predicted well,
even though system is nonlinear.
Validating numerical models.
Understanding the solution of the governing
equations in particular regimes.
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Method
Characterise regime using appropriate
asymptotic limit
Derive asymptotic limit equations satisfying basic
conservation properties (e.g. mass, energy)
Show that they can be solved.
Prove that solutions of the Euler or NavierStokes equations converge to them at the
expected rate-validating the scale analysis.
Include the error estimate when making
predictions.
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The semi-geostrophic scaling
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Euler-Boussinesq system
Illustrate with Euler-Boussinesq system with
constant rotation in plane geometry and a free
upper boundary.
𝐷𝑡 𝑢 + 𝑓𝑒3 × 𝑢 + 𝛻𝑝 + 𝜌𝑔𝑒3 = 0
𝐷𝑡 𝜌 = 0
𝛻∙𝑢 =0
𝜕ℎ
𝜕ℎ
𝜕𝑡 ℎ + 𝑢1
+ 𝑢2
= 𝑢3
𝜕𝑥1
𝜕𝑥2
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Initial and boundary
conditions
Solve on [0,τ)xΩ(t) with
Ω 𝑡 = Ω2 × 0, ℎ 𝑡, 𝑥1 , 𝑥2 ⊂ ℝ3 ; Ω2 ⊂ ℝ2
𝑢 ∥ 𝜕Ω 𝑡 𝑥3 = ℎ
𝑝 𝑡, 𝑥1 , 𝑥2 , ℎ 𝑡, 𝑥1 , 𝑥2 = 𝑝ℎ )
𝑢 0, 𝑥 = 𝑢0 𝑥 , 𝜌 0, 𝑥 = 𝜌0 𝑥 , ℎ(0, 𝑥) = ℎ0 (𝑥)
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Scaled equations
Consider rotation dominated limit, ε=U/(fL)
(Rossby number)=(H/L)2
𝜀𝐷𝑡 𝑢 1,2 𝑢 + −𝑢2 , 𝑢1 + 𝛻 1,2 𝑝 = 0
𝜀 2 𝐷𝑡 𝑢3 + 𝛻3 𝑝 + 𝜌 = 0
The other equations are unchanged.
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Semigeostrophic equations
Define the geostrophic wind by
𝛻 1,2 𝑝 + −𝑢𝑔2 , 𝑢𝑔1 = 0, 𝑢𝑔3 = 0
then the geostrophic wind by
𝛻 ∙ 𝑢𝑔 = 0; 𝑢 = 𝑢𝑔 + O 𝜀
𝜀𝐷𝑡 𝑢𝑔 + −𝑢2 , 𝑢1 , 0 + 𝛻𝑝 + 𝜌 = O 𝜀 2
The other equations are unchanged
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Properties of solutions
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Domain of validity
This (Philips type II) scaling requires the Froude
number to be O(Rossby½). This implies that the
horizontal scale is larger than the Rossby
radius LD, or the aspect ratio is less than f/N.
Disturbances confined to the troposphere satisfy
this for length scales>~1000km in mid-latitudes.
In this regime, PV anomalies are dominated by
static stability anomalies, and the energy by the
APE.
Rossby waves are only weakly dispersive.
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Variable Coriolis effect needed whenever SG
applicable (but not in ocean where LD smaller)
Properties
The total velocity u=(u,v,w) is computed
diagnostically, not prognostically.
It can take any size. If Du/Dt is large compared to
Dug/Dt the scale analysis will be inconsistent
and the solutions unphysical.
If Q has negative eigenvalues the state is
unstable and the flow unbalanced. SG cannot
be solved in this case.
Can be proved that if Q is positive definite at t=0,
there is always a solution of SG that preserves
this indefinitely.
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Properties II
While det Q is conserved with constant f, and only
changes slowly with variable f, individual
eigenvalues can deteriorate. This is a
mechanism for extreme reaction to forcing.
Moisture reduces the effective static stability, so
reaction to forcing is much stronger.
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Properties III
Q is positive definite regardless of the sign of f.
The condition that Q is positive definite is very
severe at the equator; p cannot vary in the
horizontal.
The ageostrophic flow maintains this against
forcing that varies in the horizontal (Hadley and
Walker circulations).
In general the ageostrophic flow filters small
scales from the forcing and limits the effect on
the large scales.
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This is only realistic if the forcing is on a slow time
scale.
Persistent of eddies
Consider the difference between SG shallow water
flow, with depth h, and 2d incompressible flow.
In SG, a vortex in ug is naturally an anomaly in h.
There is no induced flow outside the vortex. In
effect the vortex is shielded. In 2d Euler a vortex
has an effect for a long distance unless shieldedbut this is not natural.
This prevents an upscale energy cascade.
The 2d turbulence scaling argument for an upscale
energy cascade does not apply in this regime
because PV and energy anomalies both scale with
h.
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Observed spectra
If 2d turbulence theory applies, expect -3
spectrum at large scales, and -5/3 where 3d
effects take over.
Examples shown for 2010-11 winter. Observed
spectra do not show any uniform behaviour on
largest scales. Beyond wavenumber 7 there is
a systematic energy decrease with k (-5/3 law).
Data are 3 day averages of 200hpa geopotential,
spectra are in longitudinal direction averaged
from 30N-60N.
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Evolution of spectra at
200hpa
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Evolution of spectra at
200hpa
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Evolution of spectra at
200hpa
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Validation of numerical models
(with Abeed Visram and Colin Cotter)
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Numerical test
The SG solution is invariant (in this problem) to
rescaling x1 to βx1, u1 to βu1 and f to β-1f. Then ε
becomes βε.
Solve the Euler equations using a fully implicit
semi-Lagrangian method.
SG solution computed using fully Lagrangian
particle method. The latter has been proved to
converge to the SG solution.
SG solution maintains Lagrangian conservation
laws and exact geostrophic and hydrostatic
balance.
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Euler obeys the same Lagrangian conservation
laws.
Plot rms value of u2
Effect of resolution
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Convergence as β reduced
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Converence to geostrophic
balance
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Comments
Model gives the correct rate of convergence to
balance.
It reproduces the periodic lifecycles rather better
than Nakamura and Held (1989) who used
Eulerian advection and artificial viscosity.
Peak amplitude not predicted. This is because
nonlinearity stops the linear growth too quickly.
Implicit diffusion due to the limiters in the
advection scheme balances the frontogenesis.
Lagrangian conservation under advection is
badly violated.
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Enforcing conservation
Illustrate the difference between (ρa)2 and (ρ2)a
normalised by ρ2 under advection for one
timestep using smooth prefrontal flow fields.
ρ
V
1
1.4E-3
1.0E-4
2
5.3E-4
3.0E-5
4
4.3E-4
9.8E-6
Conv
rate
0.88
1.70
L2
1
2.9E-5
6.2E-6
2
6.4E-6
4
rate
L∞
Conv
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1
2.2E-4
2.2E-5
2
1.1E-4
6.6E-6
4
3.1E-5
2.1E-6
1.41
1.70
1
6.1E-6
1.8E-6
9.9E-7
2
1.6E-6
3.2E-7
2.9E-6
2.2E-7
4
4.2E-7
7.2E-8
1.67
2.41
1.92
2.33
L∞
L2
Enforcing conservation
Illustrate the difference between (ρa)2 and (ρ2)a
normalised by ρ2 under advection for one
timestep using sharp post frontal flow fields.
ρ
V
1
8.7E-2
1.7E-1
2
8.7E-1
9.5E-1
4
1.9E0
1.9E-0
Conv
rate
-2.21
-1.70
L2
1
3.1E-3
4.3E-3
2
1.4E-2
4
rate
L∞
Conv
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1
1.3E-2
1.5E-2
2
3.3E-2
3.2E-2
4
5.4E-2
8.1E-2
-1.02
-1.21
1
4.0E-4
3.9E-4
1.7E-2
2
7.0E-4
7.9E-4
1.3E-2
2.1E-2
4
6.8E-4
4.7E-4
-1.05
-1.16
-0.07
-0.40
L∞
L2
Comments
If balance and Lagrangian conservation both
enforced, should get convergence to SG which
imposes these constraints.
Explanation of failure to get adequate lifecycle is
that variance is dissipated at the front, There is
no reason why Euler solutions should not be
able to maintain Lagrangian conservation.
Obvious remedy is to improve Lagrangian
conservation (ideally enforce it-but this is very
hard in practice).
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Extension to the atmospheric
boundary layer
(with Bob Beare)
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Basic idea
Consider flow in 2d cross section with
realistic boundary layer. In particular, mixing
is strongly stability dependent.
Seek to derive scaled equations that give SG
in the free atmosphere and Ekman balance
within boundary layer. These equations
should have negative definite energy
tendency in absence of thermal forcing.
Seek to explain observed phenomena
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Equations
2d cross-section as before
p
 
u1 
 K m

Dt u1 
 fu 2 
x1
x3 
x3 
 
u2 
 K m

Dt u2  fU  fu1 
x3 
x3 
p
Dt u3 
 g  0
x3
Dt   Fb
 u  0
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Scaled equations
Let Ek=Km/fh2, h is boundary layer depth. Assume
Ek=O(1) in boundary layer, 0 elsewhere. Do not
assume u1=εu2.
p
  ˆ u1 
 K m

Dt u1 
 u2 
x1
x3 
x3 
  ˆ u2 
 K m

Dt u2  U  u1 
x3 
x3 
p
2
 Dt u3 
 0
x3
D   Fˆ
t
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 u  0
b
Geostrophic
Ekman (geotriptic) balance
balance
Steady
state
balances
Geostrophic
wind
Pressure
gradient
Coriolis
Ekman
balanced
wind
Pressure
gradient
Coriolis
Boundary layer drag
Prognostic
models
Planetary geostrophic
(PG)
Semi-geostrophic (SG)
Quasi-geostrophic (QG)
Planetary-geotriptic
(PGT)
Semi-geotriptic (SGT)
Leading order balance
Ekman balance
p
  ˆ u1 
 K m

 u2 
x1
x3 
x3 
  ˆ u2 
 K m

 U  u1 
x3 
x3 
In general, define Ekman balanced wind by
p
  ˆ u1e 
 K m

 u2e 
x1
x3 
x3 
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  ˆ u2 e 
 K m

 U  u1e 
x3 
x3 
SG consistent balance
Replace u by ue in Dt term, get to O(ε2)
p
  ˆ u1 
 K m

Dt ue1 
 u2 
x1
x3 
x3 
  ˆ u2 
 K m

Dt ue 2  U  u1 
x3 
x3 
p
 0
x3
D   Fˆ
t
 u  0
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b
Sustainable balance
These equations do not have a negative definite
energy integral. Instead to O(ε) set

  ˆ u1
 K m
ue1  u1 
Dt ue1  ue 2  u2 
x3 
x3


  ˆ 
 K m
ue 2  u2 
Dt ue 2  ue1  u1  
x3 
x3

p
 0
x3
D   Fˆ
t
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 u  0
b
Comments
These (SGT) equations are no more accurate in
the boundary layer than just imposing u=ue.
However, that does not give a negative definite
energy integral either. SGT is consistent with SG
in the free atmosphere.
Does not appear possible to get O(ε2) accuracy
sustainably with models of this type. Probably the
boundary layer cannot be ‘balanced’ to this order.
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Diagnostic equation for u
Calculate u required to maintain Ekman balance
relation for ue (Sawyer-Eliassen equation in SG
case).
Gives diagnostic equation for stream function
which determines (u1,u3). Then deduce u2.
Diagnostic equation is elliptic if the state is
statically stable, and satisfies an inertial stability
condition reinforced by friction.
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Low level jet simulation
Use analytically generated jet profile, wind in x2
direction. Profiles of u2 and ρ-1 (illustrating
boundary layer structure):
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Diagnosed u3
Positive values bold. Max negative value much
bigger than given by Ekman pumping.
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Diagnosed u2
u2 in bold, ue2 feint.
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Comments
Enhanced low level jet seen, as often observed.
This is a different mechanism from the nocturnal
collapse of the boundary layer-also often
observed.
This model very useful in the tropics, where
friction can support a horizontal pressure gradient,
while geostrophic dynamics cannot.
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Questions
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