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Modelling large-scale atmospheric circulations using semi-geostrophic theoryMike 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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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 Navier-Stokes 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.

 

Initial and boundary conditions

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Solve on [0,τ)xΩ(t) with

)

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Scaled equations

Consider rotation dominated limit, ε=U/(fL) (Rossby number)=(H/L)2

 

The other equations are unchanged.

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Semigeostrophic equations

Define the geostrophic wind by

𝛻 ∙𝑢𝑔=0 ;𝑢=𝑢𝑔+O (𝜀 )  𝜀𝐷𝑡 𝑢𝑔+(−𝑢2 ,𝑢1 ,0 )+𝛻𝑝+𝜌=O (𝜀2 )  

The other equations are unchanged

𝛻 { 1,2}𝑝+(−𝑢𝑔2 ,𝑢𝑔1 )=0 ,𝑢𝑔 3=0

then the geostrophic wind by

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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.

Variable Coriolis effect needed whenever SG applicable (but not in ocean where LD smaller)

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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.

This is only realistic if the forcing is on a slow time scale.

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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 shielded-but 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.

Euler obeys the same Lagrangian conservation laws.

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Plot rms value of u2Effect 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.

V ρ

1 1.4E-3 1.0E-4 1 2.2E-4 2.2E-5

L∞ 2 5.3E-4 3.0E-5 L∞ 2 1.1E-4 6.6E-6

4 4.3E-4 9.8E-6 4 3.1E-5 2.1E-6

Conv rate 0.88 1.70 1.41 1.70

L2 1 2.9E-5 6.2E-6 L2 1 6.1E-6 1.8E-6

2 6.4E-6 9.9E-7 2 1.6E-6 3.2E-7

4 2.9E-6 2.2E-7 4 4.2E-7 7.2E-8

Conv rate 1.67 2.41 1.92 2.33

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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.

V ρ

1 8.7E-2 1.7E-1 1 1.3E-2 1.5E-2

L∞ 2 8.7E-1 9.5E-1 L∞ 2 3.3E-2 3.2E-2

4 1.9E0 1.9E-0 4 5.4E-2 8.1E-2

Conv rate -2.21 -1.70 -1.02 -1.21

L2 1 3.1E-3 4.3E-3 L2 1 4.0E-4 3.9E-4

2 1.4E-2 1.7E-2 2 7.0E-4 7.9E-4

4 1.3E-2 2.1E-2 4 6.8E-4 4.7E-4

Conv rate -1.05 -1.16 -0.07 -0.40

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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

0

03

3

3

2

312

3

1

32

11

u

FD

gx

puD

x

uK

xfufUuD

x

uK

xfu

x

puD

bt

t

mt

mt

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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.

0

ˆ

0

ˆ

ˆ

33

2

3

2

312

3

1

32

11

u

FD

x

puD

x

uK

xuUuD

x

uK

xu

x

puD

bt

t

mt

mt

Geostrophic balance

Ekman (geotriptic) balance

Steadystatebalances

Prognosticmodels

Planetary geostrophic (PG) Semi-geostrophic (SG) Quasi-geostrophic (QG)

Planetary-geotriptic (PGT) Semi-geotriptic (SGT)

Geostrophicwind

CoriolisPressuregradient

Boundary layer drag

Ekmanbalancedwind

CoriolisPressuregradient

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Leading order balance

Ekman balance

3

2

31

3

1

32

1

ˆ

ˆ

x

uK

xuU

x

uK

xu

x

p

m

m

In general, define Ekman balanced wind by

3

2

31

3

1

32

1

ˆ

ˆ

x

uK

xuU

x

uK

xu

x

p

eme

eme

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SG consistent balance

Replace u by ue in Dt term, get to O(ε2)

0

ˆ

0

ˆ

ˆ

3

3

2

312

3

1

32

11

u

FD

x

p

x

uK

xuUuD

x

uK

xu

x

puD

bt

met

met

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Sustainable balance

These equations do not have a negative definite energy integral. Instead to O(ε) set

0

ˆ

0

ˆ

ˆ

3

2233

112

113

1

3221

u

FD

x

p

uux

Kx

uuuD

uux

uK

xuuuD

bt

emeet

emeet

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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 simulationUse 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