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© Crown copyright 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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© Crown copyright Met Office 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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© Crown copyright Background

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© Crown copyright 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 10 30 times faster than now available Therefore cannot guarantee that numerical model solutions will be useful

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© Crown copyright 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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© Crown copyright 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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© Crown copyright The semi-geostrophic scaling

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© Crown copyright Euler-Boussinesq system Illustrate with Euler-Boussinesq system with constant rotation in plane geometry and a free upper boundary.

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Initial and boundary conditions © Crown copyright Solve on [0,τ)xΩ(t) with

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© Crown copyright Scaled equations Consider rotation dominated limit, ε=U/(fL) (Rossby number)=(H/L) 2 The other equations are unchanged.

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© Crown copyright Semigeostrophic equations Define the geostrophic wind by The other equations are unchanged then the geostrophic wind by

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© Crown copyright Properties of solutions

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© Crown copyright Domain of validity This (Philips type II) scaling requires the Froude number to be O(Rossby ½ ). This implies that the horizontal scale is larger than the Rossby radius L D, 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 L D smaller)

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© Crown copyright SG evolution The SG evolution equations can be written in the form: S is the diabatic heat source. Q measures the sensitivity to forcing.

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© Crown copyright SG evolution II In unscaled variables, thus allowing variable rotation, they are:

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© Crown copyright 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 Du g /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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© Crown copyright 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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© Crown copyright 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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© Crown copyright Persistent of eddies Consider the difference between SG shallow water flow, with depth h, and 2d incompressible flow. In SG, a vortex in u g 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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© Crown copyright 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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© Crown copyright Evolution of spectra at 200hpa

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© Crown copyright Evolution of spectra at 200hpa

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© Crown copyright Evolution of spectra at 200hpa

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© Crown copyright Validation of numerical models (with Abeed Visram and Colin Cotter)

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© Crown copyright Eady model In a 2d domain (x 1,x 3 )(-L,L)x(0,H), and with D t, and ∂ operating only in 2d:

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© Crown copyright Eady model scaled With ε=U/fL, L/H can be O(1). u 1 =εu 2.

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© Crown copyright Semigeostrophic limit

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© Crown copyright Numerical test The SG solution is invariant (in this problem) to rescaling x 1 to βx 1, u 1 to βu 1 and f to β -1 f. 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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© Crown copyright Plot rms value of u 2 Effect of resolution

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© Crown copyright Convergence as β reduced

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© Crown copyright Converence to geostrophic balance

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© Crown copyright 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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© Crown copyright Enforcing conservation Illustrate the difference between (ρ a ) 2 and (ρ 2 ) a normalised by ρ 2 under advection for one timestep using smooth prefrontal flow fields. Vρ 11.4E-31.0E-412.2E-42.2E-5 L∞L∞ 25.3E-43.0E-5L∞L∞ 21.1E-46.6E-6 44.3E-49.8E-643.1E-52.1E-6 Convrate0.881.701.411.70 L2L2 12.9E-56.2E-6L2L2 16.1E-61.8E-6 26.4E-69.9E-721.6E-63.2E-7 42.9E-62.2E-744.2E-77.2E-8 Convrate1.672.411.922.33

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© Crown copyright Enforcing conservation Illustrate the difference between (ρ a ) 2 and (ρ 2 ) a normalised by ρ 2 under advection for one timestep using sharp post frontal flow fields. Vρ 18.7E-21.7E-111.3E-21.5E-2 L∞L∞ 28.7E-19.5E-1L∞L∞ 23.3E-23.2E-2 41.9E01.9E-045.4E-28.1E-2 Convrate-2.21-1.70-1.02-1.21 L2L2 13.1E-34.3E-3L2L2 14.0E-43.9E-4 21.4E-21.7E-227.0E-47.9E-4 41.3E-22.1E-246.8E-44.7E-4 Convrate-1.05-1.16-0.07-0.40

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© Crown copyright 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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© Crown copyright Extension to the atmospheric boundary layer (with Bob Beare)

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© Crown copyright 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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© Crown copyright Equations 2d cross-section as before

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© Crown copyright Scaled equations Let E k =K m /fh 2, h is boundary layer depth. Assume E k =O(1) in boundary layer, 0 elsewhere. Do not assume u 1 =εu 2.

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Geostrophic balance Ekman (geotriptic) balance Steady state balances Prognostic models Planetary geostrophic (PG) Semi-geostrophic (SG) Quasi-geostrophic (QG) Planetary-geotriptic (PGT) Semi-geotriptic (SGT) Geostrophic wind Coriolis Pressure gradient Boundary layer drag Ekman balanced wind Coriolis Pressure gradient

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© Crown copyright Leading order balance Ekman balance In general, define Ekman balanced wind by

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© Crown copyright SG consistent balance Replace u by u e in D t term, get to O(ε 2 )

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© Crown copyright Sustainable balance These equations do not have a negative definite energy integral. Instead to O(ε) set

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© Crown copyright Comments These (SGT) equations are no more accurate in the boundary layer than just imposing u=u e. 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(ε 2 ) accuracy sustainably with models of this type. Probably the boundary layer cannot be ‘balanced’ to this order.

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© Crown copyright Diagnostic equation for u Calculate u required to maintain Ekman balance relation for u e (Sawyer-Eliassen equation in SG case). Gives diagnostic equation for stream function which determines (u 1,u 3 ). Then deduce u 2. Diagnostic equation is elliptic if the state is statically stable, and satisfies an inertial stability condition reinforced by friction.

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© Crown copyright Low level jet simulation Use analytically generated jet profile, wind in x 2 direction. Profiles of u 2 and ρ -1 (illustrating boundary layer structure):

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© Crown copyright Diagnosed u 3 Positive values bold. Max negative value much bigger than given by Ekman pumping.

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© Crown copyright Diagnosed u 2 u 2 in bold, u e2 feint.

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© Crown copyright 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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© Crown copyright Questions

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