understanding physics-dynamics coupling in weather and climate...

Understanding physics-dynamics coupling in

weather and climate models

Bob Beare

1and Mike Cullen

1University of Exeter.

2Met O�ce, Exeter.

7 July 2015

R. J. Beare

(a)400 m grid length

-4800 -2400 0 2400 4800x (m) x (m)

-5.5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5 5.5

Vertical velocity (m s-1) Vertical velocity (m s-1)

(b)200 m grid length

-4800 -2400 0 2400 4800

-4800 -2400 0 2400 4800x (m) x (m)

-4800 -2400 0 2400 4800

-5.5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5 5.5

(c)100 m grid length

-5.5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5

(d)50 m grid length

-5.5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.54.5 5.5 4.5 5.5

Fig. 2 Horizontal cross-sections of vertical velocity at height 500 m at time 3.5 h for horizontal grid lengthsof: a 400 m, b 200 m, c 100 m and d 50 m. Contour interval 1 m s−1

grid length of 400 m, the subgrid flux has become a significant proportion (46 %) of theresolved flux, confirming that the simulation is in the grey zone at this height.

At the inversion, however, the total buoyancy flux is not constant with resolution (Fig. 3).As the horizontal grid length is decreased, the entrainment flux decreases, although notmonotonically, to the converged value at 25 m, in agreement with Sullivan and Patton (2011).Whilst the total flux is relatively insensitive to resolution in the middle of the boundary layer,the resolved fields are likely to be more sensitive. It is the resolved fields that define whetherthe simulations are in the grey zone and are the focus of of our study.

Author's personal copy

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Introduction

IWeather and climate models partitioned into resolveddynamics and parametrized sub-grid processes (‘physics’).

IUnderstanding coupling of dynamics and physics central to

improving weather and climate models e.g. tropical

performance and diurnal cycle over land.

IComplex models should converge to a state of geostrophic

balance at small Rossby number (Ro). Convergence rate Ro

Cullen (2008).

IDoes this idea extend to the physics? For boundary-layer

parametrization, use Ekman balance.

IUnphysical deviation from Ekman balance could be used to

diagnose numerical coupling problems.

ITest this approach for an idealised baroclinic wave.

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Introduction

Cullen (2008).

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Introduction

Cullen (2008).

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Introduction

Cullen (2008).

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Introduction

Cullen (2008).

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Introduction

Cullen (2008).

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Introduction

Cullen (2008).

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Slaving concept

Boundary layer (function of Ro)

Dynamics (function of Ro)

Slaving

Figure: Illustration of the slaving principle between the dynamics and

boundary layer for Ekman-balanced states. Given the dynamics is a

function of Ro, slaving means that the boundary layer is also a function

of Ro.

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Momementum equation

Scaling momentum equation assuming the boundary layer is slaved

to the dynamics. Slaving means the boundary layer shares the

timescales of the dynamics.

Material Derivativez }| {Ro

Coriolisz }| {k⇥ b

Pressure gradientz }| {Ro

2brb� =

Dragz }| {EkBbu,

b are dimensionless values.

fLFr =

B = h2 @

✓bKm

U horizontal velocity, L length, f Coriolis parameter, NBrunt-Vaisalla frequency, H troposphere depth, K typical

boundary-layer di↵usion, h mean boundary-layer depth.

Full details in Beare and Cullen 20155 / 13

Simplified momentum equation

For pressure gradient O(1) and geostrophic limit at small Ro

Ro = Fr

For boundary-layer depths small compared to troposphere, pressure

gradient constant across the boundary layer. Coriolis and drag

terms have the same magnitude

Ek = O(1).

Momentum equations simplifies to

Material Derivativez }| {Ro

Pressure gradientz}|{brb� =

Dragz}|{Bbu .

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Ekman-balanced states

Ekman balance In the limit of Ro ! 0

Dragz}|{Bbu

is the Ekman momentum.

SGT model Additional trajectory (

) variable. The SGT

momentum balance is

Balanced material derivativez }| {Ro

DsbueDbt

Balanced dragz }| {B(2bu

� bu

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Baroclinic wave.

bu � bug at Ro = 0.15, contour interval

Primitive equations

-0.5 0 0.5

!z 0.15

Ekman balance SGT model

-0.5 0 0.5

!z 0.15

-0.5 0 0.5

!z 0.15

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bu � bug at Ro = 0.04, contour interval 0.02.

Primitive equations

-0.5 0 0.5

!z 0.15

Ekman balance SGT model

-0.5 0 0.5

!z 0.15

-0.5 0 0.5

!z 0.15

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Convergence with Ro

Rossby no.10

Ekman (ue)

SGT (us)

Figure: Log-log plot of deviation of primitive equation model from either

Ekman balance ( filled circles) or SGT model (crosses) against Ro.

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Boundary-layer numerical instability

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Disruption of convergence with Ro

Rossby no.0.04 0.06 0.08 0.1

ControlK-updateWood et al.

Figure: A log-log plot of deviation of the primitive equation model from

the SGT model against Ro for the di↵erent boundary-layer time-stepping

schemes and a larger time step. 12 / 13

Summary

IConvergence to Ekman balance challenging test for

physics-dynamics coupling in weather and climate models.

IConvergence with Rossby number shown for a boundary layer

coupled to a baroclinic wave. Varies as Ro with respect to

Ekman balance and Ro

1.7with respect to the SGT model.

INot quite as fast convergence as dynamics-only models (Ro

IConvergence disrupted by unstable numerical time-stepping

methods.

IApproach can be used to design idealised testbeds including

physics.

IExtend ideas to tropical convection.

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Summary

methods.

physics.

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Summary

methods.

physics.

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Summary

methods.

physics.

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Summary

methods.

physics.

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Summary

methods.

physics.

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Cullen, M. J. P. (2008). A comparison of numerical solutions to

the Eady frontogenesis problem. Quart. J. Roy. Meteorol. Soc.,134, 2143–2155.

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understanding physics-dynamics coupling in weather and climate...

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