msc module mtmw14 : numerical modelling of atmospheres and oceans

(c) 2004 d.b.stephenson@reading.ac.uk 1

MSc Module MTMW14: Numerical modelling of atmospheres and oceans

Week 3: Modelling the Real World

3.1 Staggered time schemes (semi-implicit)3.2 Physical parameterisations3.3 Ocean modelling

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Course content

Week 1: The Basics1.1 Introduction1.2 Brief history of numerical weather forecasting 1.3 Dynamical equations for the unforced fluid

Week 3: Modelling the real world3.1 Physical parameterisations3.2 Ocean modelling3.3 Staggered time schemes and the semi-implicit method

Week 4: More advanced spatial methods4.1 Staggered horizontal and vertical grid discretisations4.2 Lagrangian and semi-lagrangian schemes4.3 Series expansion methods

Week 5: Final thoughts5.1 Revision5.2 Test5.3 Survey of state-of-the-art coupled climate models

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3.2 Physical parameterisations

],;[][ tFQt

Q is the fluid dynamics term (dynamical core)

F is forcing and dissipation due to physical processes such as:

• radiation• clouds• convection• horizontal and vertical mixing/transport• soil moisture and land surface processes• etc.

The atmosphere and oceans are FORCED DISSIPATIVE fluid dynamical systems:

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“Earth System” Science, NASA 1986

From Earth System Science – Overview, NASA, 1986

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In the Hollow of a Wave off the Coast at Kanagawa", Hokusai

3.1 Unresolved process: horizontal mixing

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3.1 Small-scale mixing

Two approaches:

• Direct Numerical Simulation (DNS)try to resolve all spatial scales using a very high resolution model.

• Large-Eddy Simulation (LES)model the effects of sub-grid scale eddies as functions of the large-scale flow (closure parameterisation).

][]'[

'

FQ

''

0'

'

babaab

a

aa

aaa

:averaging sReynold' on Tips

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3.1 Horizontal mixing in vorticity eqn

)(''

0)''.().(

0))')('.(('

0).(

fu

uut

uutt

ut

:by eddies grid-sub to due fluxvorticity mean the separameteri totry then

:boxes grid over average

Or more correctly model fluxes due to unresolved scales as f(.) + random noise rather than just f(.). (STOCHASTIC PHYSICS)

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1.3 Some mixing parameterisations

DiffusionAssume that vorticity flux due to eddies acts to reduce the large-scale vorticity gradient:

Note: Mixing isn’t always down-gradient. For example, vorticity fluxes due to mid-latitude storms act to strengthen the westerly jets!

)viscosity! molecular (NOT(d) viscosity"eddy "

2)''.(

''

u

u

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1.3 Some mixing parameterisations

Hyper-diffusion

Diffusion is found to damp synoptic features too much. More scale-selective damping can be obtained using:

Note: Widely used in spectral models.

klmm

m

lk

etcm

u

~)()(

.,4,3,2

)()''.(

222

2

: space spectral in form Simple

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1.3 Some mixing parameterisations

Strain-dependent viscosity (Smagorinsky 1963)Put more diffusion in regions that have more large-scale strain:

Note: Diffuses more strongly in high strain zones (e.g. between cyclones) but is computationally expensive to implement.

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2

2

][

2

1)2.0(

TL

Dimensions

y

v

x

u

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3.1 Spectral blocking and instabilityIn 2-d turbulence and the atmosphere and oceans, enstrophy cascades to ever smaller scales:

This cascade can’t continue for ever in numerical models and so in the absence of any mixing parameterisations, energy builds up on the smallest scales. This is known as spectral blocking and can lead to non-linear instability:

Phillips, N. A., 1959: An example of nonlinear computational instability. In: B. Bolin (Editor), The Atmosphere and the Sea in Motion. Rockefeller Inst. Press in association with Oxford Univ.Press, New York, 501-504.

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3.1 Convection schemes

Why parameterise cumulus convection?

Precipitation is caused by rising of air due to: Local convective instability Large-scale ascent (slantwise convection) Orographic influences

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3.1 Convection schemes

Convection occurs on small spatial scales

(less than 10km) not explicitly resolvable

by weather and climate models.

It is important for

producing precipitation and releasing latent heat (diabatic heating of atmosphere).

producing clouds that affect radiation (e.g. cumulonimbus

Cb, cumulus Cu, and stratocumulus Sc)

vertical mixing of heat and moisture (and horizontal momentum)

Further reading:

J. Kiehl’s Chapter 10 in Climate System Modelling (Editor: K. Trenberth)

Emanuel, K.A. 1994: Atmospheric convection, Oxford University Press.

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3.1 Summary of main points

Modelling physical processes is an important aspect of weather prediction and climate simulation. Not as simple as modelling the fluid dynamics parts.

Physical schemes take roughly the same amount as computer time as the fluid dynamics schemes.

Forcing (e.g. radiation) and dissipative processes (e.g. mixing) need to be correctly modelled.

Unresolved processes that feedback onto the large-scale flow need to be parameterised in terms of the the large-scale flow.

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3.1 Summary of main points

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Basic idea:

Represent time derivative by centered time finite difference (CT) and then put different terms of the equation on different time levels. (mixed scheme)

Why? Different equations have different stability

properties (e.g. diffusion equation is unstable for CT and advection equation is unstable for FT scheme).

Treat some of the terms implicitly in order to slow down simulated waves and get round CFL criterion (semi-implicit)

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3.1 Staggered time schemes (semi-implicit)

Example:

2.

ut

)1(2)()()1()1(

.2

nnn

nn

uh

• stable leapfrog (centred) for advection• stable forward for diffusion• Note: both are conditionally stable

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3.1 Semi-implicit gravity wave schemes

Key concept: slow down fast gravity waves by treating them implicitly while treating other terms explicitly

Example: shallow water equations

T

xx

x

x

vu

f

f

Q

Qt

),,(

0

0

0

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3.1 Semi-implicit gravity wave schemes

Split the operator:

Centered scheme (explicit)

Backward scheme (fully implicit)

Mixed scheme (semi-implicit)

)()(

0

00

00

000

00

00

ravityGossbyR

f

f

Q

xx

x

x

)()1()1( )(2 nnn GRh

)1()1()1( )(2 nnn GRh

)(2 )()()1()1( nnnn GRh

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3.1 Semi-implicit gravity wave schemes

The sparse operator 1+hG is a lot easier to invert than is 1+2hQ – obtain a simple Helmholtz equation for height field. Rank of matrix to be inverted is 1/3 that of fully implicit scheme.

Fast sparse matrix techniques can be used to invert (1+hG)

Causes gravity waves to be treated implicitly and so larger time steps can be used without running into CFL problems

ECMWF example:Ritchie …

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3.1 Summary of main points

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Winnighoff 1968, Arakawa and Lamb 1977.)(

2

1)(

)(2

1)(

)(

,2/1,2/1

,2/1,2/1

,2/1,2/1

jiy

jiy

ij

xy

jijiij

x

jijiijx

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ECMWF algorithm

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3.1 Summary of main points

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If the lapse rate exceeds dry adiabatic, a convective adjustment restores the lapse rate to dry adiabatic, with conservation of dry static energy in the vertical. A moist convective adjustment scheme after Manabe et al. (1965) also operates when the lapse rate exceeds moist adiabatic and the air is supersaturated. (For supersaturated stable layers, nonconvective large-scale condensation takes place--see Precipitation). In moist convective layers it is assumed that the intensity of convection is strong enough to eliminate the vertical gradient of potential temperature instantaneously, while conserving total moist static energy. It is further assumed that the relative humidity in the layer is maintained at 100 percent, owing to the vertical mixing of moisture, condensation, and evaporation from water droplets. Shallow convection is not explicitly simulated.

Manabe, S., J. Smagorinsky, and R. F. Strickler, 1965: Simulated climatology of a general circulation model with a hydrologic cycle. Mon. Wea. Rev., 93, 769-798.

Arakawa, A. and W.H. Schubert, 1974: Interaction of cumulus cloud ensemble with the largescale environment, Part I. J. Atmos. Sci., 31, 674-701.

Betts, A.K. and M.J. Miller, 1986: A new convective adjustment scheme. Part I: Observational & theoretical basis. QJRMS, 112, 677-691.

Kuo, H.L., 1974: Further studies of the parameterization of the influence of cumulus convection on large scale flow. J. Atmos. Sci., 31, 1232-1240.

Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev., 117, 1799-1800.

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