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Chebyshev spectral methods for quasigeostrophicshallow-water flow

Duncan Sutherland

School of Mathematics and Statistics University of Sydney

ANZIAM 2010

Duncan Sutherland (Usyd) Chebyshev Methods ANZIAM 2010 1 / 13

Motivation

Formation and maintenance of jetsI A jet is a high energy elongated

flow. Eg: Jetstreams on Earth

Potential vorticity (PV) staircasesI PV profiles of alternating steep

and gentle gradients. By theinvertibility principle, thiscorresponds to thin, high velocityeasterly jets and wide, lowvelocity westerly jets.

Investigation of stability of jetsover topography.

I Ocean (or atmospheric) currentsare often unstable and the role ofthe ocean floor topography is notfully understood.

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Averaged potential vorticityS

patia

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n−4 −2 0 2

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Average velocity

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Top: http://cubanology.com/JetStream.jpg,found by Theo Vo.


Quasigeostrophic flow in a channel

A channel geometry is a simple approximation to the motivating problems,in the case of a shallow atmosphere or considering ocean flow.Stream-function vorticity equations with added terms due to the rotatingframe, called the β-effect.

q = ∇2ψ − L−2D ψ + f0 + βy potential vorticity,

∇2ψ − L−2D ψ = −ω ω is the relative vorticity

∂ω

∂t= J(ψ, ω) + β

∂ψ

∂y, J(ψ, ω) =

∂ψ

∂x

∂ω

∂y− ∂ψ

∂y

∂ω

∂x(1)

ψ(x ,−1) = α1, ψ(x , 1) = α2

ψ(−L

2, y) = ψ(

L

2, y) ω(−L

2, y) = ω(

L

2, y)

LD is the Rossby deformation length, the length scale at which rotationaleffects are as significant as gravitational (or buoyancy effects).


Spatial Collocation and discretisation

The equations are Fourier transformed in streamwise x−direction using Mpoints, with wavenumbers k. Chebyshev collocation with N points is usedin the channel y−direction. This gives the following equations:

∂ω

∂t= J(ψ, ω) + β

∂ψ

∂y

∂ψ

∂y− (k2 − L−2

D )ψ = −ω (2)

ψ(x ,−1) = α1, ψ(x , 1) = α2

The hats denote Fourier transform. The nonlinear term is evaluated by thestandard pseudospectral technique. The derivatives are evaluated byFourier or Chebyshev transform and then the product is evaluated in thephysical domain. Timestepping is by Adams-Bashforth O(2) orRunge-Kutta methods O(4).


Chebyshev discretisation

This section examines the Poisson’s equation relating ψ and ω

∂2ψ

∂y2− (k2 − L−2

D )ψ = −ω upon transforming into the Chebyshev domain:

N∑n=0

a(2)n Tn −

N∑n=0

K 2anTn = −N∑

n=0

fnTn where: K 2 = k2 − L−2D (3)

N∑n=0

(−1)nan = α1,

N∑n=0

an = α2.

The an and fn are the Chebyshev-Fourier coefficients of ψ and ω

respectively. a(2)n is the Chebyshev-Fourier coefficient of the second

derivative.


Fast Helmholtz solver

Using the recurrence relation between the Chebyshev polynomials it can beshown that the discretisation of Poisson’s equation gives aquasipentadiagonal matrix system:

cn−1a(p)n−1 = a

(p)n+1 + 2na

(p−1)n n ≥ 1, Recurrence relation

a(2)n + K 2an = −fn n = 1, · · · ,N − 1 from Poisson’s equation∑

n even

an =α1 + α2

2,

∑n odd

an =α2 − α1

2Simplifying BCs

K 2cn−2an−2

4n(n − 1)−(

1 +K 2en+2

2(n2 − 1)

)an +

K 2en+4an+2

4n(n + 1)

= − cn−2fn−2

4n(n − 1)+

en+2fn2(n2 − 1)

− en+4fn+2

4n(n + 1), (4)

where en =

0 n > 0

1 n = 0and cj =

2 j = 0,

1 j = 1, 2, · · ·.


Fast Helmholtz solver

The matrix equation considered inthis section is:

γ0a0 + γ2a2 + · · ·+ γNaN = g0,

γ1a1 + γ3a3 + · · ·+ γN−1aN−1 = g1,

pnan−2 + qnan + rnan+2 = Fn

n = 2, · · · ,N − 2,

pNaN−2 + qNaN = FN ,

pN−1aN−3 + qN − 1aN−1 = FN−1.

In the case of Dirichlet conditions,γn = 1 ∀n, for Neumann conditionsγn = n2.

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Thual algorithmTo solve this matrix equation efficiently a recursion relation algorithm dueto Thual is employed, see Peyret 2001.

Consider the pure tri-diagonal case using the Thomas algorithm. Thismethod is Gaussian elimination followed by back substitution. Aforward sweep eliminates one diagonal and a simplified backwardssubstitution method gives the solution. Because of the structure, theoperations count is linear.

This structure is similar, notice that the even indexed and oddindexed elements are independent.

Substituting a recurrence relation, with starting values from the rowswith only two variables allows a first sweep and eliminates onediagonal.

A second recurrence relation with starting values deduced from thetwo full rows allows a second sweep, giving the solution.

So that numerical errors do not propagate the matrix is required to bediagonally dominant.


Hyperviscosity

Motivated by the doubly periodic case, the high modes are subject to aGaussian like filter approximating viscous diffusion. This is done forstability reasons.

ωn = ωn exp(−ν(k2 + l2)pδt)

kij =2πi

Mi = 0, · · · ,N − 1 ∀j ,

lij =πij

Ni = 0, · · · ,N − 1 j = 0, · · · ,M − 1 0

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40

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800

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1

Chebyshev wavenumberFourier wavenumber

Am

plitu

de o

f filt

er

The tilde denote Fourier-Chebyshev coefficient. p is the hyperviscouspower, typically 2. ν is the hyperviscous coefficient.


Test case: Rossby waveA Rossby wave satisfying the boundary conditions is an exact solution ofthe problem.

ω = sin(kπx − σt) sin(lπy), σ =−kβ

π(k2 + l2)− L−2D

wavespeed

Time integration over one period of the Rossby wave on a 64× 64 pointsquare grid,the looping is artificial. Black line is exact solution. The errorover several periods is approximately 6× 10−8.


Test case: Modon

A modon is an exact solution in the infinte domain, which have interestingbehaviour in collisions.

Head on collision between two modons, one half the amplitude of theother. Note that this interaction is similar to soliton behaviour. A squaregrid of 128 points was used.


Limitations and extensions

The Chebyshev code is diffusive. There is a high resolution near theboundary but less resolution in the middle of the channel. Comparedto contour dynamics methods, where contour values rather than gridvalues are evolved, a gridded scheme will poorly resolve filamentarybehaviour.

The hyperviscous filtering isn’t rigorously justified.

This is a single geometry, extension to irregular domains is difficult.

Extension to viscous problemI Stability problems. Typically the solution is influence matrix

techniques, using O(2) linear multistep methods in timeI Would be nice to have O(4) single step schemesI Hybrid contour advection method


References and Acknowledgements

Pedlosky, J. (1986) Geophysical fluid dynamics, 2nd ed.Springer-Verlag New York

Peyret, R. (2001) Spectral methods for incompressible viscous flows.Springer-Verlag New York.

Gottlieb, D. and Orszag, S. A. (1977) Numerical analysis of spectralmethods SIAM Philadelphia

Dang-Vu,H. (1993), J. Comp. Phys. V104, pp211-220.

Canuto, C, Hussaini, M, Quarteroni, A, Zang, T.A,(1988) Spectralmethods in fluid dynamics, Springer-Verlag New York.

Thual, O. (1986) PhD Thesis.

Trefethen, L.N, (2000) Spectral methods in Matlab. SIAMPhiladelphia.

Thanks to my supervisor A/Prof Charlie Macaskill, and also to Theo Vo.


Questions


Appendix slides


The β-plane

The Coriolis parameterf = 2Ω sin(ϕ) is the localcomponent of the Earth’s rotation,for a particular latitude, ϕ.

A simple approximation to f is toignore latitudinal variation, givesf = 2Ω sin(ϕ0), this is called thef−plane approximation.

The β-plane approximation is toTaylor expand f around some y0

and keep the linear termf = f0 + β(y − y0)

Modified from http://mitgcm.org/public/pelican/online documents/img194.png


Chebyshev collocation

Chebyshev polynomials Tn = cos(n cos−1(x)) are used to satisfy theboundary conditions

Runge phenomenon is minimised by selecting the collocation points asx = cos( jπ

N−1 ) for j = 0, · · ·N − 1

FFT can be used to transform back and forth between physical spaceand Chebyshev coefficient space


Matrix equation solution

This may be solved in O(N) operations by using the following recurrencerelations:

an+2 = Xnan + Yn n = 0, · · · ,N − 2 (5)

Xn−2 =−pn

qn + rnXn; Yn−2 =

fn − rnYn

qn + rnXn

XN−2 = −pN

qN,

YN−2 =fNqN,

XN−3 = −pN−1

qN−1,

YN−3 =fN−1

qN−1.


Matrix equation solution

Now write:

an = θnal + λn n = 1, · · · ,N l =

0 n even

1 n odd. (6)

Notice that if n = 0, 1 then θl = 1 and λl = 0. Substituting equation (6)into equation (4) yields:

θn+2 = Xnθn; λn+2 = Xnλn + Yn (7)

a0 =g0 −

∑Nn=0

n evenγnλn∑N

n=0 γnθn, a1 =

g1 −∑N

n=0n odd

γnλn∑Nn=0 γnθn

. (8)

For numerical errors not to propagate, the matrix is required to bediagonally dominant (such as this case).


Details of simulations

Inital condtions: Two modons.Adams-Bashforth timesteppingPlots every 100 timestepsGrid size 128 squareLength of channel 2Timestep constant 0.01Number of timesteps 10400β = 1Fr = 0.4p = 4‘Width’ of modon 0.3 units.ν = 1.7× 10−18

Homogenous BCs.

Inital condition: Rossby wave.Runge-Kutta timesteppingPlots every:100Grid size: 64Length of channel: 2Timestep constant: 0.01Number of timesteps 2500β = 10Fr = 0.01No hyperviscous filtering.Homogenous BCs.


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