consistent numerical modelling of linear inertial wavesobokhove/edinb2013.pdf · waves conclusions...

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics

LinearRotatingCompressibleWaves

LinearRotatingIncompressibleWaves

Conclusions

References

Consistent Numerical Modelling of LinearInertial Waves

Onno Bokhove

AGFD, School of Maths, University of Leedswith Shavarsh Nurijanyan & Jaap van der Vegt (Twente)

ACM Seminars, School of Maths, Edinburgh 28-02-2013

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

1 Introduction

2 Hamiltonian Dynamics

3 Linear Rotating Compressible Waves

4 Linear Rotating Incompressible Waves

5 Conclusions

6 References

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

1. Introduction

Linear and nonlinear waves are ubiquitous in GeophysicalFluid Dynamics (GFD).

GFD concerns hydrodynamics of atmosphere and oceandynamics,

including weather prediction, climate modelling, wave andflood predictions.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

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Nonlinear Surface Waves & Inertial Waves

Two types of waves in GFD caught my attention:

Nonlinear breaking surface waves and currents at beaches:coastal m-to-km scales.

We study these via Computational Fluid Dynamics (CFD)& laboratory scale experiments.

Linear internal/inertial waves in oceans/lakes: all scales.

We study these consistently via Hamiltonian CFD.

Mathematically, discretisation of 3D incompressible linearinertial or free surface waves with vorticity is similar.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

Nonlinear Surface Waves: Hele Shaw Beach

Laboratory experiment & mathematical design (nano scale analog?):

wave!maker

l

B0

0 Lx

p!w

z

H0

g

particles

free surface

wedge

lwx

w

forstetalfig1.png

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

Nonlinear Surface Waves: Hele Shaw Beach

Mathematical design & analysis:

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

Nonlinear Surface Waves: Bore Soliton Splash

Tank experiment (singularity?):

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

Internal Waves in Paraboloid

Driving mechanism: stratification (Hazewinkel, Maas, Dalziel2010).

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

Internal Waves in Paraboloid

Localised attactors: role energy balance oceans’ climate.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

Inertial Waves in Rotating Cuboidal Tank

Driving mechanism: Coriolis force. Set-up & rays.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

Inertial Waves in Cut Cuboidal Tank

Rotating tanks (Manders & Maas 2003): attractors.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

Challenges CFD of Inertial Waves

Inertial waves are solutions of 3D rotating Euler equations:

@u

@t= �2⌦⇥ u�rP , (1a)

r · u = 0, r2P = �r · 2⌦⇥ u (1b)

with

Variables: pressure P = P(x , y , z , t) & 3D velocityu = u(x , y , z , t)

Constant background rotation 2⌦

Constrained perturbation density ⇢ = 0 & total density ⇢0

Closed domain D

u · n = 0 at boundary @D

Initial conditions: u(x, 0).

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

Challenges CFD of Inertial Waves

Discontinuous Galerkin Finite Element Method for:

Discretisation divergence free velocity field r · u = 0.

Discretisation geostrophic boundary conditions.

Preservation Hamiltonian dynamics: energy & phasevolume conservation.

Large computational demands: 3D narrow attractors,complex shaped domains. hpGEM.

The goal is to create a Hamiltonian DGFEM for inertial waves.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

2. Hamiltonian Dynamics

Roadmap ODEs:

Newtonian dynamics 3D particle: variational principle &bracket.

Constrained dynamics 3D particle: Dirac bracket.

Idea: discretise compressible fluid system (easy), then applytheory of Dirac bracket to ODEs to get incompressible case.

Roadmap linear ODEs/discretised PDEs:

3D rotating, compressible, acoustic waves: bracket.

3D rotating, incompressible, inertial waves:

- analyse time discretisation.- Dirac bracket.

HamiltonianDGFEM

OnnoBokhove

Introduction

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Conclusions

References

3D particle: Newtonian dynamics

Variational principle (VP) particle in 3D & potential V :

0 =�

ZT

0u · x� (

1

2

��u|2 + V (x)

�dt (2)

with x = x(t) = (x , y , z)T & u = u(t) = (u, v ,w)T

Equations of motion:

x = u and u = �@V /@x (3)

initial conditions.

HamiltonianDGFEM

OnnoBokhove

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3D particle: Poisson bracket

Poisson Bracket (PB) dynamics:

dF

dt={F ,H} (4)

=@F/@x · @H/@u� @F/@u · @H/@x (5)

with Hamiltonian or energy

H =1

2|u|2 + V (x). (6)

Equations follow from PB by choosing F = x & u, resp.

Conclusion: instead of VP use bracket dynamics, forskew-symmetric, bilinear bracket.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

3D particle: constrained dynamics

Take V (x , y , z) = P(x , y , z) + gz with acceleration g s.t.particle wants to stay near z = 0: w ⇡ 0.

Constrained variational principle:

0 =�

ZT

0u · x� 1

2(u2 + v2)� V (x)� �wdt

=�

ZT

0u · x� (H + �w)dt (7)

with constraint w = 0 imposed by Lagrange multiplier �.


x =u, y = v , z = �

u =� @x

P , v = �@y

P , w = �@z

P � g , w = 0.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

3D particle: constrained dynamics

Consistency requires constraint w = 0 over time

Hence, secondary constraint:

�w = �{w ,H + �w} = @z

V = @z

P + g = 0, (8)

which needs to be fixed in time, such that for suitablepotential V :

0 =d@

z

V

dt= u@

xz

V + v@yz

V + �@zz

V (9)

Simple example: for

V (x , y , z) =1

2(x + y + z)2 + gz ,

we find (surprise)� = �u � v . (10)

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

3D particle: Dirac bracket

Using Dirac Bracket (DB), process summarizes to:

F ={F ,H + �w} = {F ,H}+ �{F ,�1}0 =�1 = {�1,H}+ �{�1,�1} = {�1,H} = ��2 (11)

0 =�2 = {�2,H}+ �{�2,�1}

for constraints �1 = w (primary) and �2 = @z

V(secondary).

Conclusion: instead of using VP, start directly withbracket formulation.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



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References

3. DGFEM: Linear Compressible Inertial Waves

No VP: use PB-bracket formulation ito u & ⇢ = Rj

(t)'j

(x).

Hamiltonian dynamics via bracket:

dF

dt=[F ,H]

d

= �2⌦⇥ @H

@Ui

· @F

@Uj

M�1ij

+

✓@H

@Uj

@F

@Ri

� @F

@Uj

@H

@Ri

◆·DIV

kl

M�1jk

M�1il

(12a)

with coe�cients for perturbation density Rj

= Rj

(t) &velocity vector U

j

= U

j

(t) & Hamiltonian

H =1

2M

ij

(Ui

·Uj

+ Ri

Rj

) . (12b)


U

j

=�M�1jk

DIV

kl

Rl

� 2⌦⇥U

j

Mkl

Rl

=DIV

jk

·Uj

. (13)

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

DGFEM Incompressible: time discretisation

Mid-point integration/Crank-Nicolson in time:

(Un+1j

�U

n

j

)

�t= �⌦⇥ (Un+1

j

+U

n

j

)

�M�1jk

DIV

kl

(Rn+1l

+ Rn

l

)/2

Mkl

(Rn+1l

� Rn

l

)

�t=

1

2DIV

jk

· (Un+1j

+U

n

j

). (14)

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

4. DGFEM: Linear Incompressible Inertial Waves

Primary constraints: constant discrete density field

Dk

= Mkl

Rl

= 0. (15)

Preservation in time leads to discrete zero divergenceconstraint

0 = Dk

= [Dk

,H + �l

Dl

]d

= DIV

lk

·Ul

= Lk

Analogous to particle case, start with DB from discretecompressible case.

0 = Lk

=[Lk

,H]d

+ �l

[Lk

,Dl

]d

(16a)

=�DIV

jk

· 2⌦⇥U

j

�DIV

jk

M�1jm

·DIV

ml

�l

,

discrete equivalent of Poisson equation.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

DGFEM Incompressible: discrete Dirac

Method I: ensure time scheme fixes 2 constraints in time.

Introduce discrete Lagrange multiplier �

(Un+1j

�U

n

j

)

�t= �⌦⇥ (Un+1

j

+U

n

j

)�M�1jm

DIV

ml

�l

Mkl

(Rn+1l

� Rn

l

)

�t=

1

2DIV

jk

· (Un+1j

+U

n

j

). (17)

Assume DIV

jk

·Un

j

= 0.

By inspection: Rn+1l

� Rn

l

= 0 if DIV

jk

·Un+1j

= 0.

Note that DIV

jk

·Un+1j

remains zero when

DIV

jk

·M�1jm

DIV

ml

�l

=�DIV

jk

·⌦⇥ (Un+1j

+U

n

j

).

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

DGFEM Incompressible: discrete Dirac

Summary time & space discrete system:

Zero divergence (initially): DIV

jk

·Un

j

= 0.

Velocity update

U

n+1j

=U

n

j

��t⌦⇥ (Un+1j

+U

n

j

)��tM�1jk

DIV

kl

�l

Discrete Poisson equation for pressure �l

:

DIV

jk

·M�1km

DIV

ml

�l

=�DIV

jk

·⌦⇥ (Un+1j

+U

n

j

)

Solve together.

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

DGFEM Incompressible: Dirac bracket

Method II: via Dirac bracket:

time continuous

dF

dt=[F ,H]

inc

= [F ,H]d

+ �l

[F ,Dl

]d

⌘� @F

@Uj

·�2⌦⇥ @H

@Ui

M�1ij

+M�1jk

DIV

kl

�l

�(18a)

with energy function

H =1

2M

ij

U

i

·Uj

. (18b)


U

j

= �2⌦⇥U

j

�M�1jk

DIV

kl

�l

, (19)

combined with DIV

lk

·Ul

= 0 & (16):

DIV

jk

M�1jm

·DIV

ml

�l

= �DIV

jk

· 2⌦⇥U

j

. (20)

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

DGFEM Incompressible: time discretisation

Mid-point integration in time:

Mij

(Un+1j

�U

n

j

)

�t= �M

ij

⌦⇥ (Un+1j

+U

n

j

)�DIV

ij

�j

DIV

jk

M�1jm

·DIV

ml

�l

= �DIV

jk

·⌦⇥ (Un+1j

+U

n

j

). (21)

Theorem: Numerical scheme given by (21) exactly conservesdiscrete zero-divergence in time:

DIV

jk

·Un+1j

= 0 when DIV

jk

·U0j

= 0 (22)

and (kinetic) energy:

Mij

U

n+1i

·Un+1j

= Mij

U

n

i

·Un

j

. (23)

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

DGFEM Incompressible: numerical verification

Test against semi-analytical solutions in cuboids:

Poincare waves (compared with exact solution)

Inertial waves (FEM better than semi-analytical solutions)

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References


Energy and zero divergence:

HamiltonianDGFEM

OnnoBokhove

Introduction

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IVP starting with semi-analytical solution in tilted cuboid:

rotation aligned with cuboid

tilted cuboid (hint of focussing in “attractors”).

HamiltonianDGFEM

OnnoBokhove

Introduction

HamiltonianDynamics



Conclusions

References

DGFEM Incompressible: note on modulated flows

Modulated inertial waves (Manders & Maas 2003):

@u

@t= �2⌦3v + y

@⌦3

@t� @P

@x, (24a)

@v

@t= +2⌦3u � x

@⌦3

@t� @P

@y, (24b)

@w

@t= �@P

@z, (24c)

r · u = 0, r2P = �r · 2⌦⇥ u (24d)

with ⌦3 = ⌦3(t) = ⌦30 + ✏⌦31(t) & ✏ ⌧ 1; ⌦1 = ⌦2 = 0Note that r · (y ,�x , 0)T@⌦3/@t = 0.Modify numerical projection q

h

= Qj

'j

(t) ofq ⌘ (y ,�x , 0)T on basis functions, such that:

DIV

kl

Ql

= 0 (25)

Discretize @⌦3(t)/@t with modified midpoint rule.

HamiltonianDGFEM

OnnoBokhove

Introduction

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References

5. Conclusions

Established compatible DGFEM based on Dirac theory.

Numerical method stable: bypasses FEM inf-sup stability.

Method likely extends to 3D linear waves & currents withfree surface.

Method likely extends to 3D internal waves (climate).

Hamiltonian nonlinear case nontrivial: potential flow.

Applications with attractors & waves/currents pending:Royal Neth. Inst. Sea Res. & MARIN.

http://www.fast.u-psud.fr/~ppcortet/inertial_wave.php

HamiltonianDGFEM

OnnoBokhove

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References

B. 2002: Balanced models in Geophysical Fluid Dynamics:Hamiltonian formulation, constraints.url In “Large-ScaleAtmosphere-Ocean Dynamics 2”. CUP.

Nurijanyan, Van der Vegt & B. 2013: HamiltonianDGFEM for rotating linear incompressible Euler equations:inertial waves.url In press JCP.

Nurijanyan, B. & Maas 2012: Inertial waves in a cuboid.Subm. PoF.

Pesch et al. 2007: hpGEM– Software framework forDGFEM. ACM Trans. Softw.

http://wwwhome.math.utwente.nl/~bokhoveo/ini01bokhove.pdf

http://eprints.eemcs.utwente.nl/21124/




consistent numerical modelling of linear inertial wavesobokhove/edinb2013.pdf · waves conclusions...

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