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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
Nonlinear Surface Waves: Hele Shaw Beach
Mathematical design & analysis:
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
Nonlinear Surface Waves: Bore Soliton Splash
Tank experiment (singularity?):
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
Internal Waves in Paraboloid
Driving mechanism: stratification (Hazewinkel, Maas, Dalziel2010).
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
Internal Waves in Paraboloid
Localised attactors: role energy balance oceans’ climate.
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
Inertial Waves in Rotating Cuboidal Tank
Driving mechanism: Coriolis force. Set-up & rays.
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
Inertial Waves in Cut Cuboidal Tank
Rotating tanks (Manders & Maas 2003): attractors.
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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 �.
Equations of motion:
x =u, y = v , z = �
u =� @x
P , v = �@y
P , w = �@z
P � g , w = 0.
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
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)
Equations of motion:
U
j
=�M�1jk
DIV
kl
Rl
� 2⌦⇥U
j
Mkl
Rl
=DIV
jk
·Uj
. (13)
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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)
Equations of motion:
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
DGFEM Incompressible: numerical verification
Energy and zero divergence:
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
DGFEM Incompressible: numerical verification
IVP starting with semi-analytical solution in tilted cuboid:
rotation aligned with cuboid
tilted cuboid (hint of focussing in “attractors”).
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
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
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
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.
HamiltonianDGFEM
OnnoBokhove
Introduction
HamiltonianDynamics
LinearRotatingCompressibleWaves
LinearRotatingIncompressibleWaves
Conclusions
References
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.