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Physics and Numerics in GETMHans Burchard1,2 and Karsten
Bolding2
[email protected],
[email protected]
1. Baltic Sea Research Institute Warnem̈unde, Germany2. Bolding
& Burchard Hydrodynamics, Denmark
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 1/37
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Contents• Physics
• Standard physics for transport• Complex statistical mixing
schemes• Drying & flooding
• Numerics• Mode splitting• Horizontal grids• Vertical grids•
High-order monotone advection schemes• Various pressure gradient
schemes
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 2/37
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Model Requirements I• Algorithm for drying and flooding for
simulating Wadden Sea dynamics.• Bottom-fitted coordinatesfor
better
representation of near-bottom flows.• Surface-fitted
coordinatesfor high vertical
near-surface resolution and with large tidalamplitude.
• General vertical coordinatesfor better fitting ofthe model
grid to the internal flow structures.
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 3/37
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Model Requirements II• Curvilinear horizontal coordinates for
better
representation of complex bathymetry and higherresolution of
narrow regions without nesting.
• Monotone high-order advection schemesforbetter representation
of fronts and stratification.
• High-order turbulence modelsfor goodrepresentation of vertical
mixing.
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 4/37
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PhysicsIn GETM, standard physical laws for transport infairly
shallow natural waters (small aspect ratio) areimplemented:
• 3D primitive equations• hydrostatic approximation• Boussinesq
approximation• Free surface
Extentions (non-Boussinesq, non-hydrostatic)
seemfeasable.Non-standard are thehigh-order turbulence modelsand
the simplifications fordrying & flooding.
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 5/37
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Physics - Turbulence ModellingGETM uses the turbulence module of
the watercolumn (1D) model GOTM, which includes thefollowing
parameterisations:
• Zero-equation models (algebraic TKE)• One-equation models
(algebraic length scales)• Two-equation models as work horses such
as:
• k-ε model• Mellor-Yamada model• Generic two-equation model
(e.g.k-ω)
• Various algebraic second moment closures• Non-local KPP model
coming soon
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 6/37
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k-ε model I
Turbulent Fluxes (velocity & temperature):
〈ũw̃〉 = −νt∂zū, 〈w̃T̃ 〉 = −ν′t∂zT̄
Eddy Viscosity / Eddy Diffusivity:
νt = cµk2
ε, ν ′t = c
′µ
k2
ε.
k: Turbulent kinetic energy (TKE) [J/kg]ε: Dissipation of TKE
[W/kg]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 7/37
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k-ε model II
k-ε model (Launder and Spalding [1972]):
∂tk − ∂z
(
νt
σk∂zk
)
= P + B − ε,
∂tε − ∂z
(
νt
σε∂zε
)
=ε
k(cε1P + cε3B − cε2ε) .
P : Shear production of TKE [W/kg]B: Buoyancy production
[W/kg]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 8/37
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Total equilibrium ( k-ε)
k̇ = ε̇ =⇒ Ri =− g
ρ0∂zρ
(∂zu)2 + (∂zv)2= Rsti =
cµ
c′µ·c2ε − c1εc2ε − c3ε
.
Rsti ≈ 0.25: Steady-state Richardson number.
Burchard & Bolding [2001]1. GETM Users Workshop, Båring
Hojskole, Denmark, June 6-8, 2004 – p. 9/37
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Kato-Phillips experimentWind-induced mixed-layer depth (MLD)
Burchard & Bolding [2001]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 10/37
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GOTM: Liverpool BaySST from space and location of station
(•)
Courtesy to School of Ocean Sciences, UBW, Wales
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 11/37
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GOTM: Liverpool BaySection of Temperature and Salinity
Rippeth, Fisher, Simpson [2001]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 12/37
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GOTM: Liverpool BayObserved and simulated temperature and
salinity
Simpson, Burchard, Fisher, Rippeth [2002]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 13/37
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GOTM: Liverpool BayObserved and simulated current velocity
Simpson, Burchard, Fisher, Rippeth [2002]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 14/37
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GOTM: Liverpool Bay
Observed and simulated dissipation rates
Simpson, Burchard, Fisher, Rippeth [2002]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 15/37
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Drying: Physical mechanismMomentum equation:
∂tu + ∂z(uw) − ∂z ((νt + ν)∂zu)
+α
(
∂x(u2) + ∂y(uv) − ∂x
(
2AMh ∂xu)
− ∂y(
AMh (∂yu + ∂xv))
−fv −
∫ ζ
z
∂xb dz′
)
= −g∂xζ,
α = min
{
1,D − Dmin
Dcrit − Dmin
}
, Dmin = 2cm, Dmin = 10cm.
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 16/37
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Drying: Numerical mechanism
Virtual sea surface elevation
Actual sea surface elevation
Bathymetry approximation
−Hi,j
ζi,j
−Hi,j + Dminζ̃i+1,j
ζi+1,j
−Hi+1,j
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 17/37
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Drying: Sylt-Rømø-Bight I
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 18/37
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Drying: Sylt-Rømø-Bight II
-3
-2
-1
0
1
8 10 12 14 16 18 20
220.000.010.020.030.040.050.060.070.080.090.10
x / km
z/m
Eddy viscosityνt along cross-section during high
waterνt/(m2s−1)
-3
-2
-1
0
1
8 10 12 14 16 18 20
220.000.010.020.030.040.050.060.070.080.090.10
x / km
z/m
Eddy viscosityνt along cross-section during low
waterνt/(m2s−1)
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 19/37
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Drying: Sylt-Rømø-Bight III
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5u velocity component in point 2 u/(m s−1)
z/m
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.00.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
z/m
Eddy viscosityνt in point 2 νt/(m2s−1)
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8
1.00.00000.00020.00040.00060.00080.00100.00120.00140.00160.00180.00200.00220.0024
z/m
Turbulent kinetic energyk in point 2 k/(m2s−2)
t / T4
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
z/m
t / T4
Turbulent dissipation rateε in point 2 log10(ε/(m2s−3))
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 20/37
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Drying: East Frisia
Stanev et al. [2002]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 21/37
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Mode splittingGETM time stepping is based on conservative mode
splitting.Fast time step for external (vertically-integrated
mode),slowtime stepping for internal (vertically-resolved) mode,
necessityof mode coupling terms.
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 22/37
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General vertical coords.S-equation in Cartesian coordinates:
∂∗t S + ∂∗
x(uS) + ∂∗
y(vS) + ∂∗
z (wS) − ∂∗
z (ν′
t∂∗
zS) = 0. (0)
Coordinate transformation:
γ = γ(t∗, x∗, y∗, z) ⇔ z = z(t, x, y, γ). (1)
Jacobian of the transformation:
J := ∂γz = (∂∗
zγ)−1
. (2)
S-equation in transformed coordinates:
∂t(JS)+∂x(JuS)+∂y(JvS)+∂γ(w̃S)−∂γ
(
ν ′tJ
∂γS
)
= 0. (3)
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 23/37
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General vertical coords.The same the discrete vertical layer
distribution maybe obtained
by equidistantly discretising the transformedequationsorby
discretising the equations in Cartesian coordinatesby means of
layers with moving interfaces usingkinematic boundary
conditions.
The result is the same: Layers which are basicallynewly
distributed after each time step, guaranteeingmass
conservation.
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 24/37
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General vertical coords.Cross-section through Dogger Bank area
with variouscoordinate transformations:
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 25/37
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Adaptive vertical coordinatesFLEX’76 simulation (water column in
NorthernNorth Sea).
Shear-squared (left) and buoyancy frequency (right)
Burchard and Beckers [2004]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 26/37
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Adaptive vertical coordinatesFLEX’76 simulation (water column in
NorthernNorth Sea).
Layer interface evolution forN =10, 20, 40 and 80 layers
Burchard and Beckers [2004]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 27/37
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Adaptive vertical coordinates
Burchard and Beckers [2004]
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 28/37
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Adaptive vertical coordinatesInternal seiche with fixed grid
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 29/37
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Adaptive vertical coordinatesInternal seiche with adaptive grid
refining at gradients
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 30/37
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Adaptive vertical coordinatesInternal seiche with semi-Lagranian
adaptive grid
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 31/37
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Advection schemesGETM has implemented various different
advectionschemes for tracers and momentum (turbulenceadvection
under development).
• One-dimensional schemes are used indirectional-split mode
(Pietrzak 1998).
• These schemes are e.g. First-order upstream,ULTIMATE QUICKEST
and the TVD-schemesSuperbee, MUSCL and P2-PDM.
• Iteration of vertical advection (CFL-criterium).• As
2D-horizontal schemes we have first-order
upstream and FCT, which may be combined withvertical 1D
scheme.
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 32/37
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2D test case: P2 split schemeCube resulting after one solid-body
rotation with∆x = ∆y = 1 m and a Courant number ofc = 0.5.Left:
unlimited P2 scheme; right: limited P2-PDMscheme
TVD-Verfahren führen zu monotonenAdvektionsverfahren höherer
Ordnung.
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 33/37
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Freshwater eddy ILeft: surface salinity and current vectors;
right:bottom current vectors. Momentum advection:multidimensional
upwind scheme; salinity advection:TVD-Superbee directional-split
scheme.
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 34/37
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Freshwater eddy IILeft: surface salinity and current vectors;
right:bottom current vectors. Momentum advection:momentum and
salinity advection: TVD-Superbeedirectional-split scheme.
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 35/37
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Pressure gradient problemWhen sloping coordinate surfaces
intersect withisopycnal surfaces, truncation errors occur due to
thebalance of two large terms:
1
2(hi,j,k + hi,j,k+1) (m∂
∗
Xb)i,j,k
≈1
2(hui,j,k + h
ui,j,k+1)
1
2(bi+1,j,k+1 + bi+1,j,k) −
1
2(bi,j,k+1 + bi,j,k)
∆xui,j
− (∂xzk)x
i,j,k
(
1
2(bi+1,j,k+1 + bi,j,k+1) −
1
2(bi+1,j,k + bi,j,k)
)
(4)
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 36/37
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HCCThe hydrostatic consistency condition (HCC) saysthat the
relative slope of the coordinates should not belarger than
unity:
|∂xzk|∆x
1
2(hi,j,k + hi+1,j,k)
≤ 1. (5)
It is not always possible to avoid violation of HCC.The problem
may be relaxed by smoothingbathymetry, having coarse vertical
near-bedresolution, applying adaptive grids, increasinghorizontal
resolution . . .
1. GETM Users Workshop, Båring Hojskole, Denmark, June 6-8,
2004 – p. 37/37
ContentsModel Requirements IModel Requirements IIPhysicsPhysics
- Turbulence Modelling$k$-$eps $ model I$k$-$eps $ model IITotal
equilibrium ($k$-$eps $)Kato-Phillips experimentGOTM: Liverpool
BayGOTM: Liverpool BayGOTM: Liverpool BayGOTM: Liverpool BayGOTM:
Liverpool BayDrying: Physical mechanismDrying: Numerical
mechanismDrying: Sylt-Ro mo {}-Bight IDrying: Sylt-Ro mo {}-Bight
IIDrying: Sylt-Ro mo {}-Bight IIIDrying: East FrisiaMode
splittingGeneral vertical coords.General vertical coords.General
vertical coords.Adaptive vertical coordinatesAdaptive vertical
coordinatesAdaptive vertical coordinatesAdaptive vertical
coordinatesAdaptive vertical coordinatesAdaptive vertical
coordinatesAdvection schemes2D test case: P$_2$ split
schemeFreshwater eddy IFreshwater eddy IIPressure gradient
problemHCC