physics and numerics in getm - iow · freshwater eddy ii left: surface salinity and current...

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

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


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.


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.


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.


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


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]


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]


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

Kato-Phillips experimentWind-induced mixed-layer depth (MLD)

Burchard & Bolding [2001]


GOTM: Liverpool BaySST from space and location of station (•)

Courtesy to School of Ocean Sciences, UBW, Wales


GOTM: Liverpool BaySection of Temperature and Salinity

Rippeth, Fisher, Simpson [2001]


GOTM: Liverpool BayObserved and simulated temperature and salinity

Simpson, Burchard, Fisher, Rippeth [2002]


GOTM: Liverpool BayObserved and simulated current velocity



GOTM: Liverpool Bay

Observed and simulated dissipation rates



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.


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


Drying: Sylt-Rømø-Bight I


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)


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))


Drying: East Frisia

Stanev et al. [2002]


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.


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)


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.


General vertical coords.Cross-section through Dogger Bank area with variouscoordinate transformations:


Adaptive vertical coordinatesFLEX’76 simulation (water column in NorthernNorth Sea).

Shear-squared (left) and buoyancy frequency (right)

Burchard and Beckers [2004]


Adaptive vertical coordinatesFLEX’76 simulation (water column in NorthernNorth Sea).

Layer interface evolution forN =10, 20, 40 and 80 layers



Adaptive vertical coordinates



Adaptive vertical coordinatesInternal seiche with fixed grid


Adaptive vertical coordinatesInternal seiche with adaptive grid refining at gradients


Adaptive vertical coordinatesInternal seiche with semi-Lagranian adaptive grid


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.


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.


Freshwater eddy ILeft: surface salinity and current vectors; right:bottom current vectors. Momentum advection:multidimensional upwind scheme; salinity advection:TVD-Superbee directional-split scheme.


Freshwater eddy IILeft: surface salinity and current vectors; right:bottom current vectors. Momentum advection:momentum and salinity advection: TVD-Superbeedirectional-split scheme.


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)


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 . . .


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

physics and numerics in getm - iow · freshwater eddy ii left: surface salinity and current...

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