Hans Burchard

Leibniz Institute for Baltic Sea Research Warnemünde

hans.burchard@io-warnemuende.de

How to make a three-dimensional numerical model that works for lakes and estuaries?

Essential problem in ocean models:

To move a continuous property through a fixed grid without changing its form.Example: solid body rotation:

Result of a high-order linear scheme: new artificial maxima and minima.

Two strategies for solutions:

Non-linear limiter schemes.

Vertically adaptive coordinates.

+ a numerical mixing analysis to quantify the numerical mixing.

Principle of non-linear schemes: combine the advantages of two linear schemes in a non-linear way.

Use monotonicity from the diffusive first-order upstream and the 2nd order of accuracy from the non-diffusive Lax-Wendroff scheme.

What is mixing ?

Salinity equation (no horizontal mixing):

Salinity variance equation:

?

Mixing is dissipation of tracer variance.

Principle of numerical mixing diagnostics:First-order upstream (FOU) for s:

FOU for s is equivalent to FOU for s² with variance decay :

numerical diffusivitySalinity gradient squared

See Maqueda Morales and Holloway (2006)

1D advection equation for S:

1D advection equation for s2:

Transport pathways in the Baltic Sea

heating a t sum m er

overflow s

outflow s

seasonal therm ocline

coo ling a t w inte r

in ternalm ix ing

perm anent ha locline

uplift

in terleaving

in terna lw ave m ixing

bottomcurrenten tra inm ent

surface w avem ixing

boundarym ix ing

convectiveentra inm ent

shear-inducedentra inm ent

d ifferen tia ladvection

riverrunoff

w ind stresscoasta lupw elling

sun

Reissmann et al. (2009)

Pressure gradient problem of sigma coordinates

Sigma coordinate problem

Inflows

Inflow approximation problem of geopotential coordinates

Geopotential coordinate problem

Inflows

Additionally, both coordinate types share the problem of numerical mixing.

Adaptive vertical grids in GETM

hor. filteringof layer heights

Vertical zoomingof layer interfaces towards:

a) Stratification

b) Shear

c) surface/ bottom

z

bottom

Vertical direction

Horizontal direction

hor. filteringof vertical position

Lagrangiantendency

isopycnaltendency

Solution of a vertical diffusion equation for the coordinate position

Burchard & Beckers (2004); Hofmeister, Burchard & Beckers (2010)

www.getm.eu

The philosophy behind GETM

GETM is a coastal and shelf sea (and lake?) hydrodynamic model.

GETM is a Public Domain Community Model.

GETM is released under the Gnu Public Licence.

GETM is Open Source.

GETM has a modular structure (open for extentions).

GETM has an international developer and user community.

GETM started in 1997 and has been steadily developed since then.

The physics & numerics of the GETM core

GETM is based on the 3D shallow water equations has been extended to represent non-hydrostatic pressure is using GOTM as turbulence closure model is using bulk formulae to calculate surface fluxes is a finite volume (i.e., conservative) model uses Cartesian, spherical or curvilinear horizontal coordinates uses general vertical coordinates (including adaptive) uses high-resolution TVD advection schemes is based on explicit mode splitting is fully paralellised using MPI and domain decomposition works with netCDF input and output

Baltic slice with adaptive vertical coordinates

Fixed coordinates Adaptive coordinates

Hofmeister, Burchard & Beckers (2010)

Numerical mixing Numerical mixingPhysical mixing Physical mixing

Adaptive vertical coordinates

along transect in 600 m Western Baltic Sea model

Gräwe et al. (in prep.)

Adaptive coordinates in Bornholm Sea

1 nm Baltic Sea model with adaptive coordinates- refinement partially towards isopycnal coordinates

- reduced numerical mixing- reduced pressure gradient errors- still allowing flow along the bottom

salinity

temperature

km

Hofmeister, Beckers & Burchard (2011)

Feistel et al., 2004

Observations

November 2003

Channelled gravity current in Bornholm Channel

sigma-coordinates

adaptive coordinates

- stronger stratification with adaptive coordinates- larger core of g.c.- salinity transport increased by 25%

- interface jet along the coordinates

Hofmeister, Beckers & Burchard (2011)

Gotland Sea time series

3d baroclinic simulation

50 adaptive layers vs. 50 sigma layers

num. : turb. mixing80% : 20%

num. : turb. mixing50% : 50%

Hofmeister, Beckers & Burchard (2011)

Multi-scale applicationsStructured

modelsUnstructured models

Gräwe et al. (in prep.) de Bauwere et al. (2009)

Does this model also work for lakes?

Yes, but you have to resolve the slopes which are generally steeper in lakes than in the shelf

sea.

Example: seiches in Lake Alpnach:

Study of boundary mixing (Lake Alpnach, Switzerland)

Becherer & Umlauf (2011)

Simulation Lake Alpnacher (Switzerland)

Becherer & Umlauf (2011)

Take home:It is a demanding numerical task to obtain efficient and accuratediscretisations for the advection terms.

Mass conservation can be obtained by finite-volume schemes, butvariance conservation would only work in Lagrangean (particle tracking) models.

Vertically adaptive coordinates are one efficient method to reducenumerical variance decay due to advection schemes.

Question: What is the 3D type of model for the future? Will in 20 years still structured and unstructured models co-exist, orwill one or the other or a third method rule out the others?