a conservative scheme for the shallow-water system on a staggered geodesic grid based on a nambu...

QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETYQ. J. R. Meteorol. Soc. 135: 485–494 (2009)Published online 9 February 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/qj.368

A conservative scheme for the shallow-water system on astaggered geodesic grid based on a Nambu representation

Matthias Sommer* and Peter NevirInstitute for Meteorology, Freie Universitat Berlin, Germany

ABSTRACT: A conservative spatial discretization scheme is constructed for a shallow-water system on a geodesic gridwith C-type staggering. It is derived from the original equations written in Nambu form, which is a generalization ofHamiltonian representation. The term ‘conservative scheme’ refers to one that preserves the constitutive quantities, heretotal energy and potential enstrophy. We give a proof for the non-existence of potential enstrophy sources in this semi-discretization. Furthermore, we show numerically that in comparison with traditional discretizations, such schemes canimprove stability and the ability to represent conservation and spectral properties of the underlying partial differentialequations. Copyright c© 2009 Royal Meteorological Society

KEY WORDS NWP; conservative operators; Nambu dynamics

Received 22 August 2008; Revised 12 December 2008; Accepted 17 December 2008

1. Introduction

Even if real atmospheric dynamics always includes dis-sipative effects, it can be of interest to study the prop-erties of the conservative core of a nonlinear dynamicalsystem. It has long been known that errors in the numer-ical representation of this conservative part of the sys-tem can lead to incorrect results. Phillips (1956, 1959)first pointed out that a spatial discretization scheme cancause what he termed ‘nonlinear computational instabil-ity’. Later Arakawa (1966) located the reason for thisinstability in the numerical noncompliance with energyand enstrophy conservation laws. He showed that thisshortcoming will eventually lead to an incorrect inverseenergy cascade, an unnatural accumulation of kineticenergy at the smallest resolved scales and finally theinstability discovered by Phillips. Arakawa (1966) andArakawa and Lamb (1972) solved this problem tem-porarily by defining a conservative discretization thatindeed avoided such complications. However, their ideais applicable only to simple systems (barotropic vortic-ity equation, shallow-water equations) with a restrictedgrid complexity. In more sophisticated models, this kindof instability is usually eliminated by adding numericaldamping.

It is in general not an easy task to find conserva-tive schemes for a certain system on a given grid.On the basis of Nambu representation and energy–vorticity theory (Nevir and Blender, 1993; Nevir andSommer, 2008), Salmon (2005, 2007) has elucidated

∗Correspondence to: Matthias Sommer, Institute for Meteorology, FreieUniversitat Berlin, Carl-Heinrich-Becker-Weg 6–10, 12165 Berlin,Germany. E-mail: [email protected]

how to construct such schemes for very general set-tings. In this publication we present the applicationof his idea to the shallow-water equations on a stag-gered geodesic grid (more precisely the grid usedfor the ICOsahedral Non-hydrostatic (ICON) model;(Under development at the German Weather Serviceand Max Planck Institute.) Bonaventura et al. 2005)and show the main numerical results. They confirmSalmon’s expectation about the advantages of conser-vative schemes and encourage researchers to apply thismethod to more complex systems (Gassmann and Herzog,2008).

Section 2 gives a short overview of the theory in acontinuous formulation. In section 3 the general methodis explained and applied to the ICON grid. Section 4presents some numerical output from the constructedscheme.

2. Energy–vorticity theory of the shallow-waterequations

The notion of ‘energy–vorticity theory’ has been givento the concept of writing hydrothermodynamical systemsin the Nambu representation (Nevir and Blender, 1993;Nevir and Sommer, 2008). The latter is a generalizationof the Hamiltonian representation, disposing of multipleHamiltonians (Nambu, 1973). Adopting this idea, energy-and vorticity-based conserved quantities (e.g. potentialenstrophy) can be kept on the same level, with theoreticaland numerical benefits. Following Salmon (2005), a briefsketch of this theory for a shallow-water system is givenhere.

Copyright c© 2009 Royal Meteorological Society

486 M. SOMMER AND P. NEVIR

Consider the shallow-water equations on a rotatingsphere:

∂v∂t

= −ζ ak × v − ∇�, (1)

∂h

∂t= −∇ · (hv), (2)

where v is the horizontal velocity, h the thickness ofthe fluid layer, ζ a = k · ∇ × v + f the absolute vorticity,f the Coriolis parameter, k the vertical (i.e. normal tothe (x, y) plane under consideration) unit vector and� = 1

2 v2 + gh the specific energy.The cross product is a short form for writing

k × v = k ×(

u

v

)=

( −v

u

).

The vector equation (1) can be brought into scalarform using the two prognostic quantities vorticity anddivergence µ := div v:

∂ζ

∂t= −∇ · (ζ av), (3)

∂µ

∂t= k · ∇ × (ζ av) − ��. (4)

Kinetic, potential and total energy and potential enstro-phy are defined as

Hkin =∫

1

2hv2 dA, (5)

Hpot =∫

1

2gh2 dA, (6)

H = Hkin + Hpot, (7)

E =∫

1

2hq2 dA = 1

2

∫ζ 2

a

hdA, (8)

with absolute potential vorticity (PV) q := ζ a/h. Thecorresponding functional derivatives read

δHδζ

= −χ,δHδµ

= −γ ,δHδh

= �,

δEδζ

= q,δEδµ

= 0,δEδh

= −1

2q2.

Here χ and γ are the streamfunction and velocity poten-tial of the layer momentum defined by the Helmholtzdecomposition:

hv = k × ∇χ + ∇γ . (9)

Employing these results, the Nambu representation ofthe shallow-water model for an arbitrary functional Fof the prognostic variables vorticity ζ , divergence µ andthickness h can be given as follows (Salmon, 2005):

∂tF[ζ , µ, h] = {F,H, E}= {F,H, E}ζζ ζ + {F,H, E}µµζ + {F,H, E}ζµh,

with the bracket definitions

{F,H, E}ζζ ζ =∫

J

(δFδζ

,δHδζ

)δEδζ

dA, (10)

{F,H, E}µµζ =∫

J

(δFδµ

,δHδµ

)δEδζ

dA + cyc(F,H, E),

(11)

{F,H, E}ζµh

=∫

1

∂xq

(∂x

δFδζ

∂x

δHδµ

− ∂x

δFδµ

∂x

δHδζ

)∂x

δEδh

dA

+∫

1

∂yq

(∂y

δFδζ

∂y

δHδµ

− ∂y

δFδµ

∂y

δHδζ

)∂y

δEδh

dA

+cyc(F,H,E). (12)

Here J (α, β) = k · rot(α∇β) denotes the Jacobian andcyc() stands for the sum of cyclic permutations of thearguments. x and y are horizontal rectangular coordinates.It should be noted that the last bracket (12) is not writ-ten in a coordinate-free manner. Furthermore, it featuresa PV gradient in the denominator, which cancels ana-lytically with the functional derivatives. The momentumform of the shallow-water equations can also be writtenin the Nambu representation; however, the singularitiesarising (i.e. denominators in the bracket expression) areeven more complex so that no feasible discretization canbe found (Salmon, 2005). These facts complicate workwith the Nambu formalism of a shallow-water systemcompared with, for example, the non-hydrostatic prim-itive model, which is physically and numerically muchmore challenging but at least lacks the aforementioneddifficulties. Most notably, a singularity-free Nambu rep-resentation for the non-hydrostatic system can be givenfor the momentum form, so no vorticity-divergence for-malism is necessary there.

3. Conservative discretization for the shallow-waterequations on the ICON grid

3.1. General procedure

The main idea of Salmon (2005, 2007) is to use theNambu representation of a given dynamical system toexpress numerically the conservation of the constitutiveglobal quantities. Instead of directly defining a spatialdiscretization of the operators (equations (2)–(4)), heproposes defining discrete expressions for the constitutivequantities (5)–(8) and the bracket expressions (10)–(12). Finally the discrete brackets are evaluated withthe prognostic quantities inserted. It should be stressedthat any discretization can be chosen here – one isfree to chose one of the desired accuracy. As long asthe antisymmetry property of the bracket is preserved,the resulting spatially discrete system will conserve thecorresponding global quantities algebraically exact. Thisis a consequence of the method of construction but anexplicit proof of this fact is also given in appendix A.Indeed, such schemes not only obey global conservation

Copyright c© 2009 Royal Meteorological Society Q. J. R. Meteorol. Soc. 135: 485–494 (2009)DOI: 10.1002/qj

A CONSERVATIVE STAGGERED SHALLOW-WATER SCHEME 487

laws but also exclude any local sources of potentialenstrophy or non-convective fluxes.

Friction terms can then be added at will and the systemwill dissipate energy and potential enstrophy. However,it exhibits the remarkable feature of reaching a state ofcomplete conservation of energy and enstrophy as thefriction coefficient approaches zero.

3.2. The ICON grid

It is well known that geodesic grids have some advantagesover the classical latitude–longitude grids (Heikes andRandall, 1995). Recently Ringler and Randall (2002) pro-posed a conservative spatial discretization scheme for theshallow-water equations on an unstaggered geodesic grid.Later, the use of C-type staggered geodesic Delaunay–Voronoi grids was suggested by Bonaventura and Ringler(2005). A combined weather and climate forecast modelon this kind of grid, the so-called ICON model, is cur-rently being developed by the German Weather Serviceand the Max Planck Institute for Meteorology. For athorough description of the ICON grid refer to Bonaven-tura et al. (2005) or Bonaventura and Ringler (2005). Itsglobal structure is that of an icosahedron and its dualsolid, the dodecahedron, which are mutually refined untilreaching the desired resolution. Its tiles are made of tri-angles, hexagons and pentagons. For the global structureand the local staggering see Figure 1. Mass points i arelocated at the centre of triangle faces, vorticity points ν

at triangle vertices and the wind edges l connect the vor-ticity points. The notation for the grid is summarized inTable I.

The algebraic operators ∂l and ∂⊥l act as central second-

order difference quotients on quantities defined at (oraveraged to) vorticity points and mass points respectively.

3.3. Spatial averaging on the ICON grid

Several possibilities for the averaging of variables todifferent places on the stencil exist. It will be shown laterthat it is not the simplest one that leads to the desiredresult. For later use, the following averaging is defined.

n

li

Figure 1. The ICON grid at low resolution and its local stencil: vorticitypoint ν, mass point i and wind edge l. This figure is available in colour

online at www.interscience.wiley.com/journal/qj

Table I. Grid notation.

i Mass pointν Vorticity pointl Wind edgeN(.) Set of neighbouring points of relative dual gridE(.) Set of edges of a cellAi Area of mass cell (triangle)Aν Area of vorticity cell (hexagon/pentagon)Al = 1

2δlλl Area of edge cellλl Length of edge l

δl Length of normal of edge l

∂l Difference quotient along edge l

∂⊥l Difference quotient normal to edge l

• Averaging from triangle vertices to faces:

αi := 1

3

∑ν∈N(i)

αν.

• Averaging from triangle faces to vertices:

αν := 1

Aν

∑i∈N(ν)

Ai

3αi. (13)

• Averaging from triangle vertices or faces to edges:

αl := α◦1l + α◦2

l

2. (14)

• Alternative averaging from triangle vertices toedges:

αl := α�1l + α�2

l

2. (15)

• Averaging from edges to triangle faces:

αi := 1

Ai

∑l∈E(i)

Al

2αl.

The choice of averaging onto vorticity points (13)allows for a uniform definition for hexagons and pen-tagons, takes into account the varying triangle size andperforms the desingularization of the mixed ζµh bracket,as explained later. The average onto edges (14) is definedas the arithmetic mean of the two nearest vertices or facesto that edge (marked by circles in Figure 1). The alterna-tive average from triangle vertices to edges (15) is takenas the arithmetic mean of the two nearest triangle verticesnot adjacent to that edge (marked by diamonds). The dis-tinction between these two definitions will be importantfor the conservation property.

3.4. Local and global quantities on the ICON grid

The ICON model uses a finite volume discretization forthe horizontal curl and divergence operators:

div i (v⊥l ) = 1

Ai

∑l∈E(i)

v⊥l λl,

rot ν(v⊥l ) = 1

Aν

∑l∈E(ν)

v⊥l δl .



The definition of these operators makes use of theVoronoi–Delaunay property of the ICON grid and there-fore the specific scheme developed here also relies onthis property. The method of discretizing Nambu brack-ets is, however, far more general and can in principle beapplied to any spatial discretization technique. The localand global quantities used here are defined in Table II.In the ICON shallow-water prototype, only the normalvelocity v⊥

l at each edge is predicted; the other compo-nent is reconstructed. The discretization of kinetic energysums twice the energy associated with the normal velocityat each edge. As the orientation of the edges is randomlydistributed, we obtain an overall isotropic expression.

Furthermore we make use of the streamfunction χν

and velocity potential γ i of the momentum. Theydefine a Helmholtz decomposition of the momentum intosolenoidal and irrotational parts according to equation (9):

hlv⊥l = ∂lχ︸︷︷︸

=:hvχl

⊥

+ ∂⊥l γ .︸︷︷︸

=:hvγl

⊥

(16)

3.5. Functional derivatives on the ICON grid

Keeping their definition in mind, the functional deriva-tives introduced in section 2 are replaced in the discreteformulation as

δ

δζ→ 1

Aν

∂

∂ζ ν

,δ

δµ→ 1

Ai

∂

∂µi

,δ

δh→ 1

Ai

∂

∂hi

.

By explicitly computing the partial derivatives of theconstitutive quantities H = Hkin + Hpot and E, we obtainthe following expressions:

1

Aν

∂H

∂ζ ν

= −χν,1

Aν

∂E

∂ζν

= ζ ν

hν

= qν,

1

Ai

∂H

∂µi

= −γ i,1

Ai

∂E

∂µi

= 0,

1

Ai

∂H

∂hi

= �i,1

Ai

∂E

∂hi

= −1

2q2

i .

(17)

Table II. Local and global quantities.

Field variable Grid variable

Height h hi

Velocity v v⊥l

Streamfunction χ χν

Velocity potential γ γ i

Vorticity ζ ζ ν := rot ν(v⊥l )

Divergence µ µi := div i (v⊥l )

PV q qν := ζ ν/hν

Specific energy � �i := (v⊥

l · v⊥l

)i+ ghi

Potential enstrophy E E := 12

∑ν Aνhνq

2ν

Kinetic energy Hkin Hkin := ∑l Alhl

(v⊥

l

)2

Potential energy Hpot Hpot = (g/2)∑

i Aih2i

3.6. Nambu brackets on the ICON grid

The first step towards a conservative scheme has beenperformed in the previous section by defining discreteexpressions for the constitutive quantities and computingtheir derivatives. As a second step, we now derive anappropriate discretization for the bracket operations.

3.6.1. The ζµh bracketWhen discretizing the mixed bracket (12), it has to beensured that the PV gradient in the denominator cancelsin order to avoid a spurious singularity. This can beachieved by choosing appropriate averaging as definedin section 3.3 and taking the local x- and y-directionalderivatives in (12) normal to triangle edges. It turnsout that the triangular grid structure strongly facilitatesthis task compared with a C-grid discretization on arectangular grid.

With this choice, for the three permutation terms of(12) (indexed π1–π3) one obtains

{F, H,E}π1ζµh

= 2∑

l

Al

1

∂⊥l q

(∂⊥l

[1

Aν

∂F

∂ζ ν

]i

∂⊥l

[1

Ai

∂H

∂µi

]

−∂⊥l

[1

Ai

∂F

∂µi

]∂⊥l

[1

Aν

∂H

∂ζ ν

]i

)∂⊥l

[1

Ai

∂E

∂hi

],

{F, H,E}π2ζµh

= 2∑

l

Al

1

∂⊥l q

(∂⊥l

[1

Aν

∂E

∂ζ ν

]i

∂⊥l

[1

Ai

∂F

∂µi

]

−∂⊥l

[1

Ai

∂E

∂µi

]∂⊥l

[1

Aν

∂F

∂ζ ν

]i

)∂⊥l

[1

Ai

∂H

∂hi

],

{F, H,E}π3ζµh

= 2∑

l

Al

1

∂⊥l q

(∂⊥l

[1

Aν

∂H

∂ζ ν

]i

∂⊥l

[1

Ai

∂E

∂µi

]

− ∂⊥l

[1

Ai

∂H

∂µi

]∂⊥l

[1

Aν

∂E

∂ζ ν

]i

)∂⊥l

[1

Ai

∂F

∂hi

].

The factor of 2 in front of the sums takes into accountthat the two arbitrary directional derivatives ∂x and ∂y inthe continuous bracket (12) can be replaced by 2∂⊥

l , asexplained earlier.

3.6.2. The ζζ ζ bracketFor this bracket a discrete form of the antisymmetricJacobian is needed:

J (α, β) = k · rot (α∇β) → J (α, β)i = 1

Ai

∑l∈E(i)

λlαl∂lβ.

Using this operator, we obtain for the vorticity bracket(10)

{F, H,E}ζζ ζ

=∑

i

Ai

[1

Aν

∂E

∂ζ ν

]i

1

Ai

∑l∈E(i)

[1

Aν

∂F

∂ζ ν

]l

λl∂l

[1

Aν

∂H

∂ζ ν

],



which turns out to be already twofold antisymmetric asrequired. This is in contrast to the discretization on therectangular grid, where a sum over cyclic permutations(each one corresponding to a term in the ArakawaJacobian) has to be carried out in order to satisfy theantisymmetry condition.

3.6.3. The µµζ bracket

The Jacobian on the hexagons and pentagons is definedas

J (α, β)ν = 1

Aν

∑l∈E(ν)

δlαl∂⊥l β,

and the first permutation of the bracket (11) reads

{F,H, E}π1µµζ =

∑ν

Aν

[1

Aν

∂E

∂ζ ν

]

× 1

Aν

∑l∈E(ν)

[1

Ai

∂F

∂µi

]l

δl∂⊥l

[1

Ai

∂H

∂µi

].

The other two cyclic permutations vanish naturally.

3.7. Resulting conservative operators

In a last step, the prognostic quantities ζ ν , µi and hi attheir respective grid point and the functional derivativesof the energy and potential enstrophy (17) can be insertedinto the previously defined discrete brackets. This willfinally give the conservative prognostic equations.

3.7.1. Vorticity equation

Taking F = ζ ν yields

{ζ ν, H,E}ζζ ζ = −div i

(qlhlv

χ

l

⊥)ν.

For the vorticity in the mixed bracket one obtains

{ζ ν, H,E}π1ζµh = −div i

(qlhlv

γ

l

⊥)ν.

It becomes apparent that the discrete divergence forsolenoidal vorticity flow differs from the irrotational partin the type of PV averaging chosen.

3.7.2. Divergence equation

For F = µi we obtain

{µi, H, E}µµζ = − 1

Ai

∑l∈E(i)

λl∂lqγ l = −rot i

(γ l∂lq

),

(18)

{µi, H, E}π1ζµh = 1

Ai

∑l∈E(i)

λl ql∂⊥l χ = div i

(ql∂

⊥l χ

),

(19)

{µi, H, E}π2ζµh = − 1

Ai

∑l∈E(i)

λl∂⊥l � = −div i

(∂⊥l �

),

where the curl on triangles is defined as

rot i

(γ l∂lq

):= 1

Ai

∑l∈E(i)

λl∂lqγ l. (20)

Just as in the previous section, the operators for thesolenoidal (19) and the irrotational (18) parts of the floware not the same.

3.7.3. Continuity equation

For the height hi in the bracket we obtain

{hi0, H, E}π3ζµh = − 1

Ai

∑l∈E(i)

λl∂⊥l γ = −div i

(∂⊥l γ

),

which uses just the normal divergence operator and thusensures mass conservation as well.

3.8. Summarized operators and prediction process

The equations of motion read, in a summarized form:

∂t ζ ν = −div i

(hl

(qlv

χ

l

⊥ + qlvγ

l

⊥))ν

, (21)

∂tµi = −rot i

(γ l∂lq

) + div i

(ql∂

⊥l χ

) − div i

(∂⊥l �

),

(22)

∂thi = −div i

(∂⊥l γ

). (23)

Writing it this way, the only new operator is the curlon triangles (20); the other changes can be expressed bymeans of the averaging definitions. The vorticity tendencyis given by the negative of the divergence of the vorticityflow (for qh = ζ ) as prescribed by the partial vorticityequation (3). Recalling the analytic relation

−rot (γ∇q) + div (q∇χ) = rot(ζ av),

it becomes clear that the divergence equation (22) isindeed an approximation to the partial divergence equa-tion (4). The same is true for the continuity equation.

Essential for the conservation property is the correctaccomplishment of the averaging as defined in section3.3. At a given time in the prediction process, thefollowing steps are conducted.

(1) For vorticity ζ ν , divergence µi and height hi

given from the previous time level, reconstructstreamfunction χν , velocity potential γ i and thecorresponding velocity components vχ

l

⊥ and vγ

l

⊥

(16).(2) Compute tendencies using (21)–(23).(3) Step forward in time (leapfrog with optional

Asselin filtering is used here).

Two drawbacks of this scheme are obvious.

• The use of potentials requires elliptic problems tobe solved at every time step, which involves anincrease by a factor of eight in CPU time comparedwith the momentum formulation.



• The relatively large stencils make it difficult todefine an efficient implicit time integration scheme.

However, no vector reconstruction (for the normal veloc-ity part) is required as the scheme can be expressedentirely by the scalar potentials. The semi-discretization(21)–(23) will be termed the ‘Nambu scheme’ in the nextsection.

4. Numerical results

In this section a comparison between the original ICONshallow-water prototype ‘icoswp v1-1’ in its potentialenstrophy conserving mode (ICOSWP) and the Nambuscheme is presented regarding different aspects. A thirdscheme that also uses the vorticity-divergence form of theequations but is not defined according to the describedmethod, and is therefore not explicitly conservative, hasalso been evaluated but is not displayed. It is verysimilar in most aspects to the ICOSWP scheme. Grav-ity wave (atmosphere at rest, pressure displacement atthe pole), Rossby wave (test case 6 of Williamsonet al. 1992) and white noise (h = 350 m ± 100 m; ζ , µ =±5 × 10−6 s−1, chosen to allow for comparison withBonaventura and Ringler (2005)) initial conditions havebeen used for these tests. Resolution was fixed at approx-imately 1◦.

4.1. Rossby wave

To get an impression of the scheme’s forecast behaviour,a plot of the height field of a Rossby wave after 10 and16 days of inviscid integration is given in Figure 2. Someslight asymmetries are visible after 10 days, particularlyin the ICOSWP scheme. They initiate the decay ofthis structure as described in Thuburn and Li (2000).This decay is taking place slightly more slowly in theNambu scheme. After 16 days, however, the wave hasundergone a considerable change in both schemes and atransition to a more zonal state is present. This behaviour

continues until eventually the ICON scheme becomesunstable, while no instability is detected with the Nambuscheme.

4.2. Conservation aspects

Potential enstrophy conservation can be proven ana-lytically by calculating its tendency with the prognos-tic equations and writing it as an integral of a diver-gence:

∂tE = 1

2

∫∂t (hq2) dA = 1

2

∫q(2q∂t ζ − q∂th) dA

= −1

2

∫div (hq2v) dA = 0.

In semi-discretized notation, proving a given schemeto be conservative is equivalent to showing algebraicallythat the corresponding sum vanishes:

∂tE = 1

2

∑ν

Aν∂t

(hνq

2ν

)= 1

2

∑ν

Aνqν

(2∂t ζ ν − qν∂thν

), (24)

with ∂thν = (∂thi)ν = −div i(hlv⊥l )

ν.

In Figure 3 total potential enstrophy is plotted togetherwith its tendency against time for a gravity wave circlingthe globe. The fact that the tendency (diagnosed usingequation (24)) is indeed the temporal derivative is clearlyvisible, and it can also be seen that neither vanish in theICOSWP in contrast to the Nambu scheme.

It is argued in Bonaventura and Ringler (2005) thatthe ICOSWP scheme is conservative by calculating thepotential enstrophy tendency as

∂tEICON = 1

2

∑ν

Aνqν

(2∂tζ ν − qν∂thν

)(25)

with ∂thν = −div ν(hlvl),

(a)

−60°−60°

0°0°

60°60°

8500 m9000 m9500 m10000 m10500 m

−60°−60°

0°0°

60°60°

8000 m

9000 m8500 m

9500 m10000 m10500 m

−60°−60°

0°0°

60°60°

8500 m9000 m9500 m10000 m10500 m

−60°−60°

0°0°

60°60°

8000 m8500 m9000 m9500 m10000 m10500 m

(b)

Figure 2. Pressure field after 10 days (upper panels) and 16 days (lower panels). (a): ICOSWP; (b): Nambu scheme. Initial condition: Rossbywave (test case 6 of Williamson et al. 1992). This figure is available in colour online at www.interscience.wiley.com/journal/qj



5e−01 d 1e+00 d 2e+00 d

Time

0e+00

1e−05

2e−05

(a)

−1e−03

0e+00

1e−03

E−E

0E

0

5e−01 d 1e+00 d 2e+00 d

Time

0e+00

1e−05

2e−05

(b)

−1e−03

0e+00

1e−03

E−E

0E

0

tE tE

Figure 3. (a): ICOSWP; (b): Nambu. Upper panels: potential enstrophy. Lower panels: tendency of potential enstrophy diagnosed using equations(24) (solid line) and (25) (dashed line). Initial condition: atmosphere at rest, pressure displacement at the pole.

and showing that this sum vanishes. This is indeed true(dashed line in Figure 3), however the continuity equa-tion at triangle faces used in (25) is not a prognosticequation in the C-type staggered scheme applied andthis design hence does not prevent the potential enstro-phy from fluctuating. It can be seen that the draw-back of ICOSWP in reflecting the conservation prop-erty lies in the type of spatial discretization. It ispossible to overcome this shortcoming using conserva-tive operators as previously described (right panel ofFigure 3).

The findings on the ability of the schemes to conservetotal energy are similar to those discussed above regard-ing potential enstrophy.

The assertion that non-conservation in ICOSWP comesfrom spatial rather than temporal discretization incon-sistencies can also be confirmed by looking at differ-ent temporal resolutions. Figure 4 shows the integratedenstrophy error versus time step. Even after choosing atiny time step, the deviation of the ICOSWP scheme can-not be made smaller than a fixed value resulting fromspace discretization. In contrast, the error of the Nambuscheme is much smaller and depends linearly on time-stepsize.

When friction terms are added to the equation ofmotion, the system is certainly not conservative anymore,

0.00

0.50

1.00

1.50

0.00 0.05 0.10 0.15 0.20Courant number

0.00 0.05 0.10 0.15 0.20Courant number

(a)

0.00

0.01

0.02

(b)

E−E

0E

0∫d

tE

−E0

E0

∫dt

Figure 4. Integrated potential enstrophy deviation from the initial stateagainst time step size in Courant units without (solid) and with (dotted)Asselin filtering. (a): ICOSWP; (b): Nambu scheme. Initial condition:

Rossby wave.

−0.10

−0.05

0.00

0.00 5.00 10.00

Viscosity coefficient / (k −2max · d−1)

H−H

0H

0Figure 5. Energy loss after two days against viscosity coefficient forICOSWP (solid) and Nambu (dotted) schemes. Initial condition: white

noise.

as shown in Figure 5. It can, however, be clearly seenthat the Nambu scheme reproduces very well the propertyof the original system to be conservative in the inviscidlimit. In contrast, this is not the case for the ICOSWPscheme: it produces kinetic energy that can be seenas an excess at zero viscosity. It is just this energythat for very small viscosity coefficients will pile upat the smallest resolved scales, producing non-linearcomputational instability.

4.3. Stability

The energy and enstrophy produced by the ICOSWPscheme as described in the previous section will even-tually ruin any forecast if not removed by some arti-ficial friction terms. This mechanism of computationalinstability has been known since Phillips (1956, 1959)and can be observed (Figure 6) in a monotonic increaseof potential enstrophy. The Nambu scheme, in contrast,exhibits only a small loss, which is due to Asselin fil-tering and could be further reduced. No artificial fric-tion is therefore necessary here and the scheme isvery stable. This behaviour has been observed with abroad class of initial conditions and grid resolution set-tings.

4.4. Spectrums

Decaying turbulence tests have been promoted by Ringlerand Randall (2002) and Bonaventura and Ringler (2005)to test a scheme’s performance in dealing with verysmall structures and to evaluate spectral properties. The



0.00

0.05

4.00 d 8.00 dTime

4.00 d 8.00 dTime

(a)

0.00

0.05

(b)

E−E

0E

0

E−E

0E

0

Figure 6. Total energy (dotted line) and potential enstrophy (solid line)for (a) ICOSWP and (b) Nambu schemes. Initial condition: white noise.

idea is to choose a very noisy initial condition (herewhite noise) and let the system propagate with someviscosity added to initiate energy and enstrophy cascades.After a given time (here 20 days), a spectral analysisof kinetic energy in terms of spherical harmonics iscarried out. For incompressible two-dimensional flow themain theoretical results on these spectrums come fromKolmogorov (1941), Kraichnan (1967) and Kraichnanand Montgomery (1980), summarized in Salmon (1998).Assuming a large inertial range, a dimensional analysisof the cascading quantities yields a k−5/3 behaviour forlarge scales and a k−3 behaviour for small scales. Inspite of the strong prerequisites, these results can give anidea of what kind of output a numerical scheme shouldgive.

The spectral distribution of kinetic energy and thecorresponding pressure field after 20 days is depictedfor both schemes in Figure 7. The apparent differenceis that the spectrum of the Nambu scheme is steeper thanthat of the ICOSWP scheme: it has more energy in thelarge scales and less in the smallest scales, a fact that isalso visible in the pressure pattern. The ICOSWP schemeconstantly produces energy at the smallest resolvedscale, which is removed by viscosity (if not, the statewould be unstable). The slope of the kinetic energyspectrum in this scheme is probably determined by theviscosity coefficient rather than by the intrinsic dynamicalproperties. Dissipation operators were taken as ν�v in theICOSWP momentum formulation and ν�ζ and ν�µ inthe vorticity-divergence formulation. In addition, a thirdscheme in vorticity-divergence form but not conservativein the Nambu sense was tested with exactly the samedissipation operator as the Nambu scheme and was foundto reproduce the spectrum of the ICOSWP scheme.

5. Conclusion and outlook

It was shown that energy–vorticity theory, which isa generalization of the Hamiltonian formalism for

fluid dynamics equations, can be useful for construct-ing conservative schemes even on a staggered trian-gular grid. Such schemes are defined by the prop-erty to conserve algebraically exact specific quantities(here total energy and potential enstrophy) when writ-ten as a spatially discretized ordinary differential equa-tion. The tendency of potential enstrophy of such ascheme can locally be written as a divergence andthis type of representation of the nonlinear dynamicsensures a correct account of the inverse energy cas-cade. It must, however, be said that in the shallow-water case, these schemes require a relatively high com-putational cost for the reconstruction of the potentialsfrom divergence and vorticity and the extended averagingscheme.

With numerical tests it could be shown that apartfrom being conservative the schemes thus defined alsotend to be very stable and reproduce a spectral dis-tribution of kinetic energy consistent with theoreticalexpectations. Apparently the numerical scheme chosenhas an influence not only on very small scales butalso on the global scale, which might be of importancefor structure formation in long-term integrations. Thesefindings encourage researchers to apply such ideas alsoto more sophisticated models (Gassmann and Herzog,2008).

A. Algebraic potential enstrophy conservation

Proposition 1 The semidiscretization (21)–(23) conservesthe potential enstrophy E = 1

2

∑ν Aνhνq

2ν exactly, i.e.

∂tE = 0.

Proof. We show that the potential enstrophy change atone grid point depends only on the oriented edges of thatcell and sums to zero for neighbouring cells of a commonedge: [

∂t

(Aν1hν1q

2ν1

)]l= − [

∂t

(Aν2hν2q

2ν2

)]l, (26)

for neighbouring gridpoints ν1, ν2 at edge l. From thisglobal conservation follows.

We compute

∂t

(1

2Aνhνq

2ν

)=∂t

(1

2Aν

(ζ ν + fν)2

hν

)=−1

3qν

∑i∈N(ν)

∑l∈E(i)

λlqlhlvχ

l

⊥

︸︷︷︸=:A

−1

3qν

∑i∈N(ν)

∑l∈E(i)

λl qlhlvγ

l

⊥

︸︷︷︸=:B

+1

6q2

ν

∑i∈N(ν)

∑l∈E(i)

λlhlvγ

l

⊥

︸︷︷︸=:C

.



−4.0

−3.0

−2.0

log

e(l)

0.0 0.5 1.0 1.5log(wave number degree l )

(a)

−4.0

−3.0

−2.0

log

e(l)

0.0 0.5 1.0 1.5log(wave number degree l )

(b)

Figure 7. Spectral kinetic energy density ε (upper panels) and height field h (lower panels) at the end of the simulation for (a) ICOSWP and (b)

Nambu schemes (solid line). Also shown are initial distribution (dotted), k− 53 and k−3 lines (dashed). Initial condition: white noise. This figure

is available in colour online at www.interscience.wiley.com/journal/qj

Term A comes from the solenoidal part of vorticityequation (21) and can be represented as

A = −1

3qν

∑l∈E2(ν)

λlql∂lχ

= −1

2

∑l∈E(ν)

δl

(2q2

l − q2l

)∂⊥l χ. (27)

E2(ν) stands for the set of the nearest edges of grid point ν

that do not originate there. Expression (27) clearly fulfilsthe condition (26).

For the irrotational part of the vorticity equation B wehave

B = −1

3qν

∑l∈E2(ν)

λl qlhlvγ

l

⊥

= −1

3

∑l∈E2(ν)

λl q2l hlv

γ

l

⊥ + 1

2

∑l∈E2(ν)

Al∂⊥l q2

lhlvγ

l

⊥.

(28)

Finally, the term C originating in the continuity equa-tion gives

C = 1

6q2

ν

∑l∈E2(ν)

λlhlvγ

l

⊥

= 1

6

∑l∈E2(ν)

λl q2lhlv

γ

l

⊥ − 1

2

∑l∈E2(ν)

Al∂⊥l q2

lhlvγ

l

⊥.

(29)

Now, the first terms of (28) and (29) separately fulfil therequirement (26) and the second terms cancel each other

(they do not sum up to zero individually as they no longerdepend on edge orientation), which ends the proof.

To stress the analogy to the analytical expression, werewrite the sum as

∂t

(1

2Aνhνq

2ν

)=A + B + C

=Aν

2

(rotν

[(2q2

l − q2l

)∂⊥l χ

]− divi

[(2q2

l − q2l

)hlv

γ

l

⊥]ν

).

Recalling

rot(q2∇χ

) = −div(q2hvχ

)it becomes clear that locally the tendency of hq2 is indeeda divergence. With this knowledge, a discrete Gauss–Stokes theorem for a limited area ( ) model based onthe presented scheme can be formulated:

∂t

1

2

∑ν∈

Aνhνq2ν = 1

2

∑l∈∂

λl

((2q2

l − q2l

)∂⊥l χ

−(

2q2l − q2

l

)hlv

γ

l

⊥)

.

The fact of no potential enstrophy production thereforeholds not only globally but also locally: for any section with boundary ∂ a corresponding theorem holds.This means that there are no sources or non-convectivefluxes for potential enstrophy. From that, individualconservation such as

dqn

dt= 0 ∀n ∈ N

could also be derived.



Acknowledgements

The authors are indebted to A. Gassmann (MPI-M),H.-J. Herzog (DWD), S. Reich (University Potsdam)and L. Bonaventura (Politecnico di Milano) for helpfuldiscussions on general numerics and specifically on theICON project. We further thank the two reviewers fortheir comments, which helped to improve the manuscript.

This work has been funded by DFG (DeutscheForschungsgemeinschaft) as project ‘Structure PreservingNumerics’.

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