a hybrid eulerian lagrangian numerical scheme … hybrid eulerian lagrangian numerical scheme for...

A hybrid Eulerian Lagrangian numerical scheme for

solving prognostic equations in fluid dynamics

Eigil Kaasa, Brian Sørensena, Peter Hjort Lauritzenb, Ayoe Buus Hansenc

aNiels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100Copenhagen, DENMARK

bClimate and Global Dynamics Division, NCAR, 1850 Table Mesa Drive, Boulder, CO,80305, Boulder, USA

cAarhus University, Department of Environmental Science, Frederiksborgvej 399,DK-4000 Roskilde, DENMARK

Abstract

A new hybrid Eulerian Lagrangian numerical scheme (HEL) for solving prog-nostic equations in fluid dynamics is proposed. The basic idea is to keep bothan Eulerian and a fully Lagrangian representation of all prognostic variables.

The time step in Lagrangian space is obtained as a translation of irreg-ularly spaced Lagrangian parcels along downstream trajectories. Tendenciesdue to other physical processes than advection are calculated in Eulerianspace, interpolated, and added to the Lagrangian parcel values. A direction-ally biased mixing amongst neighboring Lagrangian parcels is introduced.The rate of mixing is proportional to the local deformation rate of the flow.

The time stepping in Eulerian representation is achieved in two steps:first a mass conserving Eulerian or semi-Lagrangian scheme is used to obtaina provisional forecast. This forecast is then nudged towards target valuesdefined from the irregularly spaced Lagrangian parcel values. The nudgingprocedure is defined in such a way that mass conservation and shape preser-vation is ensured in Eulerian space.

The HEL scheme is accurate, multi-tracer efficient, mass conserving, andshape preserving. In Lagrangian space only physically based mixing takesplace, i.e., the problem of artificial numerical mixing is avoided. This prop-erty is highly important in atmospheric chemical transport models since spu-rious numerical mixing can impact chemical concentrations severely.

The attractive properties of HEL are demonstrated in two-dimensionaltests. These include deformational passive transport on the sphere, andsimulations with a semi-implicit shallow water model including topography.

Preprint submitted to Journal of Computational Physics October 17, 2012

1. Introduction

Numerical chemical weather forecast systems and Earth system mod-els include components describing the chemisty, including aerosols, and theinteraction of these with cloud and radiation processes, e.g. [1, 2]. The in-troduction of many more prognostic variables, sometimes several hundred,representing the concentrations of the individual chemical species, poses somesevere challenges regarding computational methodologies.

1.1. Desirable properties

A number of desirable properties for numerical schemes solving the conti-nuity and other prognostic equations have been identified, e.g. [3, 4, 5, 6, 7].These deal with

1. Accuracy

2. Stability

3. Computational efficiency (i.e., accuracy for a given computational re-source)

4. Transportivity and locality (the solution should follow characteristics)

5. Shape preservation (positive definite and non-oscillatory)

6. Conservation of invariant quantities such as mass

7. Consistency between wind and mass fields (minimize the so-called mass-wind inconsistency problem)

8. Compatibility (mixing ratios should be bound by their upstream values)

9. Preservation of constant mixing ratios in non-linear flows

10. Preservation of linear correlations between constituents

For a brief discussion of these see [8].While several of the above listed desired properties are of particular rel-

evance in atmospheric chemical transport models, most are also highly de-sirable/necessary when it comes to simulation of geophysical dynamics, notleast those components involving water vapor, liquid water droplets and icecrystals in the atmosphere, or e.g. salinity in the ocean.

There is, however, one additional property, not listed above and less dis-cussed in the literature, which is particularly important for chemistry andchemistry-climate applications:

11. Avoidance of spurious numerical mixing/unmixing [9].

2

1.2. Mixing and unmixing in Eulerian based modelsThe 11th property above refers to the ability of a scheme to preserve

pre-existing functional relations between tracers. Mixing or unmixing canbe divided into three categories (see Figure 1), real mixing, range preservingunmixing, and overshooting. Most transport schemes operating on a fixedEulerian grid (including semi-Lagrangian schemes) will lead to numericalmixing between tracers. In a real fluid, mixing can only result in values thatlie on so called mixing lines, which in Figure 1 are straight lines that connectany two points in the convex hull [10]. A transport scheme should thereforeideally not produce any mixing that cannot be represented by a mixing line,i.e., real mixing. If mixing results in points which fall outside the domain ofreal mixing, it is spurious unmixing, which can be either range preserving,if the values fall within the original range of values, or overshooting if thevalues are outside the possible range of values. It should be noted that non-shape preserving schemes will give rise to overshooting, which can result incompletely unphysical values, such as negative concentrations.

In the case where the flow is completely linear, and therefore free ofmacro-scale deformation or turbulence, only molecular mixing will occur.Since, in geophysical problems, molecular mixing is normally several ordersof magnitude smaller than mixing due to macro-scale turbulence it can beneglected, and therefore no real mixing takes place when the flow is linear.Any functional relation between tracers will therefore remain unchanged forinert tracers, i.e., all points in a diagram like that in Figure 1 will keep theirinitial positions. However, as mentioned above, this is generally not thecase when an Eulerian based numerical scheme is used to solve this simplephysical problem of linear advection. In this case some spurious numericalmixing or unmixing will normally take place, i.e., the points in the diagramwill move away from their initial positions. In the case where the tracersare chemically active, this can potentially be a serious problem as spuriouschemical reactions are then initiated, or chemical equilibria are displaced.

Note that apart from the initial truncation numerical methods based onorthogonal series expansion functions are the only Eulerian type numeri-cal schemes, which do not introduce numerical mixing in the case of non-deforming flow. However, generally filters introducing numerical mixing mustbe applied to prevent e.g. development of negative values, when such schemesare applied for linear flow problems.

In the more realistic case of non-linear flow Lagrangian parcels will de-form into thinner and thinner filaments, which in nature are finally mixed

3

via molecular mixing. An important question is to what extend explicit nu-merical diffusion/mixing is required as a supplement to that implied by thenative version of some numerical scheme in order to mimic the cascade intosmall scales correctly. For typical grid point based methods, including semi-Lagrangian schemes, some inherent numerical mixing is almost unavoidableand this may be sufficient to control the cascade in a statistical sense. InGalerkin methods – e.g. the classical spectral method [11] – the gradual de-velopment of non-resolved filaments and structures is controlled by demand-ing the residual to be orthogonal to the resolved expansion functions (see e.g.[11] or [12]). This gives rise to an implied mixing, which, depending on thechosen expansion functions, is generally non-local in physical space. Explicithorizontal diffusion – in addition to the implied mixing – is required for mostGalerkin schemes in order to prevent so-called ”spectral blocking” [11], i.e.,spurious accumulation of energy on the shortest resolved scales. The situa-tion is different in pseudo-spectral models where the non-linear developmentof unresolved structures gives rise to aliasing, i.e., the non-linear generationof unresolved features is misinterpreted as spurious resolved features, as op-posed to the situation in traditional spectral models where aliasing is formallyavoided [11, 12]. Diffusion, which introduces explicit numerical mixing, canbe used to reduce the aliasing problem and thereby one can obtain a morerealistic spectral distribution of the prognostic variables in a pseudo-spectralmodel.

In conclusion explicit mixing, in terms of filters, diffusion, spectral damp-ing etc., is needed in both grid point/cell methods and methods based onseries expansion in order to ensure shape preservation, and in particular pos-itive definiteness, and to control the cascade into smaller scales in a realisticway. In general such mixing will, however, not represent true physical mixingalthough it may be ”real” in the sense described above.

1.3. The HEL approach

Here we present a numerical methodology, termed the hybrid Eulerian -Lagrangian (HEL) numerical scheme, which has been designed to fulfill asmany as possible of the desired properties mentioned in Section 1.1. Theaim is to combine the Eulerian and the Lagrangian approaches in such a waythat the main problems related to either of these are eliminated or at leastreduced. The ideas behind HEL have been inspired by other Lagrangianapproaches, in particular that of ATTILA (Atmospheric Tracer Transport

4

ξmin

ξ

ξmax

χχmin χmax

range preserving unmixing

range preserving unmixing

real mixing

overshooting

overshooting

Figure 1: Illustration of numerical mixing categories. The thick curve is the pre-existingfunctional relation between tracer ξ and tracer χ. Any new relative concentrations (ξk, χk),generated by the transport scheme, can be represented as a point. If the point falls withinthe shaded convex hull, it is classified as real mixing. If within the dashed rectangle butoutside the shaded area it is classified as range preserving unmixing, and, finally, if outsidethe dashed rectangle it is classified as overshooting. (adopted from [9])

In a Lagrangian Atmospheric model) [13, 14, 15], which is a global fullyLagrangian model.

Contrary to the problems mentioned in the previous subsection for Eu-lerian based schemes, fully Lagrangian schemes are formally exact for thepure advection problem assuming trajectories have been calculated exactly.However, with such schemes, and in general non-linear flows, the Lagrangianparcels become irregularly distributed in space and strongly deformed. Theirregular distribution is a fundamental problem for the following reasons:

1. Dynamical tendencies due to non-advective processes generally need tobe estimated on a regular grid to ensure consistency with and betweenthe governing prognostic equations.

2. In practice, it is problematic to calculate parameterized physical pro-cesses on an irregular grid.

3. There may be sub-domains with no or very few parcels.

The result of an interpolation from mass centers of Lagrangian parcels to aregular Eulerian grid will typically, after some time, show a completely unre-alistic ”spotty” distribution because neighboring parcels originate from quitedifferent positions at the initial time of the integration (see e.g. lower right

5

panel of Figure 12). In nature finite size Lagrangian parcels typically deforminto very long, very thin filaments. A Lagrangian transport scheme aimingat resolving the developing shape of individual parcels without any explicitmixing between neighboring parcels would represent the discrete Lagrangiananalogy to the pseudospectral method. The aliasing of such a Lagrangianscheme is realised when one interpolates from Lagrangian to physical space,i.e., to a fixed Eulerian grid. To avoid such aliasing one can introduce mix-ing between parcels. One may think of the difference between a mixed andunmixed Lagrangian scheme as an analogy to the difference between spectraland pseudo-spectral schemes.

In HEL the density and other optionally prognostic variables, is knownat all times via a fully Lagrangian as well as a traditional Eulerian represen-tation. At each time step a nudging technique is applied where the densityinformation in the downstream translated Lagrangian parcels is used to mod-ify or ”repair” an Eulerian based advection. In this way the non-dispersive,non-diffusive, and shape preserving advantages of the Lagrangian methodcan be adopted in an otherwise diffusive and/or strongly dispersive Eulerianbased transport scheme. Physical tendency contributions not related to pureadvection are most obviously and accurately calculated in Eulerian space andsubsequently interpolated and added to the Lagrangian parcel values. In thisway the bulk of the model history is kept in the Lagrangian representation.

With this generic design we are, however, still left with the problem ofaliasing in Lagrangian space. To reduce or eliminate aliasing we introduce a”real” mixing in Lagrangian space between neighboring parcels. We let thedegree of mixing depend on the local flow deformation rate. Not surprisingly,the introduction of such mixing turns out to be instrumental for a properdescription of the spectral distribution of prognostic variables although thisissue is not covered in the present paper. The mixing, presented in more de-tail below, is quite different from the uncontrollable, and in many cases unre-alistic, numerical mixing/unmixing mentioned above, which is introduced inmost traditional models based on an Eulerian grid/cell representation, i.e.,Eulerian and semi-Lagrangian type methods. This is because in such modelsthe degree of mixing is different from tracer to tracer, because it depends onthe spatial roughness of their density fields. The degree of mixing appliedhere between Lagrangian parcels is the same for all tracers, and it dependsonly on the actual physical deformation of the Lagrangian parcels. Thus inthe Lagrangian space representation, i.e., for the parcels, the problem of nu-merical mixing and unmixing is eliminated. It is noted that the introduction

6

of a flow-dependent mixing based on the degree of deformation is not new.[16] introduced a horizontal diffusion which was dependent on the magnitudeof the deformation rate of the flow, and later [17] used a similar approach toobtain a flow dependent degree of mixing in a semi-Lagrangian model. AlsoATTILA [15] and CLaMS (Chemical Lagrangian Model of the Stratosphere)[18] employ mixing depending on horizontal flow deformation rate.

Although we are not dealing with chemistry here it is noted that calcula-tion of chemical reactions are naturally performed in the Lagrangian parcelswhere it is known that only real mixing takes place.

1.4. Other relevant Lagrangian type numerical schemes

HEL has some similarities to particle in cell (PIC) methods, which havebeen used extensively in e.g. plasma physics [19]. However, one of the maindifferences is that in PIC methods the number of Lagrangian parcels farexceeds that of Eulerian grid points/cells, while in HEL these numbers areequal or at least similar in magnitude.

As indicated above the transport and mixing in the HEL scheme is similarto other Lagrangian parcel methods like ATTILA, CLaMS and FPIC [20].In [13], and later improved in [14] ATTILA was implemented in a generalcirculation model to simulate transport of water vapor and cloud water. Thiswas extended in [15] to an interactively coupled chemistry-climate modelversion of ATTILA, i.e., water vapor, cloud water, and mixing ratios of allchemical tracers are known for each Lagrangian parcel. ATTILA is ableto maintain steep gradients, is mass conserving, numerically non-diffusive,and has been used e.g. for studying the climate radiative forcing relatedto aircraft contrails [21]. ATTILA does not handle the ”dry dynamics” asopposed to HEL, however, besides this the HEL and ATTILA approachesare quite similar when applied for non dry-dynamical prognostic variables.One difference, though, is that ATTILA on average holds more Lagrangianparcels per Eulerian grid cell than the version of HEL presented here, whichon average only has one. More importantly, there are some differences in theway horizontal mixing between neighboring parcels is performed in ATTILAand HEL. For both schemes the degree of mixing depends on the horizontalshear deformation rate of the flow, however, in ATTILA this is a simpleanalytical expression based on [22], whereas in HEL the deformation of eachparcel is kept as an additional prognostic variable, which is increased eachtime step in proportion to the shear deformation rate, and attempted reducedvia realised mixing with neighboring parcels.

7

The approaches in the CLaMS model [18] are less similar than ATTILAto those applied in HEL. The mixing in CLaMS is based on a dynamicallyadaptive grid and it becomes active in terms of mass exchange between neigh-boring parcels when the flow deformation is high. A local, in time and space,Lyapunov exponent is used to determine the degree of mixing that takesplace, which in practice takes place via generation of new Lagrangian parcelsin strongly deformed flow, or merging of clustered Lagrangian parcels. Thisis one main difference compared to HEL and ATTILA where Lagrangianparcels survive throughout the integration. In the ATLAS model [23] theflow dependent mixing methodology of CLaMS has been modified with em-phasis on better performance in lower resolution model configurations. Alsoin FPIC [20] an implied mixing takes place via simple birth and merging ofparticles.

The so-called ”Trajectory-Tracking scheme” introduced in [24] and up-dated in [25] has some similarities to HEL. In two dimensional problemsthis scheme treats Lagrangian parcels as polygons with a finite number ofedges, and with all Lagrangian parcel polygons spanning exactly the com-plete integration domain. The Eulerian space representation is obtained viaa first-order conservative remapping so that total mass is conserved in theLagrangian as well as the Eulerian representation. A ”curvature-guard” al-gorithm is applied in order to maintain an accurate polygon representationin non-linear deformation flows. However, this algorithm does not lead toany mixing, as in HEL, between neighboring Lagrangian parcels. Therefore,in long term simulations, one should expect problems equivalent to aliasing.

1.5. Overview

The paper is organized as follows: Section 2 provides a generic descriptionof the HEL approach, i.e., HEL without any type of mixing between parcels,while Section 3 describes how mixing between adjacent parcels is achieved.Section 4 presents passive inert transport tests on the sphere. Various tra-ditional and more recently proposed error measures and evaluation statisticsare used to demonstrate the performance of HEL in both solid body rota-tion flow (Section 4.1) and in strongly non-linear flow (Sections 4.2 and 4.3).Section 5 deals with some initial attempts to implement HEL as the basisfor a dynamical in core in a geophysical fluid dynamics model. In this casethe test bed is a shallow water model in plane geometry. Finally Sections 6and 7 discuss the results and summarizes the basic findings.

8

2. The HEL approach - passive transport

To introduce the procedure followed in HEL in more detail we first con-sider the continuity equations for a set of M tracers with densities ρm:

∂ρm∂t

= −∇ · (ρmV) ,m = 1, . . . ,M , (1)

or alternatively on Lagrangian form,

d ln ρmdt

= −∇ ·V ,m = 1, . . . ,M (2)

where V is the flow velocity vector. For simplicity we have ignored anysources and sinks, and any diffusion in (1) or (2).

In geophysical fluid dynamics (1) or (2) is normally solved via finite vol-ume (FV) methods operating on a fixed Eulerian grid. Two different familiesof FV methods have been applied: flux based and cell integrated. In fluxbased methods the fluxes of mass swept through each face of pre-definedEulerian grid cells within a time step are calculated first, and the change indensity is then determined from the net inflow of mass into each grid cell.In cell integrated methods a semi-Lagrangian upstream departure cell is firstidentified. The estimated total mass in this upstream cell then determinesthe corresponding arrival (Eulerian) grid cell density. For a more detaileddescription of the difference between the two families, see [8], where it is alsodemonstrated that they are in fact numerically equivalent. For a generalreview of FV methods, we refer to [26] and [27].

It should be noted that other non-FV, yet still mass conservative, schemeshave also been used. For the present work it is of particular relevance to men-tion the so called locally mass conserving semi-Lagrangian method (LMCSL)[28] which is based on a simple partition of unity principle.

Depending on the chosen order of accuracy any numerical method - Eu-lerian or semi-Lagrangian type - applied for solving (1) or (2) on an Euleriangrid will suffer from some degree of dissipation (or possibly anti-dissipation),some numerical dispersion, and generally, for higher order schemes, the so-lution will not be shape preserving.

As mentioned above the main motivation behind the present work is touse a fully Lagrangian forecast, run in parallel, to modify the Eulerian gridforecast in such a way that the above mentioned disadvantages are reducedor eliminated. A purely Lagrangian forecast describes the temporal evolution

9

of the densities of individual Lagrangian fluid parcels as they move around.Formally it is straightforward to integrate (2) for a Lagrangian parcel fromtime t to some future time t+ ∆t:

ln ρ (r(t+ ∆t), t+ ∆t)) − ln ρ (r(t), t))

= −∫ t+∆t

t

∇ ·V(r(t′), t′

)dt′

= ∆tD , (3)

where r(t) is the position vector of the parcel at time t, and D represents theaverage divergence along the trajectory from r(t) to r(t+ ∆t). From (3) oneimmediately gets:

ρ (r(t+ ∆t), t+ ∆t)) = ρ (r(t), t)) exp(∆tD

), (4)

i.e., the effect of divergence over the period from t to t + ∆t is an expan-sion/contraction factor:

σ = exp(∆tD

)(5)

multiplied by the original parcel density at time t.We will now consider the actual numerical discretization of the prognos-

tic equations in Eulerian and Lagrangian space. The Lagrangian parcels areintroduced at the initial time, and they survive throughout the model inte-gration. In the version of HEL presented here, the total number of parcels Pand the number of Eulerian grid cells K are equal. At the initial time step,n = 0, Lagrangian parcel densities, Lρ, are initialized by the correspondingvalues in Eulerian grid cell centroids, i.e.,

Lρ0p = Eρ0

k

Lρ0m,p = Eρ0

m,k ,m = 1, . . . ,M (6)

where p, k = 1, . . . , P (= K), and m counts the individual tracers as in(1)/(2). In the following we generally use upper left superscripts L and Eto indicate Lagrangian and Eulerian space representation, respectively. Anupper right index denotes the time step. A list of all prognostic variables inHEL to be described below can be found in Appendix A.

Assume some numerical scheme has been used to solve (1) or (2) in theEulerian grid cell representation, and let

Eρn+1k

Eρn+1m,k ,m = 1, . . . ,M (7)

10

denote the forecast in Eulerian grid cells k = 1, . . . , K at time step n+ 1.In general we use the notation (·) to represent some provisional approximatevalue. This is also the case here where ρ indicates that the forecast at timestep n + 1 is only a provisional Eulerian space forecast to be modified bydensities in the Lagrangian representation.

Letting parcels follow downstream trajectories from time step n to n+ 1estimated from the actual velocity components in the Eulerian grid, oneobtains an approximate Lagrangian solution to the pure advection problem.However, in a general divergent flow the parcel volume density will of courseundergo changes. According to (4) and (5) the effect of divergence for parcelp from time step n to n+ 1 can simply be modeled as

Lρn+1p = Lσn+1/2

pLρnp (8)

where superscript n + 1/2 indicates that the expansion/contraction factorrepresents the effect of divergence from time step n to n + 1. In practiceσn+1/2 is determined in Eulerian space from the provisional Eulerian spaceforecast of the ”dry air” density, Eρn+1, i.e., including the effect of divergence,and from a corresponding purely advective forecast, i.e., not including theeffect of divergence, which we term E,advρn+1:

Eσn+1/2 =Eρn+1k

E,advρn+1, k = 1, . . . , K (9)

For cell integrated FV and the LMCSL schemes applied in the present paperthe calculation of Eσn+1/2 is straightforward since estimation of E,advρn+1 isan inherent part of these schemes.

Once new downstream parcel positions Lrn+1 have been found Lσn+1/2 canbe obtained via interpolation from Eσn+1/2. The parcel forecast for parcel p,including divergence, then simply becomes:

Lρn+1p = Lσn+1/2

pLρnp

Lρn+1m,p = Lσn+1/2

pLρnm,p ,m = 1, . . . ,M. (10)

The modification of the Eulerian densities using the parcel densities isdone by first interpolating the irregularly spaced parcel densities to theEulerian grid obtaining certain Eulerian space target values, ET and ETm,m = 1, . . . ,M , and then nudging the original Eulerian based forecast to-wards these values under the constraints of mass conservation and shapepreservation (see below).

11

A generic recipe in an atmospheric multi-tracer application of HEL, notyet considering the mixing, can be summarized as follows at a given time stepn (here omitting indices for Eulerian grid cells, k, and Lagrangian parcels,p):

1. Perform a conventional, preferably inherently mass conserving Eulerianor semi-Lagrangian time step of total ”dry” density, Eρn+1, valid on anEulerian grid. This is termed the provisional forecast.

2. Perform a corresponding purely advective time step in Eulerian spaceE,advρn+1 of the ”dry” density, and use this to calculate the divergentmultiplication factor, Eσn+1/2 = Eρn+1/E,advρn+1.

3. For all tracers, m = 1, . . . ,M , perform a provisional Eulerian spaceforecast, Eρn+1

m , using exactly the same numerical scheme as in step 1(above).

4. Perform a pure downstream displacement of the irregularly spaced La-grangian parcels, i.e., calculate downstream trajectories and repositioneach parcel from Lrn to Lrn+1.

5. Interpolate Eσn+1/2 from the Eulerian grid cells to the positions Lrn+1

resulting in values Lσn+1/2. Then calculate the new parcel densities forboth the ”dry air” and all the tracers, including the effect of divergence:Lρn+1 = Lσn+1/2 · Lρn, and Lρn+1

m = Lσn+1/2 · Lρnm, m = 1, . . . ,M .

6. Interpolate Lρn+1 from the Lagrangian grid to obtain the target val-ues, ET , in the Eulerian grid. In Eulerian grid cells with no nearbyLagrangian parcels ET n+1 is set equal to Eρn+1 from step 1. Detailsconcerning interpolations can be found below.

7. As step 6 but for all tracers m = 1, . . . ,M .

8. Nudge Eρn+1 towards ET n+1 under constraints of mass conservation andshape preservation for the density. Details of the nudging procedureare given below. The result is the final HEL forecast, Eρn+1, for the”dry air” in Eulerian space.

9. As step 8 but for all tracers m = 1, . . . ,M . However, now the con-straints are mass conservation and shape preservation for tracer mixingratios.

2.1. The underlying numerical scheme

As outlined above some numerical scheme must be chosen in order toobtain the provisional forecast in Eulerian grid space. For applications in

12

HEL it would be reasonable to use a semi-Lagrangian type scheme sincetrajectory calculations can be partly re-used for estimation of the downstreamparcel trajectories, also it is then possible to take long steps not subject toclassical CFL conditions for advective processes. Therefore relevant schemesinclude, e.g., flux based multidimensional schemes such as [5] and [29], theDeparture area Cell Integrated Semi-Lagrangian (DCISL) scheme [30], theConservative Semi-LAgrangian Multi-tracer transport scheme (CSLAM) [31],the Semi-Lagrangian Inherently Conserving and Efficient scheme (SLICE)[32], or the Locally Mass Conserving Semi Lagrangian scheme (LMCSL)[28].

Note, however, that any mass conserving scheme for solving continuityequations can in principle be used as underlying scheme for HEL. In fact,relaxing the mass conservation property, any consistent numerical schemecan be used.

In the present paper we have tested the use of first and third order versionsof the CSLAM and LMCSL schemes to obtain the first guess forecast inEulerian space.

2.2. Estimation of trajectories

The downstream displacement of parcel locations obviously is an essen-tial component in HEL. In simple numerical tests such trajectories can becalculated analytically, or, as in dynamical models, they can be calculatedvia an iterative procedure, which is equivalent to that used in traditionalsemi-Lagrangian models for estimating the upstream departure points/cells.In the present applications we have generally used analytical or approximateanalytical trajectories for idealized numerical tests, and iteratively estimatedtrajectories in dynamical model implementations. This is described furtherin Sections 4.1, 4.2, and 5.0.1.

2.3. Update of the parcel volumes

Each Lagrangian parcel represents a certain volume LV of the fluid which,at the initial time step, is simply initialized as the volume represented by thevolume of the relevant Eulerian grid cell.

Once Lρn+1 has been calculated one can update the volume of each parcel.Omitting the upper index ”L” the parcel volume V n+1

p at time step n+ 1 isdetermined diagnostically by the constraint that, in the absence of mixing,the total mass of each Lagrangian parcel is conserved, i.e.,

V n+1p ρn+1

p = V np ρ

np (11)

13

whereby

V n+1p = V n

p

ρnpρn+1p

=V np

σn+1/2p

(12)

using (10).The volume of the parcels is a necessary ingredient for mixing between

neighboring parcels - see below.

2.4. Interpolations from Eulerian to Lagrangian representation

Interpolations between the Eulerian and Lagrangian representations arerequired as part of the HEL scheme. For example, as explained above, itis necessary to interpolate σ to the Lagrangian representation. Similarly,when HEL is used in a dynamical model, tendencies related to other physicalprocesses must be interpolated.

In the present formulation of HEL all interpolations from the Eulerian gridcells to the Lagrangian parcel locations are fourth order Lagrange polynomialinterpolations.

2.5. Target values

Provisional target values, ET n+1, for the Eulerian space ”dry air” densitycan be obtained when Eρn+1 and Lρn+1 have been calculated. The provisionaltarget value in Eulerian grid cell k is composed as a weighted sum of threeestimates:

ET n+1k =

w0Eρn+1k + w1

Eρn+1k + w2Rk

w0 + w1 + w2

(13)

where Eρn+1k is the provisional Eulerian based forecast, and Eρn+1

k is a spatiallysmoothed estimate of Eρn+1

k , which is determined as an average of linearlyinterpolated values in two points both lying on a line passing through theEulerian grid cell centroid k and parallel to the instantaneous asymptoticdilatation axis (see Section 3 on mixing for details). The locations of the twopoints is explained in Figure 2.

14

In (13) Rk is defined as

Rk =1

Wk

P∑p=1

wp,kEρn+1p,k (14)

with Wk =P∑p=1

wp,k

where wp,k is a simple bi-linear interpolation weight given to an estimate ofthe density Eρn+1

p,k in Eulerian grid cell k which is based on the density at thelocation of parcel number p:

Eρn+1p,k = Lρn+1

p + (Lrn+1p − rk) · gp,k . (15)

gp,k is a second order numerical approximation to the gradient ∇ (Eρ)n+1

atthe location 0.5(Lrn+1

p + rk) and rk is the position vector of the k’th Eulerian

grid cell. Formally, if Wk = 0, Rk is simply set equal to Eρn+1k .

A

B

k

Asymptotic (Lagrangian) dilatation axis

Figure 2: Smoothing along the asymptotic dilatation axis: a smoothed estimate in Euleriangrid cell centroid k is obtained as an average of linearly interpolated values in points Aand B. The dashed line shows the direction of the asymptotic dilatation axis

15

The weights w1 and w2 in (13) are determined as follows:

w1 = (1−Hk)Wk

w2 = HkWk (16)

where Hk is a measure of the homogeneity the distribution of Lagrangianparcels around the Eulerian cell k. Here we have used the following estimateof Hk

Hk =2×min[Wl]

max[Wl] + min[Wl]l ∈ Lk (17)

where Lk is a subset of Eulerian cells including k and its nearest eight sur-rounding neighbors (for a regular grid as used here). In (13) w0 is a fixednumber, which depends on the order of accuracy of the underlying Eule-rian based forecast. If this is a low order forecast the accuracy is low, andtherefore the relative importance of the parcel values should be as high aspossible, i.e., w0 must be low. Here we have, rather arbitrarily, used a valueof w0 = 10−5 for a 1st order underlying forecast and w0 = 1 for a 3rd orderunderlying forecast (the only two orders considered). A main motivation forletting w0 be higher in the last case was a suspicion that noise problems re-lated to high frequency modes (gravity and sound waves) could appear whenHEL is used in a dynamical model. Since, in any case, a higher than firstorder underlying scheme is required in a dynamical model it would not be tooharmful for accuracy considerations to increase w0. It is noted, however, thatwe have not experienced problems in the tests carried out with the simpledynamical model in Section 5 even if a small value of w0 is used.

In practice the calculations of Rk and Wk are not performed as sums overall parcels as indicated in (14) since this would be very inefficient numericallybecause it is known that only four of the parcel weights, wp,k, for a givenparcel p are different from zero. Instead Rk and Wk are calculated in asingle loop over all parcels where information is distributed (summed) to theneighboring four Eulerian cells, followed by a second loop where the result isdivided by the sum of weights for each cell.

Provisional target values for the tracers ET n+1m , m = 1, . . . ,M are ob-

tained via the same technique as outlined for the ”dry air”, i.e., ET n+1 above.To obtain an Eulerian space forecast which is shape preserving and com-

patible we must identify minimum and maximum permitted mixing ratios,q− and q+, respectively, for each tracer m in each Eulerian grid cell. The

16

mixing ratio for a tracer m is defined as qm = ρm/ρ, implying that we canalways deduce mixing ratios in both the Eulerian and Lagrangian represen-tation when ρm and ρ are known, and we can always convert a mixing ratioqm back to volume density when ρ is known. If a shape preserving underlyingEulerian based forecast is used for grid cell k the provisional minimum andmaximum values are set equal to the mixing ratio of the provisional Eulerianbased forecast. If the underlying Eulerian based forecast is non-shape pre-serving, upstream mixing ratios at time level n are used instead. In this casewe have used minimum and maximum mixing ratios in the four grid cellsk1, . . . , k4 surrounding the location of the upstream departure point for thetrajectory which at time level n + 1 is located at the cell centroid of cell k,i.e., the departure point in a classical semi-Lagrangian context (see Figure3).

k4

k2k1

k3

n+1n

Figure 3: Traditional upstream semi-Lagrangian departure point.

Thereby the provisional minimum and maximum mixing ratios for tracerm become:

Monotonic Eulerian.

Eq−m,k = Eq+m,k = Eqn+1

m,k (18)

Non-monotonic Eulerian.Eq−m,k = min[Eqnm,ki ]Eq+m,k = max[Eqnm,ki ]

, i = 1, . . . , 4 (19)

The final limits are obtained by using additional information from mixingratios in the Q parcels pi, i = 1, . . . , Q that have Euclidian distance to the

17

Eulerian cell k at time level n+ 1, which is less than 1.5 grid distances:

Eq−m,k = min[Eqm,k,Lqn+1m,pi

]Eq+m,k = max[Eqm,k,

Lqn+1m,pi

], i = 1, . . . , Q (20)

Once the Eulerian space forecast for ”dry” air density Eρ has been ob-tained the maximum and minimum permitted values of volume density fortracer m = 1, . . . ,M can easily be obtained:

Eρ−m,k = Eq−m,k · Eρk (21)Eρ+m,k = Eq+

m,k · Eρk (22)

2.6. Nudging of first guess towards target values

After the (shape preserving) target values have been calculated, the pro-visional, ”first guess”, Eulerian forecast can be corrected. The correction,or nudging, is done in two steps. First we calculate the total mass, M =∑K

k=1Eρk ·EVk, and the total mass of the target values, MT =

∑Kk=1

ETk · EVk,as well as the discrepancy, ∆M = MT −M , between the two. This will leadto three different possibilities:

• ∆M < 0: Mass of target field is too small.

• ∆M = 0: Mass of target field is correct.

• ∆M > 0: Mass of target filed is too large.

In the (extremely unlikely) event that the mass of the target field is exact(∆M = 0), the Eulerian field is simply replaced by the target field. In thetwo remaining possibilities the target field has to be modified to ensure massconservation. If the mass of the target field is less that the actual mass(∆M < 0), we calculate the maximum possible mass of the field, i.e., M+ =∑K

k=1 ρ+k · EVk, where shape preservation is still fulfilled. The target field

and the maximum field can then be combined to produce a final correctedEulerian field, which is both mass conserving and shape preserving.

w+ =∆M

M+ −MT(23)

Eρk = (1− w+) · ET k + w+ · Eρ+k (24)

18

This is always possible as M+ ≥M . The procedure is the same if the mass ofthe target field is too large, then the target field is weighed with the minimummass field, M− =

∑Kk=1 ρ

−k · EVk, to acquire mass conservation.

w− =∆M

MT −M− (25)

Eρk = (1− w−) · ET k + w− · Eρ−k (26)

The procedure is then repeated for all tracers m = 1, . . . ,M .The nudging employed in the current version is global, but since all chem-

istry is calculated in the Lagrangian parcels, it will not introduce errors dueto the inevitable numerical mixing in the Eulerian domain. A local nudgingmethod has also been tested, but it will lead to poorer results, as the nudgingmethods ability to correct the Eulerian values will be significantly lessenedby hard locality constraints. The traditional concerns when using globalmethods, i.e., mass redistribution and unphysical mixing, is fully controlledby the correct values being preserved in the Lagrangian parcels.

3. Mixing between parcels

As discussed in Section 1 densities/mixing ratios or other invariants willgenerally develop into thin filaments as part of the cascade into smaller andsmaller scales in non-linear geophysical flows. At some point a model atgiven resolution can not represent the spatial scale of the filaments, andexplicit horizontal diffusion may therefore be required to prevent spectralblocking. Considered in discrete Lagrangian space, i.e., a model, the analogueto spectral blocking, is realised as a gradual development into unrealisticallylarge differences between densities/mixing ratios in neighboring parcels.

An example of this is presented below in Section 5. Therefore, due to theirnon-dissipative nature, explicit mixing must be introduced in Lagrangianmodels. We introduce a directionally biased mixing as an a posteriori opera-tion applied each time step after the generic HEL forecast, described above,has been obtained.

In the present paper we only consider two-dimensional flow. In this casethe degree of mixing between neighboring parcels is based on a modifiedinstantaneous and local two-dimensional rate of deformation:

19

Dn = max

[0,

√(∂vn∂x

+∂un

∂y

)2+(∂un∂x− ∂vn

∂y

)2

−∣∣∣∣∂un∂x +

∂vn

∂y

∣∣∣∣ ] (27)

where un and vn, respectively, are the velocity components in the two direc-tions spanned by coordinates x and y at time step n. It can be seen that theeffect of divergence (last term) is subtracted from the traditional expressionfor deformation rate. This is done because the mixing we want to introduceshould not lead to excessive damping when the scheme is applied to the fullmass field in a dynamical model. Not introducing such a modification wouldtend to damp dynamically important gravity waves.

The calculation of D is estimated in Eulerian space via centered, i.e.,second order accurate, differences. The expression in (27) is only valid inCartesian geometry. Therefore, in the applications on the sphere presentedbelow, metric factors have been applied.

In addition to the prognostic variables discussed above a set of threeprognostic variables are introduced in Lagrangian space (only) in order toperform the mixing. For each Lagrangian parcel p these include the actualparcel deformation, Lδp, and the two coordinates of a position vector, Lrpa ,of a passive auxiliary Lagrangian parcel, Lpa, that is used to identify theasymptotic dilatation axis for shear, i.e., the direction of ”long term” parcelstretching due to shear considered in a Lagrangian sense, see e.g. [33].

For each parcel p its deformation is initialized to zero at the first timestep: Lδ0

p = 0, and it is updated as part of the mixing procedure describedbelow. The location of the auxiliary parcel pa for main parcel p is initializedone grid-distance, ∆, away from p in an arbitrary direction on the integrationplane. At each time step the auxiliary parcel is translated downstream usingthe same trajectory algorithm as for the main parcels.

The mixing operates as follows:

1. Once the modified deformation rate, Dn, has been determined in Eu-lerian space at a given time step it can be interpolated to the parcelpositions enabling calculation of new provisional parcel deformations

Lδn+1p = Lδnp + ∆tLDnp (28)

20

Time step n

an

an

an+1

an+1True asymptotic

shear dilatation axis

an+1

an+1

Time step n+1 Normalize at time step n+1

True asymptotic

shear dilatation axis True asymptotic

shear dilatation axis

Δ

Δ

Δ

Δ

Pa

P

PP

Pa Pa

Pa

P P

P

Figure 4: Downstream advection of two different Lagrangian parcels and their auxiliaryparcels in a pure shear flow. For each parcel P and Pa denotes, respectively, its positionand the position of the auxiliary parcel. The associated auxiliary vectors are marked blackand gray, respectively, to distinguish between the two parcels. In this example the (true)asymptotic / Lagrangian dilatation axis, indicated with dashed lines, is the same all overthe small domain shown, although its direction rotates clockwise from time step n to timestep n + 1. Note that it is the relative movements between parcels and auxiliary parcelsthat are relevant. After several timesteps the auxiliary vectors will align approximatelywith the true asymptotic dilatation axis

2. Omitting for simplicity the Lagrangian and parcel indices L and p, andthe time index n+1, let ra denote the pure downstream position vectorof the auxiliary parcel for a main parcel, which has downstream positionvector r. The final downstream position vector ra of the auxiliary parcelis then defined as

ra = r + ∆ra − r

‖ra − r‖ (29)

i.e., the distance between a parcel and its auxiliary parcel is simply nor-malized to one grid distance. Two examples of the identification of thea vector are shown in Figure 4. No formal proof is given here that thevectors ra will actually converge toward the Lagrangian shear dilatationaxes. Since the additional parcels a are initialized randomly a certaintime will elapse from the model initial state until realistic dilatationaxes are identified for each parcel. However this time is proportional tothe deformation rate of the flow, and therefore, when the deformationrate is large, the dilatation axes is also quickly approached. Obviouslyif the flow is linear, i.e., no mixing is needed, the dilatation axes of themodel parcels maintain their initial random orientations.

21

Asymptotic (Lagrangian) dilatation axis

Δ

Δ

Δt ⋅D ⋅ Δ

Figure 5: This figure describes why we use a factor of 0.25 between the deformationwithin one time step ∆t · D and the fraction of the parcel area ∆2 that should be mixed.The underlying assumption is that the deformation is mainly due to shear, and the figureshows how cells are then deformed within one time step from regular squares into irregularparallelograms with the same area (assuming zero divergence). In order to re-shape a cellinto squares the mass in the two shaded triangles must be given off via interpolationsto the neighboring cells. The factor comes because the sum of the triangular areas is0.25∆t · D∆2.

3. For each parcel, p, let µp,k represent a fraction of the parcel volume,LVp, which is assigned to a neighboring Eulerian grid cell centroid k.µp,k is defined as:

µp,k = 0.25 min[1, Lδn+1

p

]exp(−κd2

p,k) (30)

where dp,k is the distance in units of grid distances from the line parallelto Lan+1

p , which passes through parcel p. The constant κ determinesthe degree of directional bias for the mixing. In the present work thisconstant has been set to 15. For all Eulerian points k with distancesto p larger than ∆, µp,k is set to zero. I.e., in a regular grid µp,k is onlydifferent from zero for a maximum of four individual values of k. It isensured that the sum of these four weights do not exceed unity. Thefactor is obtained from geometrical considerations of the relationshipbetween deformation rate and the fraction of the parcel volume whichshould be mixed with neighboring parcels - see Figure 5.

4. Once µp,k is calculated a total mass contribution, µp,kLρm,pVp, for each

tracer m is transferred from the Lagrangian parcel p to Eulerian gridcell k. In other words the average density of mass contributions fromall parcels ”neighboring” k is

ρm,k =

∑p µp,k

Lρm,pVp∑p µp,kVp

, (31)

22

where it is noted that µp,k represents elements in a sparse matrix withnon-zero contributions from only none or a few parcels.

5. Now the mixing can be realized by transferring the mixed densities inthe Eulerian cells back to the Lagrangian parcels. For parcel p the finalmixed density Lρm,p becomes:

Lρm,p = (1− µp,k) u,Lρm,p + µp,kρm,k (32)

where we have formally used the notation u,Lρm,p to indicate the un-mixed forecasted density in parcel p resulting from the generic HELrecipe.Note, that not only tracers are mixed using (32). To ensure full con-sistence between prognostic variables also the ”dry air” is mixed.It can easily be shown that the total parcel mass for each tracer and forthe complete ”dry parcel mass of the atmosphere” are conserved whenapplying (32).

6. Based on the amount of actual mixing, Mp, that has taken place forparcel p via the above operations the final parcel deformation is calcu-lated. The actual mixing, not including trivial mixing of parcels withthemselves, is

Mp =∑k

µp,k

∑p′ wp′,kVp′ − µp,kVp∑

p′ wp′,kVp′(33)

where it is again noted that for regular grids the sum over k onlyincludes four grid cells surrounding parcel p. The parcel deformationis finally reduced according to the degree of mixing that has actuallytaken place:

Lδn+1p = Lδn+1

p (1− 4Mp) (34)

where the factor of ”4” is a constant determining how much Lδp isreduced per unit change in relative area. This factor is obtained fromthe same geometrical considerations mentioned above (see Figure 5)of the relationship between deformation rate and the fraction of theparcel volume which is actually mixed with neighboring parcels. Notethat the factor of 0.25 in (30) ensures that Lδn+1

p cannot be less thanzero.

23

Most computational operations in the above list are common for all tracersand therefore, in multi-tracer applications, the total number of operations islimited.

4. Passive tracer numerical simulations on a cubed sphere

To validate HEL we perform inert passive tracer transport on the spheredriven by both solid body rotation and two types of deformational flow.

For the passive transport tests presented here the underlying Eulerianbased scheme required in HEL is a first order (i.e., shape preserving bydefinition) version of CSLAM (Conservative Semi-LAgrangian Multi-tracertransport scheme) [31]. Where necessary this is referred to as CSLAM-1st.The performance of HEL is compared to that of a third order accurate versionof CSLAM in combination with a simple shape preserving filter [31], referredto as CSLAM-M.

In the tests shown here both HEL and CSLAM have been implementedon a so-called cubed sphere grid - see [31] for details.

4.1. Solid body rotation

In standard solid body rotation tests on the sphere a certain spatial dis-tribution is revoluted to its original position after one or more rotations. Inthis case HEL should perform with high accuracy since the parcels end up inexactly the same grid cell centroids they were initialized in, and no mixingtakes place because the deformation rate of this flow is zero. Thereby thetarget values will be almost exactly equal to their initial value, except forthe weight factor w0 (see Equation (14)) which is small when a first orderunderlying scheme is used. This again implies that the final HEL forecastshould be almost exactly equal to the corresponding analytic solution sinceglobal nudging is used.

The test example we present here is the solid body advection of a cosinebell with radius Rc = R/3, where R is the radius of the Earth. The angleof rotation is π/4 relative to the Earth rotation axis, i.e., the bell passesover the edges of the cubed sphere. One full revolution is completed in 576time-steps of 1800s each. These settings are identical to those in [34].

To validate the results obtained with this simple setup we use the tradi-tional l2 and l∞ error norms defined in, e.g., [31]. Results in terms of l2 andl∞ after one revolution are plotted in Figure 6. As expected the HEL is veryaccurate due to our low choise of the w0 weight. The l2 and l∞ convergence

24

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

24 48 96 192 384

l2 e

rro

r

ncube

Solid body rotation of cosine hill on cubed sphere

2nd/3rd OrderHEL (CSLAM-1st)

HEL (CSLAM-1st, w0=1)HEL (CSLAM-M, w0=1)

CSLAM-M

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

24 48 96 192 384

l2 e

rro

r

ncube

Solid body rotation of cosine hill on cubed sphere

2nd/3rd OrderHEL (CSLAM-1st)

HEL (CSLAM-1st, w0=1)HEL (CSLAM-M, w0=1)

CSLAM-M

Figure 6: Test of convergence for linear advection on cubed sphere using error norms l2(left) and l∞ (right). ncube refers to the number of grid cells in each direction on each ofthe 6 faces of the cubed sphere, i.e., ncube = 24 corresponds to an equatorial angular griddistance of 2π/(4× 24) = 0.065 radians.

rates for CSLAM-M are 2.82 and 2.04, respectively, while they are 2.24 and2.16 for HEL. It is noted that the convergence rate for HEL increases signif-icantly if a fixed Courant number is used in such tests (not shown) becausethis implies that, as the time step decreases, the weight per time unit on theparcels increases relative to that on the underlying Eulerian based forecast.While the accuracy increases the shorter the time step in HEL the oppositeis generally the case in semi-Lagrangian models such as CSLAM because thenumber remappings increases when the time step is reduced.

Since, in the dynamical model test below, we use a non-shape preservingthird order accurate semi-Lagrangian remapping as our underlying Eulerianbased forecast we have calculated the convergence rate for a setup whereCSLAM at third order resolution is used as underlying scheme in the solidbody rotational test. In this case we use w0 = 1. The results are also shownin Figure 6 and it can be seen that HEL is still more accurate than CSLAM,although not as exceptional. The l2 and l∞ convergence rates in this case are2.19 and 2.18, respectively.

The temporal growth of error in HEL and in CSLAM-M are quite dif-ferent: running over several revolutions the error norms continue to grow inCSLAM-M, although slowly, while in HEL the error norms do not grow withtime, as one should also expect from the way HEL is designed.

25

4.2. Deformational flow testsFor the deformation flow tests we have used the two types of analytic flow

fields, the density shapes, and the validation diagnostics suggested in [35].For the diagnostics this means that, in addition to the l2 and l∞ error normsand related convergence rates used for the simple solid body rotation testsabove, we have calculated an additional set of diagnostics, briefly describedbelow.

The two analytical flow fields used have originally been proposed by [36],and they include a non-divergent as well as a divergent flow. In both casesthe Lagrangian parcels follow relatively complex trajectories, and the flowis composed of a non-linear deformation component, which is different inthe two cases, and an overlaid translational flow. The translational part isdesigned to perform exactly one rotation around the sphere (along equator)during the entire simulation. After a half complete period of simulationthe non-linear flow component goes to zero and this part of the flow is thenreversed so that the final exact solution equals the initial condition. Half waythrough the simulation, at the time when the non-linear flow component goesto zero and starts to reverse, the initial distributions are deformed into thinfilaments, particularly for the non-divergent flow.

Three initial, i.e., t = 0, distributions consisting of two isolated Gaussianhills, two slotted cylinders, and two cosine hills are shown in Figure 7. Detailsfor these distributions are described in [35].

As an example Figure 8 shows the result of simulations at times t = T/2,i.e., the most deformed time, and at t = T , i.e., the final time, using theslotted cylinder initial condition and the non-divergent deformation flow.The maximum Courant number is 5.5, and the equatorial resolution 1.5 inthese simulations. At this resolution the final distribution at time t = T HELwith mixing is considerable closer to the analytic solution than CSLAM-M.At the time t = T/2 HEL appears rather smooth as compared to CSLAM-M. This is because the density of Lagrangian parcels at this time is quiteinhomogeneous, and therefore the parameter Hk in (17) is considerably lessthan one in a number of grid cells. This results in a relatively large smoothing(in Eulerian space only) along the asymptotic dilatation axis since the rawparcel based estimate Rk of the Eulerian value is given a small weight.

For illustrative purposes Figure 8 also includes the result of a simulationwithout any parcel mixing, and where the parameterHk deliberately has beenset to unity. In this case, as can be seen, the distribution at time t = T/2becomes unrealistic since the initial distribution has been cut into small parts

26

Figure 7: Initial (and final analytic) distributions for deformational test cases: Gaussianhills (upper left panel), slotted cylinders (upper right panel), and cosine hills (lower panel).

(represented by the individual parcels). However, as expected since time isreversed, the final distribution in this aliased model setup is very close to theanalytic solution. It is only small errors in the parcel trajectories, and thefact that w0 is different from zero, that prevents the final field from beingequal to the analytical solution.

Using the non-divergent flow, basic error norms l2 and l∞ for CSLAM-M and HEL, and two maximum Courant numbers (1.0 and 5.5), are listedin Tables 1, 2, and 3 for each of the three initial distributions. In generalone can conclude that HEL is very accurate at low resolution for the moresmooth distributions, i.e., the Gaussian and cosine hills, whereas CSLAM-M and HEL are comparable at the high Courant number and at the twofinest resolutions. Another feature is that, as compared to CSLAM-M, HELis less sensitive to the maximum Courant number. This is because HEL isinfluenced much less by the number of semi-Lagrangian re-mappings neededto finalize each simulation than CSLAM-M.

The convergence rates for each of the three initial distributions and for thetwo Courant numbers are listed in Table 4. In general CSLAM-M convergesfaster than HEL for the smooth distributions while the difference is small forthe rough slotted cylinder distribution where the convergence rates in anycase are low.

27

Figure 8: Simulated distributions at times t = T/2 (left panels) and t = T (right panels),based on the slotted cylinder initial distribution. From top to bottom the plots showresults obtained with CSLAM-M, HEL and HEL without any parcel mixing, respectively,all run with a maximum Courant number of 5.5 and an equatorial resolution of 1.5.

28

Table 1: Statistics for the Gaussian hill problem. The columns show maximum Courantnumber (C), Equatorial resolution in degrees (∆λ), and the error norms l2 and l∞, re-spectively, for both CSLAM-M and HEL

CSLAM-M HELC λ l2 l∞ l2 l∞

1.0

3.000 2.422E-01 3.434E-01 4.593E-02 6.934E-021.500 7.606E-02 1.576E-01 1.397E-02 3.284E-020.750 1.376E-02 5.475E-02 4.499E-03 1.431E-020.375 2.592E-03 1.850E-02 1.661E-03 6.989E-03

5.5


Table 2: As Table 1, but for the slotted cylinder problem.


1.0


5.5


Table 3: As Table 1, but for the cosine hill problem.


1.0


5.5


29

Table 4: l2 and l∞ convergence rates calculated from the error norms listed in Tables 1,2, and 3. The second column gives the initial spatial distribution with ”GH” indicatingGaussian hill, ”SL” slotted cylinder, and ”CH” cosine hill, while the convergence rates arelisted in columns three through six.

Scheme Initial distr. l2, C = 1.0 l∞, C = 1.0 l2, C = 5.5 l∞, C = 5.5CSLAM-M GH 2.21 1.42 2.53 1.52HEL GH 1.60 1.11 1.75 1.24CSLAM-M SL 0.46 0.01 0.47 0.01HEL SL 0.52 0.01 0.55 0.04CSLAM-M CH 1.68 1.14 2.44 1.56HEL CH 1.77 1.55 1.82 1.60

4.2.1. ”Minimal” resolution

Numerical schemes may be constructed to converge fast - at least forsmooth distributions. However, as pointed out by [35] increases in resolutionare often computationally expensive and therefore it is of interest to identifysome kind of measure of absolute accuracy for a numerical scheme. A diag-nostic designed for this is the ”minimal” resolution needed to obtain a certainaccuracy for a specific problem. Following the specifications in [35] we havecalculated the ”minimal” resolution as the resolution required to obtain anl2 error norm for the cosine bell distribution that is less than 0.033. In thespecifications ([35]) the result should be obtained for an unlimited scheme,i.e., no shape preserving limiter should be applied, e.g. on CSLAM, in thiscase. ”Minimal” resolutions for CSLAM and HEL are listed in Table 5.

The ”minimal” resolution for HEL is coarser than that for CSLAM, par-ticularly for a maximum Courant number of 1. We therefore conclude thatthe effective resolution for HEL is higher than for CSLAM.

The ”minimal” resolution for HEL is determined by the strength of themixing between parcels. If HEL is run without any parcel mixing the ”min-imal” resolution goes to infinity in the sense that the numerical solutionbecomes almost exactly equal to the analytic solution at any resolution, ofcourse depending on the weight w0.

4.2.2. Filament diagnostics

The filament preservation diagnostic, lf , describes the transport scheme’sability to preserve thin filaments or gradients in the concentrations. lf is

30

Table 5: ”Minimal” resolution required to obtain an l2 error norm less than 0.033 for thecosine hill problem in the non-divergent deformation flow. The columns include schemeand Courant number

Scheme C = 1.0 C = 5.5CSLAM 0.8 1.5

HEL 1.9 1.6

defined as:

lf =

100.0 · A(τ,t)

A(τ,t=0), if A(τ, t = 0) 6= 0

0.0 , else(35)

Where A(τ, t), the control volume, is a spherical area where the concentrationis equal to or larger than a given threshold value τ . The control volumesshould, without overlapping, span the entire domain. The test setup is thecosine bells initial condition in non-divergent flow where lf is calculated att = T/2 for 19 values of τ in the interval τ = (0.10...0.95). The values of lfare expected to increase for low values of τ and decrease for high values ofτ due to numerical diffusion. At τ = 0.1 the value of lf should be 100, sincethe area with the background concentration should not be increased duringthe simulation.

The lf values for CSLAM-M and HEL are presented in Figure 9 for simu-lations with maximum Courant numbers 1 and 5.5. As expected CSLAM-Mis more diffusive, i.e., it maintains filaments less well, at maximum Courantnumber 1 as opposed to 5.5. This is because more re-mappings are requiredat the lower Courant number. The HEL values are calculated for both Eule-rian and parcel, i.e., Lagrangian, representations. It can be concluded fromFigure 9 that at time t = T/2 of the simulation the Eulerian representationof HEL is generally more diffusive than CSLAM-M for the high maximumCourant number although the highest functional values are maintained to ahigher degree than for CSLAM-M. This is because a relatively large weight,w1, in (13) is given to the spatially smoothed provisional first order Eule-rian forecast, Eρn+1, due to the highly inhomogeneously spaced parcels atthis time. The Eulerian HEL representation does not change significantlywith Courant number and is generally better at preserving the maximumvalues than the CSLAM-M. Therefore the low Courant number CSLAM-Mis more diffusive than the corresponding Eulerian HEL representation. TheLagrangian lf -values are generally much closer to 100 than the correspond-ing CSLAM-M and Eulerian HEL representations, and there is almost no

31

dependency on the Courant number. The last observation is completely asexpected since the parcel mixing takes place at the time interval in the twocases. It is noted (not shown) that the lf -values are quite insensitive to themixing frequency between Lagrangian parcels. In a more general applica-tion, using non-analytic trajectories, there could in theory be some spatialoverlapping between parcels. However, such overlaps cannot be quantifieddue to deformation of the Lagrangian parcels into filaments, and HEL doesnot include information about their exact shape. The total area, however, isconserved.

It is important to note that without parcel mixing the Lagrangian lf -values would all be exactly 100. As argued above, introduction of mixing isfundamental in all Lagrangian models in order to avoid long term unphysicalaccumulation of energy at the smallest resolved scales. The same appliesto Eulerian based models (in principle, including semi-Lagrangian models)where the inherent numerical mixing is ”too weak” to properly representnon-linear scale interactions and prevent spectral blocking for a given modelresolution. This is typically the case in e.g. spectral or pseudo-spectralmodels. So, although we know that explicit mixing must be introduced insome un-diffusive models for purely physical reasons we do not know exactlyhow much mixing is required. I.e., the optimal lf -values are, unfortunately,also unknown. The fundamental idea behind the parcel mixing applied herehas been to base it on simple geometric considerations and thereby obtain asimple first principle guess on the required amount of actual physical mixing.

4.2.3. Pre-existing functional relations and mixing

To evaluate the mixing properties discussed in Section 1.2 the statisticsproposed in [9, 35] have been calculated for initial conditions consisting of twotracers: cosine bells – corresponding to χ in Figure (1) – and correspondingnon-linearly related bells – corresponding to ξ. The flow is the same non-divergent deformation flow as above.

These mixing statistics include real mixing, lr, range preserving unmixing,lu, and overshooting, lo. The more precise definitions of lr, lu, and lo areprovided in Appendix B.

The error norm, lo, should always be zero indicating that the scheme inquestion is shape preserving. However, the second norm, lu, which ideallyshould be zero as well, will generally not be zero, unless the scheme is semi-linear and monotone [10]. This was one of the motivational factors for the

32

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0τ

0

20

40

60

80

100

120

140

l f

Filament preservation, 1.5

CSLAM-CN1.0

CSLAM-CN5.5

HEL-E-CN1.0

HEL-E-CN5.5

HEL-CN1.0

HEL-CN5.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0τ

0

20

40

60

80

100

120

140

l f

Filament preservation, 0.75

CSLAM-CN1.0

CSLAM-CN5.5

HEL-E-CN1.0

HEL-E-CN5.5

HEL-CN1.0

HEL-CN5.5

Figure 9: Diagnostics for filament preservation, lf , for the two resolutions 1.5 (left) and0.75 (right). The curves show the shape preserving CSLAM-M (crosses), HEL in Euleriangrid (squares), and the HEL in Lagrangian space (circles). Each scheme is shown with amaximum Courant number of 1.0 (dotted line) and 5.5 (solid line), respectively.

development of HEL. The first norm, lr, should be a non-zero value, since”real” mixing is always present, it should however, as described in Section 3,not be artificial numerical mixing but physically based mixing.

The mixing diagnostics for CSLAM-M and HEL are listed in Table 6.Mixing diagnostics are important indicators for the influence of transportschemes on chemical reaction rates and equilibria. Since chemistry calcu-lations (not dealt with here) are performed in Lagrangian space only theparcel values of lr, lu, and lo are listed for HEL. It can be seen from Table6 that HEL behaves according to its construction: only real mixing and norange preserving un-mixing or overshooting takes place. In CSLAM-M aweak range-preserving un-mixing can be seen, and the general level of realmixing is larger than for HEL, particularly at low resolution and low Courantnumber.

4.3. Non-linear passive advection with divergence

Traditional l2 and l∞ error norms for the strongly divergent flow arepresented in Table 7 for the cosine hill initial condition. In this case thetwo maximum Courant numbers tested are 0.6 and 3.2, respectively, and itcan be seen that HEL is now considerably more accurate than CSLAM-M,particularly for the small Courant number tested. As for the non-divergentcase HEL is relatively insensitive to the Courant number.

33

Table 6: Mixing diagnostics lr, lu, and lo for CSLAM-M and HEL for two equatorialresolutions (1.5 and 0.75) and two maximum Courant numbers (1 and 5.5). The columnsshow scheme, maximum Courant number (C), Equatorial resolution in degrees (∆λ), andthe three error norms, respectively

Scheme C ∆λ lr lu loCSLAM-M 1.0 1.50 2.18E10-3 2.73E10-5 0.0CSLAM-M 5.5 1.50 6.28E10-4 6.73E10-5 0.0CSLAM-M 1.0 0.75 3.49E10-4 1.25E10-4 0.0CSLAM-M 5.5 0.75 1.05E10-4 2.57E10-5 0.0HEL 1.0 1.50 2.63E10-4 0.0 0.0HEL 5.5 1.50 2.63E10-4 0.0 0.0HEL 1.0 0.75 6.75E10-5 0.0 0.0HEL 5.5 0.75 6.75E10-5 0.0 0.0

Table 7: As Table 1, but for the cosine hill problem in strongly divergent flow. Note thatthe maximum Courant numbers C in this case are different from those of Table 1.


0.63.00 3.184E-01 4.400E-01 5.253E-02 7.607E-021.50 9.755E-02 2.048E-01 1.580E-02 2.638E-020.75 2.346E-02 7.629E-02 4.711E-03 1.034E-02

3.23.00 1.942E-01 3.034E-01 5.621E-02 8.342E-021.50 4.220E-02 1.132E-01 1.661E-02 2.578E-020.75 8.351E-03 3.965E-02 4.848E-03 1.293E-02

34

5. Implementation and tests in a shallow water dynamical model

In this section we demonstrate one application of HEL namely as trans-port scheme also for the dynamical core of a geophysical type model. Our testmodel is a simple shallow water model in Cartesian geometry. The governingdifferential equations are

du

dt= fv − g∂(h+ hs)

∂x(36)

dv

dt= −fu− g∂(h+ hs)

∂y(37)

dh

dt= −h∇ ·V +Dh + Fh (38)

dhmdt

= −hm∇ ·V + Cm + Sm +Dm ,m = 1, . . . ,M (39)

where u, v are the flow speed components in the x – y plane, f is the Coriolisparameter, g the gravitational acceleration , h is the geopotential thickness ofthe flow, and hs the stationary surface geopotential height (”topography”).hm is a quantity we can think of as the contribution to geopotential heightfrom the m’th tracer. The mixing ratio for the m’th tracer can be evaluatedfrom

qm = hm/h , (40)

i.e., formally the volume density for this tracer is ρm = ρhm/h, where ρis the density of the ”dry” fluid in the shallow water model. It has beenassumed that

∑hm << h since, otherwise, the expression in (40), which for-

mally represents specific density would not approximate mixing ratio. Theterm Fh represents a weak globally mass conserving Newtonian relaxation to-wards the initial ”zonal” average profile of h. Fh mimics the effect of diabaticprocesses. Finally the C, D, and S terms represents possible chemistry, diffu-sion/mixing, and possible source and sink terms (i.e., emissions, depositions,sedimentations), respectively.

Although u, v, and h do not depend on the hm values of the tracers themodel is formally set up as ”online coupled” (see, e.g., [37]), i.e., all equationsare solved each time step.

The integration domain covers a domain defined by x ∈ [xmin, xmax] andy ∈ [ymin, ymax], with xmin = 0 km, xmax = 20000 km, ymin = −10000 km, andymax = 10000 km. The boundary conditions are periodic in both directionsand with enforced symmetry around the line y = 0 (”Equator”) for the

35

variables u, h, hs, Fh, and hm,m = 1, . . . ,M , and anti-symmetry around thesame line for v. Also, the Coriolis parameter

f = 2Ω sin

(π

y

ymax − ymin

)is anti-symmetric around y = 0 (Ω is the angular velocity of the Earth rota-tion).

As for the inert tracer applications tested above the strategy followed forapplying HEL in the shallow water model is to use parcel densities/geopotentialthicknesses to modify an existing solution in Eulerian space. As underlyingsolution we have used a locally mass conserving, semi-Lagrangian transportscheme LMCSL (i.e., not the CSLAM as above) with a semi-implicit treat-ment of the gravity - inertial wave terms, in combination with an ArakawaC-grid staggering; see [28] for details.

In the present implementation we have only applied the HEL techniqueon the mass fields (h and the hm’s) of the model. The wind field forecast isbased on the same third order semi-Lagrangian scheme as in [28].

As for the case of passive/inert advection the divergent expansion/contractionfactors, σn+1/2, that are needed to include the effects of divergence in La-grangian space are first calculated in Eulerian space:

Eσn+1/2 =Ehn+1

E,advhn+1, (41)

where Eh is the complete provisional Eulerian space forecast including semi-implicit adjustments, and E,advh is the corresponding purely advective semi-Lagrangian forecast, i.e., also an Eulerian space forecast. The Eulerian spacevalues Eσn+1/2 are now interpolated to each parcel location (see Section 2.4)at time level n + 1 and subsequently multiplied on the parcel values of Lhand Lhm, m = 1, . . . ,M . Mixing depending on the flow deformation rate isthen performed in Lagrangian space as described in Section 3

The final Eulerian space forecasts of Eh and Ehm, m = 1, . . . ,M are ob-tained via exactly the same nudging procedure as described above in Section2.6

For the inert and passive advection tests with prescribed analytical ve-locities in Section 4 the underlying Eulerian space forecasts were all basedon a first order numerical scheme providing good numerical efficiency. Herein our dynamical model implementation we have found that it is necessary

36

to keep third order accurate remappings in the semi-Lagrangian scheme inorder to obtain sufficiently accurate estimates of pressure gradient terms andto ensure sufficiently accurate coupling between the momentum equations(36),(37), and the continuity equation (38).

Although we have used third order re-mappings it is of course still possibleto run with first order remappings for all the tracers, m = 1, . . . ,M , i.e.,when solving (39). In this way one can retain the same high multi-tracerefficiency as in the transport tests in Section 4. One may argue, however,that this violates the mass-wind consistency property, i.e., the 7th of thedesired properties listed in Section 1. It is noted, however, that since thebulk of the model memory in HEL is kept in Lagrangian space, this is nowonly a theoretical problem: If an inert and passive tracer m is initializedas hm = h, the Lagrangian space density of this variable will continue tobe exactly equal to that of h, i.e., at any time step n we have Lhnm = Lhn,independent on the order of accuracy of the underlying Eulerian space basedforecast of Ehm.

5.0.1. Estimation of trajectories

For the passive advection tests in Section 4 all trajectories were calculatedanalytically. In the shallow water model, as well as in any update to a threedimensional model, it is necessary to estimate trajectories. If the provisionalforecast in the Eulerian representation is obtained with a semi-Lagrangianmodel, as here, upstream trajectories are needed, i.e., one has to identify alldeparture points at time n∆t for trajectories which at time (n + 1)∆t endsup in each Eulerian grid cell centroid. In HEL it is furthermore requiredto calculate downstream arrival points at time (n + 1)∆t for trajectoriesbeginning at the irregularly spaced locations for each Lagrangian parcel attime n∆t.

For the upstream trajectories we use the approach described in [28]. Thisis a conventional iterative procedure using two iterations. However, here weuse third order accurate bi-cubic Lagrange interpolations of the upstreamvelocities at time level n as opposed to the first order interpolations in [28].As explained in [28], in order to ensure satisfactory behavior of the LMCSLscheme, it necessary to include the effect of accelerations in the estimate ofthe trajectories.

For the estimation of downstream parcel trajectories a procedure equiva-lent to that in [28] has been followed. This means that each parcel trajectoryis split into two segments, C1 and C2, so that the new parcel position vector,

37

rn+1p , becomes

rn+1p = rnp + C1 + C2. (42)

The C1 segment is based on velocities at time level n, and it approximatesthe forward trajectory from the departure point rnp to the trajectory midpoint

rn+1/2p :

C1 ≈(

∆t

2

)Vnp +

1

2

(∆t

2

)2

Anp (43)

where Vnp is the velocity interpolated to the parcel location rnp at time level n,

and the last term on the right hand side represents the trajectory contributiondue to the acceleration An

p also at the time space location rnp , assuming astationary velocity field in the time period from n∆t to (n+ 1/2)∆t. I.e.,

Anp = (V · ∇V)np , (44)

which is estimated via centered differences and subsequent interpolation tornp .

The second trajectory segment, C2, is based on provisional velocitiesextrapolated linearly in time from time level n− 1 and n to time level n+ 1:

Vn+1 = 2Vn −Vn−1, (45)

where (˜)n+1 indicates a quantity that has been obtained via temporal ex-trapolation. C2 approximates the forward trajectory from the midpointrn+1/2p to the arrival parcel location rn+1

p , however, obtained as minus thebackward trajectory from the arrival location:

C2 ≈ −((−∆t

2

)Vn+1p +

1

2

(−∆t

2

)2

An+1p

), (46)

where the notation ( )n+1p indicate terms that have been interpolated to the,

initially unknown, location rn+1p .

An iterative procedure, including two iterations, is used to obtain the finalestimate of rn+1

p . However, since rn+1p is unknown initially a pre-iteration is

performed to obtain a first guess of rn+1p . In the pre-iteration C2 is zero and

C1 is obtained as in (43) but with ∆t/2 replaced by ∆t.All spatial interpolations involved in the estimation of the trajectories are

third order Lagrange polynomial interpolations.

38

5.0.2. Shallow water model results

The initial state of the shallow water model is chosen rather arbitrarilyas wavy structure in the geopotential surface height field, h, shown in Figure10, with the velocity field (not shown) simply initialized to be in geostrophicbalance with this mass field. Figure 10 also shows the bottom topography,hs, consisting of a ”sharp” isolated ”mountain”, and an initial field of mix-ing ratio for a single tracer field, which here simply is a step function in abackground of zero mixing ratio.

The results obtained with HEL are compared to those obtained withthe third order LMCSL scheme [28] without introduction of any shape pre-serving filters. All simulations presented below have been performed with ahorizontal resolution of 128 × 128 points/cells. The time step was one hour,which gives a maximum Courant number slightly below one. It is informedthat LMCSL and HEL (both with and without parcel mixing) can be runstably with maximum Courant numbers up to about 3, and thereafter nu-merical mountain wave resonance, [38, 39], becomes visibly destabilizing. Nooff-centering or other techniques to control mountain wave resonance wereintroduced.

Figure 11 shows the surface height field and the mixing ratio field after 48hours of simulation for the initial conditions plotted in Figure 10. It can beseen that the geopotential height fields for each the two forecasts are almostindistinguishable, although the HEL result is based (mostly) on densities atthe locations of the irregularly spaced Lagrangian parcels. The mixing ratiofields are also similar although HEL via its design is shape preserving. It isnoted (not shown) that the LMCSL results are very similar to those obtainedwith a traditional semi-Lagrangian semi-implicit scheme based on third orderLagrange remappings.

Corresponding height and mixing ratios after 20 days of simulations areshown in Figure 12. It can be seen that HEL and LMCSL continue to producevery similar results both for the height field and the tracer mixing ratiodespite the underlying non-linear model equations (non-linear chaotic errorgrowth does not become visible until around day 30-40).

In the lower panel of Figure 12 we have also - for illustrative purposes- plotted the height and mixing ratio field obtained in an additional HEL-simulation where the parcel mixing was switched off. Surprisingly the heightfield continues to be smooth and very similar to that of LMCSL and ofthe parcel mixed version of HEL, and the unmixed HEL continues to be

39

Figure 10: Initial height field h (upper left) and surface topography (upper right) for theshallow water model. The lower panel shows an example of an initial field of a single inerttracer. Only the ”Northern hemisphere” part of the fields are plotted.

numerically stable. However, the mixing ratio for the passive inert tracer isnow highly unrealistic demonstrating the importance of the mixing we haveintroduced. In this unmixed version the initial step function type mixingratio has been ”cut into bits and pieces” determined by the location of theindividual Lagrangian parcels.

By introducing a passive tracer with initial density equal to that of h,i.e., the mixing ratio is equal to one, we have tested to what extend theHEL is ”wind-mass” consistent, and it has been found that this is indeed thecase since the density field (not shown) continued to be almost completelyidentical to that of h even in long simulations. It is noted, that for this testit was necessary to switch off the Newtonian cooling Fh since otherwise onefield would have been forced while the other was not.

5.1. Pre-existing functional relations in the shallow water model

It was shown in Section 4.2.3 that the mixing diagnostics lr, lu, andl0 obtained with HEL in strongly deforming flow were quite acceptable. Wehave performed a corresponding calculation for transport in the shallow waterflow using exactly the same initial functional shapes as in Section 4.2.3, and inFigure 13 these are shown as time series for the non-shape preserving LMCSLscheme and for HEL. As expected LMCSL produce non-zero values of bothrange-preserving unmixing, lu, and of overshooting, l0. For the Lagrangian

40

Figure 11: Results after 48 hours of simulation based on the initial conditions and bottomtopography in Figure 10. Panels to the left show the surface height field, while those to theright show the mixing ratio. The upper panels show the result obtained with the LMCSLscheme, while HEL is shown below.

representation in HEL lu and l0 are both zero via construction, while thesame is the case for l0 in the Eulerian HEL representation. The level of luin the latter is very small as compared to that in LMCSL. The degree ofreal mixing, lr, in LMCSL and in the Eulerian representation of HEL arequite similar, while lr in the Lagrangian HEL representation is about half ofthat. The fact that the overall levels of real mixing in HEL and LMCSL havethe same order of magnitude indicates that the ways we have obtained themixing and Eulerian space smoothing in HEL are not completely unrealistic.

6. Discussion

A number of issues related to the introduction of the new HEL schemedeserve some discussion provided in the following subsections.

6.1. Mixing

As noted previously, the mixing between Lagrangian parcels introducedhere is based on simple geometrical principles minimizing the need for em-pirically based tuning. However, it is of course possible to formulate suchphysically based mixing in other ways, which in practice can lead to a strongeror weaker mixing between parcels.

41

Figure 12: As Figure 11 but after 480 hours of simulation. The lower panels show resultsin a HEL-configuration without any parcel mixing.

10 20 30 40hour

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

mix

ing

erro

r

Mixing diagnostics - shallow water

lr - LMCSL

lu - LMCSL

lo - LMCSL

lr - HEL-Eulerian

lu - HEL-Eulerian

lr - HEL-Lagrangian

Figure 13: Time series of mixing diagnostics lr, lu, and l0 obtained in 48 hour simulationswith the shallow water model using the LMCSL and HEL schemes. For HEL both theEulerian and the Lagrangian representations are plotted. It is noted that in the Eulerianrepresentation of HEL l0 = 0 (not shown), while in the Lagrangian representation bothlu = 0 and l0 = 0 (not shown).

42

One potentially controversial issue is the degree of directional bias of themixing. As described in Section 3 our mixing is biased so that it is dominatedby mixing with neighbors that are aligned along the asymptotic dilatationaxis. This approach is based on the geometrical principle illustrated in Figure5: remapping of the parcels into regular squared shapes filling the integrationdomain only requires mixing along the asymptotic dilation axis. This remap-ping is needed to obtain an un-aliased representation of the Lagrangian parceldensities - a problem that is quite different from that of molecular mixing,which is generally isotropic in nature. We have tested the effect of settingthe parameter κ equal to zero in (30). The result (not shown) is an exces-sive damping and considerably larger error norms for all the inert passivetransport tests reported above. Our actual choice of κ = 15 was based on acompromise: a much smaller value would be too isotropic and damping, anda a much larger value would result in too little realised mixing, i.e. the parceldeformations Lδ would grow to unrealistic values in strongly non-linear flows.

The present paper do not investigate the influence of parcel mixing onthe distribution of energy on different wave numbers. This is the subject ofongoing research.

6.2. Local versus global nudging towards the target values

The fourth desirable property listed in Section 1.1 states that a transportscheme should be transportive and local. While this is fully achieved in theparcel (Lagrangian) representation of HEL the locality property is formallynot fulfilled in the Eulerian representation since the nudging we perform onthe Eulerian based forecast towards the target values is performed globally(see Section 2.6). We have formulated and tested a local nudging which onlyinvolves mass re-organizations between neighboring cells. This locally massconserving version of HEL performs satisfactorily with results (not shown)that are almost comparable or somewhat degraded as compared to the stan-dard HEL version. However, the local nudging is quite expensive from anumerical point of view, and therefore this version has not been investigatedfurther. In practice since the bulk memory in HEL is in Lagrangian spacethe standard version of HEL is highly local as can also be seen directly fromall the plots presented above. In fact, since the limits within which local Eu-lerian values can change due to the nudging are defined from locally definedvalues, the global nudging can be considered a localized process.

43

2 4 6 8 10 12 14 16 18 20Tracers

0

200

400

600

800

1000

1200

1400

Tim

e[s

]

Multi-tracer efficiency

CSLAM-M

HEL

Figure 14: CPU timing on a single processor.

6.3. Computational efficiency

The purpose of the present paper has been to describe HEL, and todemonstrate its accuracy. A careful investigation of the computational costof HEL would require multiple tests on a massively parallel computer system.Here we have only performed single processor CPU tests using an Intel Core2Duo, E6550 @ 2.9 GHz processor, and the Intel Fortran 13.0.0 compilerwith flags: -ipo, -O3, -no-prec-div, -static, and -xHost. The tests revealthat HEL is considerably faster than CSLAM-M for the passive tracer testpresented in Section 4, particularly when many tracers ara considered. Asan example Figure 14 shows the CPU timing required to perform the non-divergent deformation test in Section 4.2 with an equatorial spatial resolutionof 0.75, and a maximum Courant number of 5.5. The number of passive inerttracers was varied from 2 to 20 and it is seen that the multi-tracer efficiencyof of HEL is better than that of CSLAM-M.

A main reason for the faster performance of HEL is that only first orderaccurate re-mappings are needed in the underlying Eulerian forecast for pas-sive transport, while in CSLAM-M third order re-mapping have been used.

Although we have not performed parallel efficiency tests the code has beenprepared somewhat for parallelization. The most important issue relates tothe way individual parcels are transferred from the memory of one CPU/nodeto another. As usual in geophysical fluid dynamics each CPU is reservedfor a certain number of horizontal Eulerian grid cells including a halo zone.Corresponding to this, the actual physical location of each Lagrangian parceldetermines in which part of the memory it is stored. Since the number ofparcels in a given domain can vary significantly due to the divergence of

44

the flow, the actual memory allocation for Lagrangian parcel informationrequired for each CPU must be somewhat higher than that corresponding tothe average parcel density.

Although our first tests suggest that HEL is computationally efficient,particularly for multiple tracers, there is an important memory penalty. Con-sidering, e.g., two dimensional passive transport using a traditional Eulerianbased scheme with K Eulerian grid cells/points, and with M different tracersthe total number of prognostic variables is K ×M . For HEL, however, thecorresponding number is typically K × (M + M + 6) where ’6’ reflects theadditional variables of parcel deformation (Lδ), parcel volume (LV ), compo-nents of the parcel position vector (Lx and Ly) and the position componentsof the auxiliary parcel (Lxa and Lya), respectively.

6.4. Application in a three dimensional model

Arguably the most important issue is how the HEL scheme is extended foruse in three-dimensional applications. At least two fundamentally differentoptions may be investigated:

• using a Lagrangian or quasi-Lagrangian vertical coordinate as in [40]each Lagrangian parcel could stay within the same layer and have aninstantaneous vertical extension equal to that of the layer. The verticalextension of Lagrangian parcels may in fact be used as an additionalprognostic variable and used to determine the layer thickness just asit was demonstrated with the shallow water model in Section 5 (a onelayer model).

• letting parcels float freely between the Eulerian vertical levels as ine.g. ATTILA [14, 15]. For a fully dynamic model implementationinstaneous parcel densities and temperatures may be used to modifyor nudge the parcel vertical positions so that they are more consistentwith the vertical structure of the local atmosphere as represented inthe Eulerian grid. It is speculated that such nudging may be used astool to stabilize the model with regard to fast gravity and sound waves.

In a three dimensional application the mixing between parcels must be re-considered, and presumably it is necessary to separate horizontal and verticalmixing since they represent quite different processes in stratified fluids.

45

6.5. Prognostic variables in a dynamic model

In the dynamic tests in Section 5 the HEL scheme was only used fortransport of the mass field. In a future application in a baroclinic modelone may obviously use HEL as transport scheme for other invariants such astotal energy, potential vorticity, etc. Work is in progress investigating suchoptions.

6.6. Using HEL in an atmospheric chemical transport model

HEL should be suited quite well as numerical fundament for an atmo-spheric chemical transport model, and has in fact been designed with thisgoal in mind. Since the mixing we have introduced between neighboringparcels is purely ”real”, it is logical to perform chemical calculations in theparcel space. In a subsequent paper [41] it is described and demonstrated howHEL performs when it is used as underlying numerical scheme in a simpleso-called online atmospheric chemistry transport model.

7. Conclusions

The original motivation for developing HEL was to set up a numericalmethod for use in fluid dynamics, which fulfilled all the 11 desirable propertieslisted in Section 1.1.

The passive transport tests described in Section 4, the testing in thedynamical model (Section 5), and the efficiency investigations in Section 6.3clearly show that HEL fulfils all 11 properties to a high degree, and that it isvery multi-tracer efficient. Formally the convergence rates of HEL are lowerthan those of CSLAM-M, with which it has been compared. However, theabsolute level of accuracy in HEL is very high.

A fundamental component of HEL is a directionally biased mixing be-tween neighboring cells, which is proportional to the deformation rate of theflow. This mixing has been formulated in such a way that the 11th property(avoidance of spurious numerical mixing/unmixing) continues to be fulfilledin the Lagrangian representation of HEL. Thereby HEL should be ideal asunderlying scheme for chemical transport in the atmosphere particularly ifchemistry calculations are performed in Lagrangian space.

46

8. Acknowledgements

The present study was a part of the research of the Center for Energy, En-vironment and Health (CEEH), financed by The Danish Strategic ResearchProgram on Sustainable Energy under contract no 09-061417.

Appendix A. List of prognostic variables

The set of prognostic variables for solving the two-dimensional trans-port/continuity problem (passive transport) in HEL includes:

Eρ Density of ”dry” air in Eulerian spaceEρm ,m = 1, . . . ,M Density of tracers in Eulerian spaceLρ Density of ”dry” air in Lagrangian parcelsLρm ,m = 1, . . . ,M Density of tracers in Lagrangian parcelsLx x-coordinate of parcel position vectorsLy y-coordinate of parcel position vectorsLxa x-coordinate of auxiliary parcelsLya y-coordinate of auxiliary parcelsLV Area/volume of the Lagrangian parcelsLδ Deformation of Lagrangian parcels (A.1)

Each variable is represented by K Eulerian cells or P Lagrangian parcels.Note that for passive transport it has here been assumed that velocity com-ponents are known (non-prognostic) variables.

Appendix B. Mixing diagnostics

Real mixing.

lr =1

A

K∑k=1

dk∆Ak , if (χk, ξk) ∈ A0 , else

(B.1)

Range preserving unmixing.

lu =1

A

K∑k=1

dk∆Ak , if (χk, ξk) ∈ B0 , else

(B.2)

47

ξ

χχmin χmax

dk

(χk, ξk)(χroot

k , ψ(χrootk ))

ψ(χ)

Figure B.15: Illustration of the distance measure dk (arrows). The filled circles are relativeconcentrations, (ξk, χk), generated by the transport scheme and the unfilled circles are theclosest point, (χroot

k , ψ(χrootk )), on the pre-existing functional relation curve. (adopted

from [9])

Overshooting.

lo =1

A

K∑k=1

dk∆Ak , if (χk, ξk) /∈ (A ∪ B)0 , else

(B.3)

where A =∑K

k=1 ∆Ak is the total area and dk is the shortest distance betweenthe point (χk, ξk) and the functional relations curve (χ, ψ(χ)) (see FigureB.15). A is the area that is defined by the convex hull, B is the area outsideA, but within the range of the initial data (see Figure 1).

References

[1] G. A. Grell, S. E. Peckham, R. Schmitz, S. A. McKeen, G. Frost,W. C. Skamarock, B. Eder, Fully coupled online chemistry withinthe wrf model, Atmospheric Environment 39 (37) (2005) 6957 – 6975.doi:10.1016/j.atmosenv.2005.04.027.

[2] L. Pozzoli, I. Bey, S. Rast, M. G. Schultz, P. Stier, J. Feichter, Tracegas and aerosol interactions in the fully coupled model of aerosol-chemistry-climate echam5-hammoz: 2. impact of heterogeneous chem-istry on the global aerosol distributions, J. Geophys. Res. 113 (D7).

48

doi:10.1029/2007JD009008.URL http://dx.doi.org/10.1029/2007JD009008

[3] P. Rasch, D. Williamson, Computational aspects of moisture transportin global models of the atmosphere, Quarterly Royal Meteorological So-ciety 116 (1990) 1071–1090.

[4] C. Schar, P. K. Smolarkiewicz, A synchronous and iterative flux-correction formalism for coupled transport equations, J. Comput. Phys.128 (1996) 101–120. doi:http://dx.doi.org/10.1006/jcph.1996.0198.URL http://dx.doi.org/10.1006/jcph.1996.0198

[5] S. Lin, R. Rood, Multidimensional Flux-Form Semi-Lagrangian Trans-port Schemes, Monthly Weather Review 124 (9) (1996) 2046–2070.

[6] P. Jockel, R. von Kuhlmann, M. G. Lawrence, B. Steil, C. A. M. Bren-ninkmeijer, P. J. Crutzen, P. J. Rasch, B. Eaton, On a fundamentalproblem in implementing flux-form advection schemes for tracer trans-port in 3-dimensional general circulation and chemistry transport mod-els, Quarterly Journal of the Royal Meteorological Society 127 (573)(2001) 1035–1052. doi:10.1002/qj.49712757318.URL http://dx.doi.org/10.1002/qj.49712757318

[7] P. H. Lauritzen, P. A. Ullrich, R. D. Nair, Atmospheric transportschemes: Desirable properties and a semi-lagrangian view on finite-volume discretizations, in: P. Lauritzen, C. Jablonowski, M. Taylor,R. Nair, T. J. Barth, M. Griebel, D. E. Keyes, R. M. Nieminen, D. Roose,T. Schlick (Eds.), Numerical Techniques for Global Atmospheric Mod-els, Vol. 80 of Lecture Notes in Computational Science and Engineering,Springer Berlin Heidelberg, 2011, pp. 185–250.

[8] B. Machenhauer, E. Kaas, P. Lauritzen, Finite-Volume Methods in Me-teorology, in: R. Teman, J. Tribbia (Eds.), Computational methods forthe atmosphere and the oceans, ELSEVIER, Amsterdam, The Nether-lands, 2008, pp. 3–120.

[9] P. H. Lauritzen, J. Thuburn, Evaluating advection/transport schemesusing interrelated tracers, scatter plots and numerical mixing diagnos-tics, Quarterly Journal of the Royal Meteorological Society 138 (665)

49

(2012) 906–918. doi:10.1002/qj.986.URL http://dx.doi.org/10.1002/qj.986

[10] J. Thuburn, M. E. McIntyre, Numerical advection schemes, crossisen-tropic random walks, and correlations between chemical species, Journalof Geophysical Research-All Series- 120 (D6) (1997) 6775–6797.

[11] B. Machenhauer, The spectral method, in: A. Kasahara (Ed.), Numeri-cal Methods used in Atmospheric Models, GARP Publication Series No.17, Vol. II (WMO and IGSU, Geneva Switzerland), 1979, pp. 121–275.

[12] D. Durran, Numerical Methods for Fluid Dynamics: With Applicationsto Geophysics, Texts in Applied Mathematics, Springer, 2010.URL http://books.google.dk/books?id=ThMZrEOTuuUC

[13] C. Reithmeier, R. Sausen, Attila: atmospheric tracer transport in alagrangian model, Tellus B 54 (3) (2002) 278–299. doi:10.1034/j.1600-0889.2002.01236.x.URL http://dx.doi.org/10.1034/j.1600-0889.2002.01236.x

[14] A. Stenke, V. Grewe, M. Ponater, Lagrangian transport of water vaporand cloud water in the echam4 gcm and its impact on the cold bias.,Climate Dynamics 31 (5) (2008) 491 – 506.

[15] A. Stenke, M. Dameris, V. Grewe, H. Garny, Implications of la-grangian transport for simulations with a coupled chemistry-climatemodel, Atmospheric Chemistry and Physics 9 (15) (2009) 5489–5504.doi:10.5194/acp-9-5489-2009.URL http://www.atmos-chem-phys.net/9/5489/2009/

[16] R. Sadourny, K. Maynard, Formulations of lateral diffusion in geophys-ical fluid dynamics models, in: C. Lin, R. Laprise, H. Ritchie (Eds.),Numerical Methods in Atmospheric and Oceanic Modelling - The AndreJ. Robert Memorial Volume, NRC Research Press, 1997, pp. 547–556.

[17] F. Vana, P. Benard, J.-F. Geleyn, A. Simon, Y. Seity, Semi-lagrangianadvection scheme with controlled damping: An alternative to nonlinearhorizontal diffusion in a numerical weather prediction model, QuarterlyJournal of the Royal Meteorological Society 134 (631) (2008) 523–537.doi:10.1002/qj.220.URL http://dx.doi.org/10.1002/qj.220

50

[18] D. McKenna, P. Konopka, J. Grooss, G. Gunther, R. Muller, R. Spang,D. Offermann, Y. Orsolini, A new Chemical Lagrangian Model of theStratosphere (CLaMS) 1. Formulation of advection and mixing, Journalof Geophysical Research 107. doi:10.1029/2000JD000114.

[19] D. Tskhakaya, K. Matyash, R. Schneider, F. Taccogna, The particle-in-cell method, Contributions to Plasma Physics 47 (8-9) (2007) 563–594.doi:10.1002/ctpp.200710072.URL http://dx.doi.org/10.1002/ctpp.200710072

[20] E. Kaas, A. Guldberg, P. Lopez, A Lagrangian Advection Scheme UsingTracer Points, in: C. Lin, R. Laprise, H. Ritchie (Eds.), NumericalMethods in Atmospheric and Oceanic Modelling - The Andre J. RobertMemorial Volume, NRC Research Press, 1997, pp. 171–194.

[21] C. Fromming, M. Ponater, U. Burkhardt, A. Stenke, S. Pechtl,R. Sausen, Sensitivity of contrail coverage and contrail radiative forc-ing to selected key parameters, Atmospheric Environment 45 (7) (2011)1483 – 1490. doi:10.1016/j.atmosenv.2010.11.033.

[22] J. Smagorinsky, General circulation experiments with the primitiveequations: I. the basic experiment., Mon. Wea. Rev. 91 (1963) 99–164.

[23] I. Wohltmann, M. Rex, The lagrangian chemistry and transport modelatlas: validation of advective transport and mixing, Geoscientific ModelDevelopment 2 (2) (2009) 153–173. doi:10.5194/gmd-2-153-2009.URL http://www.geosci-model-dev.net/2/153/2009/

[24] L. Dong, B. Wang, Trajectory-tracking scheme in lagrangian form forsolving linear advection problems: Preliminary tests, Monthly WeatherReview 140 (2) (2011) 650–663. doi:10.1175/MWR-D-10-05026.1.URL http://dx.doi.org/10.1175/MWR-D-10-05026.1

[25] L. Dong, B. Wang, Trajectory–tracking scheme in lagrangian formfor solving linear advection problems: Interface spatial discretization,Monthly Weather Review Accepted (2012) p1–p2. doi:10.1175/MWR-D-12-00058.1.

[26] R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cam-bridge Texts in Applied Mathematics, Cambridge University Press,

51

2002.URL http://books.google.dk/books?id=QazcnD7GUoUC

[27] R. Eymard, T. Gallout, R. Herbin, Finite volume methods, in: P. Ciar-let, J. Lions (Eds.), Handbook of Numerical Analysis, Vol. 7, Elsevier,2000, pp. 713 – 1018. doi:10.1016/S1570-8659(00)07005-8.URL http://www.sciencedirect.com/science/article/pii/S1570865900070058

[28] E. Kaas, A simple and efficient locally mass conserving semi-Lagrangiantransport scheme, Tellus A 60 (2) (2008) 305–320.

[29] B. Leonard, A. Lock, M. MacVean, Conservative Explicit Unrestricted-Time-Step Multidimensional Constancy-Preserving Advection Schemes,Monthly Weather Review 124 (11) (1996) 2588–2606.

[30] B. Machenhauer, M. Olk, The Implementation of the Semi-lm, plicitScheme in Cell-lntegrated Semi-Lagrangian Models, Numerical Methodsin Atmospheric and Oceanic Modelling: The Andre J. Robert MemorialVolume.

[31] P. H. Lauritzen, R. D. Nair, P. A. Ullrich, A conserva-tive semi-lagrangian multi-tracer transport scheme (cslam) on thecubed-sphere grid, J. Comput. Phys. 229 (5) (2010) 1401–1424.doi:10.1016/j.jcp.2009.10.036.URL http://dx.doi.org/10.1016/j.jcp.2009.10.036

[32] M. Zerroukat, N. Wood, A. Staniforth, SLICE: A Semi-Lagrangian In-herently Conserving and Efficient scheme for transport problems, Quar-terly Journal of the Royal Meteorological Society 128 (586) (2002) 2801–2820.

[33] R. Cohen, D. Schultz, Contraction Rate and Its Relationship to Fronto-genesis, the Lyapunov Exponent, Fluid Trapping, and Airstream Bound-aries, Monthly Weather Review 133 (5) (2005) 1353–1369.

[34] W. M. Putman, S.-J. Lin, Finite-volume transport on vari-ous cubed-sphere grids, J. Comput. Phys. 227 (1) (2007) 55–78.doi:10.1016/j.jcp.2007.07.022.URL http://dx.doi.org/10.1016/j.jcp.2007.07.022

52

[35] P. H. Lauritzen, W. C. Skamarock, M. J. Prather, M. A. Taylor,A standard test case suite for two-dimensional linear transport onthe sphere, Geoscientific Model Development 5 (3) (2012) 887–901.doi:10.5194/gmd-5-887-2012.URL http://www.geosci-model-dev.net/5/887/2012/

[36] R. D. Nair, P. H. Lauritzen, A class of deformational flow test casesfor linear transport problems on the sphere, J. Comput. Phys. 229 (23)(2010) 8868–8887. doi:10.1016/j.jcp.2010.08.014.URL http://dx.doi.org/10.1016/j.jcp.2010.08.014

[37] G. Grell, A. Baklanov, Integrated modeling for forecasting weather andair quality: A call for fully coupled approaches, Atmospheric Environ-ment 45 (2011) 6845–6851. doi:10.1016/j.atmosenv.2011.01.017.

[38] C. Rivest, A. Staniforth, A. Robert, Spurious Resonant Response ofSemi-Lagrangian Discretizations to Orographic Forcing: Diagnosis andSolution, Monthly Weather Review 122 (2) (1994) 366–376.

[39] K. Lindberg, V. Alexeev, A Study of the Spurious Orographic Reso-nance in Semi-Implicit Semi-Lagrangian Models, Monthly Weather Re-view 128 (6) (2000) 1982–1989.

[40] B. Sørensen, E. Kaas, U. S. Korsholm, A mass conserving and multi-tracer efficient transport scheme in the online integrated enviro-hirlammodel, Geosci. Model Dev. Submitted.

[41] A. B. Hansen, B. Sørensen, P. Tarning-Andersen, J. H. Christensen,J. Brandt, E. Kaas, The hybrid eulerian lagrangian numerical schemetested with chemistry, Geosci. Model Dev. Discuss. Submitted.

53

a hybrid eulerian lagrangian numerical scheme … hybrid eulerian lagrangian numerical scheme for...

Documents