a locally mass-conserving semi-lagrangian regional nwp ...nate is understood that the pressure...

Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (0000)

A locally mass-conserving semi-Lagrangian regional NWP

model with conserved quasi-Lagrangian vertical coordinates

B. Sørensen,a⇤ E. Kaas,a P. H. Lauritzen,ba Niels Bohr Institute, University of Copenhagen

b DenmarkNational Center for Atmospheric Research, USA⇤Correspondence to: Niels Bohr Institute, University of Copenhagen

Juliane Maries Vej 30,2100 Copenhagen,Denmark. E-mail: [email protected]

In semi-Lagrangian models with quasi-Lagrangian vertical coordinates the

Lagrangian levels are, in general, remapped to Eulerian model levels each

physics time step. This remapping may introduce an undesirable tendency

to smooth sharp gradients and create numerical diffusion in the vertical

distribution. In a primitive equations model it is shown that this numerical

mixing can be reduced by introducing Lagrangian levels persisting for

multiple time steps, and by remapping only tendency information (for physical

parameterizations) between the Lagrangian and Eulerian levels at each time

step. Several methods have been tested to ensure that the deformation of

the Lagrangian layers does not degrade the accuracy of the model. After

several time steps - dynamically calculated or fixed - the Lagrangian levels

are remapped to the traditional Eulerian model levels to ensure the stability of

the model. The methodology is implemented using a Locally Mass Conserving

Semi- Lagrangian scheme for transport in the horizontal direction. A new

mass conserving 1D interpolation method for vertical interpolations, based on

a combination of non-uniform cubic interpolation and the partition of unity

principle, is applied. The method is accurate, mass conserving, and is shown

to be highly efficient compared to the traditional approach. Copyright

c� 0000

Royal Meteorological Society

Key Words: Semi-Lagrangian; Mass-conserving; ALE; Enviro-HIRLAM; Floating Lagrangian verticalcoordinates

Received . . .

Citation: . . .

1. Introduction

Numerical weather and climate modelling is under con-stant development. Semi-implicit semi-Lagrangian (SISL)models have proven to be numerically efficient for bothnumerical weather prediction (NWP) and climate mod-elling, due to the ability to use long time steps. Whilefuture generations of non-hydrostatic NWP models maybe formulated differently, it is likely that several climatemodels in a number of years from now will continue to bebased on the primitive equations employing the traditionalSISL technique.

Chemical/aerosol feedback mechanisms are becomingmore and more relevant in NWP as well as climate models,

since the biogenic and anthropogenic emissions can havea direct effect on the dynamics and radiative propertiesof the atmosphere (Arneth et al. 2009; Baklanov 2010;Baklanov and Korsholm 2008; Forkel et al. 2012). Toinclude chemical feedbacks a prognostic representation ofchemical species is neccesary, i.e. online coupling. For themodels to be able to include a large number of chemicalspecies, the advection schemes must have a high degree ofmulti-tracer efficiency, e.g Lauritzen et al. (2010).

In this paper a new semi-Lagrangian (SL) advectionmethod is introduced. It combines an inherently mass con-serving 2D semi-Lagrangian advection scheme with semi-implicit time-stepping and employing a fully Lagrangian

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2 B. Sørensen et al.

vertical coordinate (LVC). By Lagrangian vertical coordi-nate is understood that the pressure levels computed duringa time step are material surfaces, thus conserving mass inthe corresponding Lagrangian layers. This minimizes thevertical diffusion and thus potentially improves the simu-lation of the vertical profiles of moisture, clouds, and chem-ical constituents (Heikes et al. 2006). Since the Lagrangianlevels suffer from traditional Lagrangian limitations causedby the convergence and divergence of the flow, remappingsto the Eulerian model levels are generally still required,unless dynamic remapping methods are used. However, itturns out that this need only be applied after a number oftime steps. Several different remapping methods are con-sidered and a new mass conserving interpolation routine hasbeen created. This is a 1D non-uniform cubic interpolationcombined with a partition of unity principle, which modifiesthe interpolation weights. The new remapping is shownto be both mass conserving, consistent, and multi-tracerefficient.

The concept of a quasi-Lagrangian approach is notnew. It was introduced by Starr (1945) on a theoretical leveland since then several implementations of LVC in numericalmodels has been carried out, e.g. Lin (2004); Lauritzen et al.(2008); Nair et al. (2009). These implementations have,however, generally not preserved the levels for more thanone physics time step. The implementation presented heretakes a different approach by preserving the Lagrangianlevels for multiple physics time steps, either in an dynamic(adaptive) manner or fully preserved for a limited numberof time steps.

The paper is organized as follows: In section 2 and3 the traditional semi-implicit semi-Lagrangian and thelocally mass conserving semi-Lagrangian methods is brieflyintroduced, respectively. In section 4 the new Lagrangianvertical coordinate is explained in detail, and in section 5 thedifferent strategies for controlling the Lagrangian flow aredescribed as well as the new interpolation routine. In section6, 7, 8, and 9 the different setups are tested and analysed.Finally some concluding remarks in section 11.

2. Traditional semi-implicit semi-Lagrangian

formulation

In a primitive equations model both Rossby- and gravitywaves are supported. A traditional technique for eliminatingthe numerical instability associated with mainly externalgravity modes is semi-implicit time differencing. Theexplicit semi-Lagrangian continuity equation can, for a two-time level discretization, be written as,

n+1SL�exp

= { + 0.5�t r · v}n⇤

+0.5�t r · v

n+1(1)

where ()

⇤

denotes a value that has been interpolated to anupstream departure point and (

˜·) here and below denotes avalue that has been extrapolated from time levels n� 1 andn:

(

˜·)n+1= 2(·)n � (·)n�1 (2)

Following Temperton and Staniforth (1987) the corre-sponding semi-implicit semi-Lagrangian (SISL) continuityequation is,

n+1SL

=

n+1SL�exp

+

�t

2

(L

n+1

� eLn+1

) (3)

where L

and N

are the linear and non-linear parts of theright-hand side of the prognostic equation for . In SISLthe linear part is treated implicitly and the non-linear partexplicitly via the extrapolation.

3. Locally Mass Conserving Semi-Lagrangian scheme

The Locally Mass Conserving Semi-Lagrangian scheme(LMCSL) by Kaas (2008), is a semi-Lagrangian method forsolving the volume density continuity equation. It is basedon a classical 2D semi-Lagrangian advection scheme withe.g. cubic Lagrange polynomial interpolation. However,to ensure mass conservation the divergence term in thecontinuity equation is replaced by modified interpolationweights. The term local mass conservation refers to theinherent ability of a scheme to conserve mass alongcharacteristics.

One can rewrite the traditional explicit semi-Lagrangian scheme (1) as

n+1k

SL�exp

=

KX

l=1

w

k,l

n

l

� �t( r · v)

n

l

2

�

��t( r · ev)

n+1k

2

(4)

with k being an Eulerian (arrival) grid point index, K

the total number of Eulerian grid points, and w

k,l

theinterpolation weight from an upstream Eulerian grid pointl (at time level n) to the Lagrangian departure point (·)

⇤k

.Thus, the w

k,l

values define the SL remapping required foreach Eulerian arrival point. It should be noted that theseweights are independent of the actual field values and, oncecalculated, can be reused for all prognostic fields. For anytraditional SL scheme we have,

KX

l=1

w

k,l

= 1. (5)

This does however not guarantee mass conservation.

3.0.1. Modified interpolation weights

As shown in Kaas (2008) formal mass conservation canbe achieved by modifying the interpolation weights andskipping the divergence terms in (4). The modified weightsbwk,l

are defined as

bwk,l

=

A

l

A

k

w

k,lPK

m=1 wm,l

, (6)

which gives

n+1k

=

�

n

⇤⇤k

⌘KX

l=1

bwk,l

· n

l

(7)

with w

k,l

being the traditional SL-weights. The LMCSLscheme forecasts the grid cell averages and not the gridpoint values, as the traditional SL scheme, hence the barsindicating cell averages. A

l

and A

k

are, respectively, theareas or, in case of full 3D, the volume covered by the l’thand k-th Eulerian grid cells. It can easily be shown, thatwith this approach the modified weights ensure that the total

Copyright c� 0000 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (0000)Prepared using qjrms4.cls

An NWP model with quasi-Lagrangian vertical coordinates 3

mass given off by any Eulerian grid cell to all surroundingdeparture points, is equal to the actual mass representedby that particular grid cell. Thus, mass conservation inthe LMCSL scheme is obtained via a partition of unityprinciple. We will in the following continue to use the termdeparture point although, in reality, it is the location, at timelevel n, of the Lagrangian point arriving, at time level n + 1,at the centroid of an Eulerian cell.

The computation of the modified weights is straight-forward: it requires a single additional loop over all gridcells, where all the SL-weights are summed in an array, thusgetting the denominator in (6). When all modified weightshave been calculated they can be reused for any prognosticdensity variable. This makes the method very efficient if alarge number of tracer variables is introduced, since eachadditional variable does not require any new computationsof modified weights.

The underlying w

k,l

-weights can be determinedby different types of interpolation, and in the presentimplementation several are used, i.e. bilinear, biquadratic,bicubic, and mixed bilinear/bicubic interpolation. Thelocality of the scheme is dependent on the order ofinterpolation - as any traditional SL scheme the higheraccuracy the less locality. The locality of the LMCSLmethod is the same as the traditional SL-method. With bi-cubic interpolation 16 cell values are used to determinethe value at any departure point. Figure 1 illustrates thegrid cells used to compute the value in four arrival cells.The figure displays convergence as the departure points arespaced further apart than the surrounding Eulerian grid cellcentroids.

Figure 1. Illustration of the upstream bi-cubic interpolation. 16 cell valuesare used to determine the value at each Lagrangian departure point. Thefour cells to the right are the Eulerian arrival cells, and the correspondingareas to the left are the cells used in the bicubic interpolation.

The effects of divergence are implicitly included in themodified weights and following Kaas (2008) divergence inLMCSL is logically defined as:

Dn+1/2k LM

=1

�t

1 �

KX

l=1

bwk,l

!. (8)

If the sum of the modified weights is almost equal to one,meaning that the departure points are distributed equally, thedivergence will be close to zero. The actual discretizationsof the LMCSL scheme will be described further in section4.

3.1. Shape preservation

To further improve the LMCSL scheme two locallymass conserving and shape-preserving filters have been

Modified weights

’./zmhw.dat’ u 1:2:42

0 50 100 150 200 250

0

50

100

150

200

250

0.9

0.95

1

1.05

1.1

Figure 2. Instantaneous contour plot of actual normalization field,PKm=1 wm,l. Values above one indicates divergence and values below

indicates convergence.

implemented. The LMCSL scheme combined with thesefilters will be referred to as LMCSL-M and LMCSL-MS,with M indicating monotonic and S simple. The LMCSL-M filter makes an a posteriori anti-diffusive correction,where the mass is redistributed towards target values undermonotonic and mass conserving constraints. With the anti-diffusive method the filter actually improves the accuracyof the scheme. See Kaas and Nielsen (2008) for furtherdetails. The LMCSL-MS filter is not anti-diffusive, but onlyredistributes the mass under mass conserving and shapepreserving or positive definite constraints. To achieve themass conserving properties of the LMCSL scheme in anoperational model, a positive definite filter such as the Mor MS is essential, since the negative undershoots generatedby all higher order methods are non-physical and mustbe removed using a mass conserving method (Kaas andNielsen 2008).

The simple filter uses an iterative approach toredistribute the mass. When the filter encounters a valueoutside the minimum or maximum values, it will, withintheir maximum and minimum limits, remove or add themass to the neighboor cells⇤. It will do this relative to thedistributable mass present in the cells, to ensure that nonew over or under-shoots is created. If the mass is not fullydistributed in the nearest cells, the next set of neighboorswill be used, and so on, until all mass is distributed.† Thefilter is strictly a posteriori, and only depends on givenmaximum and minimum values, and is in this sense notlimited to any particular advection method.

4. Lagrangian vertical scheme

The Lagrangian vertical scheme (LMCSL-LL) is anattempt to transform the traditional 3D interpolating semi-Lagrangian advection scheme into a more Lagrangianscheme regarding vertical transport. The LMCSL-LLscheme is implemented in the primitive equations modelHIRLAM (Unden et al. 2002), but we note that there

⇤By neighboor cells is meant the cells surrounding the cell in question, thatis 2 cells for one dimensional distribution, 8 for two dimensional, and 27for three dimensional.†The radius of distribution will in general be of the same size as the stencilof the interpolation method.



are no principle obstacles implementing it into fully non-hydrostatic models.

The present implementation takes its starting pointin a special mass-conserving version, here termed CISL-HIRLAM, of HIRLAM. In CISL-HIRLAM, describedin detail in Lauritzen et al. (2008), horizontal transportincluding the effects horizontal divergence is based on theinherently mass conserving cell integrated semi-Lagrangian(CISL) scheme, (Nair and Machenhauer 2002; Lauritzenet al. 2006), while vertical transport is dealt with using aquasi-Lagrangian vertical coordinate in a fashion similar tothat in Lin (2004). The main difference between the newLMCSL-LL scheme and CISL-HIRLAM is that in LMCSL-LL the horizontal transport is based on LMCSL insteadof CISL and, more fundamentally, in LMCSL-LL full(one-dimensional) vertical remappings from the Lagrangianto the Eulerian vertical coordinate are only performed atintervals covering several time steps instead of at every timestep in CISL-HIRLAM.

Briefly, the general LMCSL-LL procedure is asfollows: after having calculated horizontal trajectories usingan iterative procedure equivalent to that in Lauritzen et al.(2008), horizontal transport is performed based on theLMCSL scheme. At the arrival column new pressures ofeach Lagrangian vertical level is defined via the hydrostaticequation. At the end of the time step the concentrations atLagrangian levels are stored so they can be used as a startingpoint for the following time step. Because the physicalparameterization of the model is designed to operate onfixed Eulerian levels, a one-dimensional vertical remappingto the Eulerian levels must still be carried out at each timestep. Once calculated there, the tendencies of the physics,i.e. not the full fields, are then remapped back to theLagrangian layer and added to the prognostic values there.This is done in a completely mass conserving way for scalarvariables, with either Piecewise Parabolic Method (PPM2)or with a 1D cubic non-uniform interpolation with modifiedweights. This is described in detail in section 5.3.

Figure 3 illustrates how the LMCSL scheme canoperate in 2 dimensions: one horizontal direction andthe vertical direction. The stippled (red) vertical arrowsrepresent the remapping of the prognostic fields fromLagrangian layers to Eulerian layers. This approach withfull vertical remapping every time step is identical tothat employed for the CISL-L scheme in Lauritzen et al.(2008). Figure 4 illustrates the new LMCSL-LL approach.

Figure 3. Flow of the traditional LMCSL scheme with remapping toEulerian model levels at every time step. The four columns are respectivelytime step n, n + 1, n + 2 and n + 3. Solid lines indicate Eulerian levelsand dashed Lagrangian levels.

Figure 4. Flow of the LMCSL-LL scheme with remapping to Eulerianmodel levels at every time step, and tendency remapping to Lagrangianlevels. The four columns are respectively time step n, n + 1, n + 2 andn + 3.

The flow between the time steps is identical to that inFigure 3 but the Lagrangian levels are re-used in thefollowing time steps, and therefore they are generallylocated at different pressure levels than those in Figure3. The solid (green) arrows represent the remapping ofphysical parameterization tendencies from Eulerian levelsto Lagrangian levels.

The Eulerian grid of the LMCSL-LL scheme isidentical to that in HIRLAM, i.e. the horizontal grid isa traditional staggered Arakawa C-grid (Mesinger andArakawa 1976), while in the vertical a staggered Lorenz-grid is used - see Unden et al. (2002) for details. The modelatmosphere is divided into K levels, with the lowest indexat the top of the atmosphere and the highest at the surface.

The pressure p

k+1/2 at the interface between levels k

and k + 1 is,

p

k+1/2 = A

k+1/2 + B

k+1/2ps (9)

where p

s

is the surface pressure, and A

k+1/2 and B

k+1/2

are predefined constants defining the terrain followinghybrid sigma-pressure vertical coordinate. Between the halflevels k � 1

2 and k +

12 are the full levels k. The pressure

level thickness of level k is defined by

�p

k

= p

k+1/2 � p

k�1/2 (10)

The uppermost half level is p1/2 = 0 and the surface isat p

K+1/2 = p

s

.

4.1. Trajectory calculations

As mentioned above the first step in the new LMCSL-LL scheme is to determine the trajectories. This is doneiteratively using two iterations, and in a way quite similarto that in CISL-HIRLAM, see Lauritzen et al. (2008) fordetails, although the LMCSL-LL calculation of upstreamhorizontal trajectories is based on velocities originallydefined in the new Lagrangian vertical levels, while theywere based on the traditional level Eulerian horizontalvelocities in the CISL-HIRLAM case. To acquire accuratetrajectories the effect of horizontal accelerations is includedas they were in CISL-HIRLAM. This was shown, in ashallow water model, to be of importance for the numericalstability when the LMCSL scheme is combined with thesemi-implicit scheme (Kaas 2008) in essentially the sameway as we will present below for the LMCSL-LL scheme.



Note that in the LMCSL-LL scheme trajectories relevant forsolving the continuity equation are those ending up in theEulerian mass points of the Arakawa C-grid, whereas for theCISL-L scheme trajectories were calculated for each cornerpoint of the Eulerian control volumes. This means thatfor the LMCSL-LL scheme velocities must be interpolatedto the mass points, defined at the center of the Euleriancells, prior to the calculation of trajectories, instead of aninterpolation to the corner points.

As for the CISL-HIRLAM two iterations have beenused to estimate the trajectories. For each guess ofhorizontal trajectories, i.e. for each iteration, an explicitupstream LMCSL interpolation is used to obtain theupdated contributions to the mass field in the arrival column.In this column the mass field is then summed from thetop of the atmosphere (p = 0) to the surface (p = p

s

) toobtain the pressure of each Lagrangian level relevant forthe actual iteration. These levels do in general not coincidewith the model levels defined at time step n. To calculatethe horizontal velocities extrapolated to time level n + 1

which is needed for the following iteration and for thefinal explicit forecast, the trajectory is therefore interpolatedvertically using cubic Lagrange interpolation to the actualLAgrangian arrival level.

4.2. Discretized equations of motion

The semi-implicit LMCSL-LL continuity equation is givenas,

�p

n+1k,L

= (�p

k

)

n+1

exp

� �t

2

�p

ref

k,L

⇥D

n+1k

�D(

evn+1k

)

⇤L

(11)

where �p

ref

k,L

is the Lagrangian reference pressure levelthickness defined in section 4.4. The last term on the right-hand side is the extrapolated LMCSL divergence, defined as(8),

Dn+1k,L

LM

=

2

�t

1 �

KX

l=1

bewk,l

!(12)

but based only on extrapolated velocities. It is obtained bycalculating the modified weights from the extrapolated halfof the trajectory and multiplying with two, as it is only ahalf-trajectory. D is the Eulerian divergence dicretized withfinite differences.

D =

1

a cos ✓

@

@�

(u cos ✓) +

@v

@✓

�(13)

The momentum equations are solved on semi-Lagrangiangrid point form, using a half-implicit-Coriolis scheme asis used in HIRLAM. Again the formulation is the same asfor the CISL-HIRLAM, see Lauritzen et al. (2008) for thedetails. Where LMCSL, and therefore LMCSL-LL, differsfrom the CISL method is the solution to the continuityequation (11). In CISL-HIRLAM the continuity equationis solved using the predictor-corrector method (Lauritzenet al. 2006). In LMCSL the Lagrangian divergence (12) isused directly to solve the continuity equation as in Kaas(2008), resulting in a somewhat simpler scheme.

4.3. Consistent Omega-P

In many NWP models the discretization of the thermo-dynamic equation is inconsistent, as is also the case forEnviro-HIRLAM and HIRLAM. The energy conversionterm is there solved with an Eulerian approach using finitedifferences, and vertical velocities are diagnostic and calcu-lated from the continuity equation, which is discretized in asemi-Lagrangian manner.

Using the approach by Lauritzen (2005) in theimplementation of the LMCSL-LL scheme, where omega-Pis defined as

✓w

p

◆

exp

=

2

�t

0

@ (p

k

)

n+1

exp

� (p

k

)

n

⇤

(p

k

)

n+1

exp

+ (p

k

)

n

⇤

1

A (14)

the vertical velocity and the energy convertion termbecomes fully consistent.

4.4. Modified pressure level thicknesses

The traditional reference pressure level thicknesses �p

ref

k

,does not apply to the Lagrangian levels, and must bemodified. This is done in a completely consistent waywith the traditional reference pressure level thicknesses.The reference pressure level thickness is defined as theEulerian pressure level thickness with a surface pressureof 1013 hP. This is very simple in the hybrid coordinatemodel levels, since the pressure levels - and therefore thelayer thicknesses - are calculated directly from the surfacepressure using (9) and (10).

The change from pressure levels to reference pressurelevels, is a change in surface pressure. The Lagrangianreference pressure level thickness, is then,

�p

ref

k,L

= �p

k,L

+ �B

k

�p

s

(15)

where

�p

s

= p

ref

s

� p

s

(16)�B

k

= B

k+1/2 �B

k�1/2 (17)

This can easily be verified by substituting �p

k,L

with �p

k

and using (9) then,

�prefk,L =�pk +�Bk�ps

=⇥(Ak+1/2 +Bk+1/2ps)� (Ak�1/2 +Bk�1/2ps)

⇤

+(Bk+1/2 �Bk�1/2)(prefs � ps)

=(Ak+1/2 �Ak�1/2) + (Bk+1/2 �Bk�1/2)prefs

=(Ak+1/2 +Bk+1/2prefs )� (Ak�1/2 +Bk�1/2p

refs )

=�prefk (18)

completes the demonstration.The surface pressure changes slightly during the

physics, which is in Eulerian model space, and the Eulerianpressure level thicknesses change accordingly. The sum ofthe pressure level thicknesses must be equal to the surfacepressure.

p

s

⌘KX

k=1

�p

k

(19)



The Lagrangian pressure level thicknesses are pre-served, and should ideally not change. This is however notpossible, since their sum will be different than the updatedsurface pressure. To modify the Lagrangian pressure levelthicknesses in such a way that they are consistent with theupdated surface pressure can be done in many ways, butit must be coherent with the underlying hybrid levels. Bycoherent it is understood, that in the special case wherethe Lagrangian levels are identical to the Eulerian levels,the updated Lagrangian levels must also be identical tothe updated Eulerian levels. This can easily be done bysubstituting p

s

with p

0s

and p

ref

s

with p

1s

in (15) thus getting,

�p

1k,L

= �p

k,L

+ �B

k

�p

s

(20)

with�p

s

= p

1s

� p

0s

(21)

where p

0(·) is the pressure before the physics and p

1(·) is the

modified pressure after the physics.

4.5. Tendencies

When the full semi-implicit forecast has been performed,the forecasted fields are at Lagrangian pressure layers.These fields are saved in temporary Lagrangian variables, inaddition to the traditional prognostic variables the pressurelevel thicknesses are kept as well, as they define theLagrangian pressure levels.

p

k+1/2,L =

kX

l=1

�p

l

(22)

A full remapping of all variables from the Lagrangianpressure levels, to the Eulerian pressure levels is performed,as is done traditionally in the LMCSL scheme. Thevelocities and temperature are remapped using 1D cubicinterpolation and the scalar fields are remapped with the1D cubic LMCSL interpolation with a shape-preservingfilter‡. The fields are then stored before the model physicsis performed. After the model physics and boundaryrelaxation, the new Eulerian fields at time level n + 1 areknown. The tendencies can then be calculated.

�

ten

k

= �

0k

� �

1k

(23)

where �tenk

is the tendency at Eulerian levels, �0k

are theremapped Eulerian values before the physics and �1

k

is theEulerian values after the physics.

The tendencies are remapped from the Eulerian levelsback to the Lagrangian pressure levels. There is nomonotonicity constraints applied to the tendencies, whichare negative as well as positive. However, as the tendenciesare added to the Lagrangian fields, negative values can beobserved, and it is thus necessary to run the simple filter,here in one dimension and with only a positive definiteconstraint.

‡Alternatively the piecewise parabolic method (PPM2) with a monotonicconstraint can be used.

4.6. LAM boundaries

The model system is a limited area model (LAM), itis therefore necessary to acquire boundary data from anexternal source such as nesting in a larger domain or from aglobal model. In the boundary zone surrounding the modeldomain, the levels are always Eulerian. This means that thebuilt in boundary relaxation routine can be used, and notendencies are calculated. A simple linear relaxation of theLagrangian levels towards the Eulerian model levels is usedclose to the boundaries.

5. Remapping methods

Conserving the Lagrangian levels indefinetely is notpossible, since fully Lagrangian methods face severalproblems. Just as in the horizontal, the vertical flow willcause the levels to converge and diverge with time, untilthey are spaced either too close to each other or too farfrom each other. The solutions presented here is either touse a limited number of Lagrangian time steps or to use adynamic remapping method.

5.1. Fixed remapping

Figure 5. Flow of the LMCSL-LL scheme with fixed remapping toEulerian model levels at time step n + 2, and tendency remapping toLagrangian levels. The four columns are respectively time step n, n + 1,n + 2 and n + 3.

A variable called the Lagrangian time step (LTS)has been introduced, it is defined as the number offully Lagrangian time steps taken, before the model willcontinue, not from the Lagrangian levels, but from theEulerian model levels. Since the Lagrangian fields alreadyhave been remapped to Eulerian levels, all computationshas already been carried out, so what is done in praxis, isthat all Lagrangian fields and variables are overwritten withthe Eulerian fields. This is a very simple, but effective andhighly variable method, it will even as LTS approaches zeroconverge toward the underlying semi-Lagrangian model,and if LTS is equal to zero, the scheme is identical tothe standard LMCSL scheme as it should. This remappingmethod is very fast since all fields already exists, and thetendency interpolation is not needed.

5.2. Dynamic remapping

In addition to the fixed remapping several different kindsof dynamic remapping schemes have been tried. The



dynamic remapping only remaps individual levels if theyviolate certain conditions. This can reduce the number ofremappings in some areas, but remapping in other areasmight be increased compared to the fixed remapping.

Method I

The simplest dynamic remapping, that ensures theLagrangian levels do not deviate excessively from theEulerian levels, only has one condition which must not beviolated. A particular Lagrangian level, may not deviatemore than a given distance (in pressure coordinates), fromthe surrounding Eulerian levels. E.g. � =

13 where � is

normalized pressure level distance.

p

k+1/2 � ��p

k+1 < p

k+1/2,L < p

k+1/2 + ��p

k

(24)

If the Lagrangian level violates this, the level is remappedto the corresponding Eulerian level. Since the pressurelevels in question is the half levels k +

12 , and the

prognostic values are defined at full levels, the full levelvalue k and k + 1 must be remapped. This method

Figure 6. Flow of the LMCSL-LL scheme with dynamic remappingmethod I at individual levels, and tendency remapping to Lagrangianlevels. The four columns are respectively time step n, n + 1, n + 2 andn + 3.

has some disadvantages, mainly that when a Lagrangianlevel is remapped, the horizontally surrounding Lagrangianlevels are in general not remapped. This can introducediscontinuities in the flow. To prevent this from affectingthe stability of the model, only minor deviations can betolerated, such as �1 1

3 . In practice, it also adds someconstraints on the time step, which cannot be too long,as it will generate larger deviations and the discontinuitiesincreases.

Method II

To address the issues in method I, a simple solutionis to remap levels which violate the condition, not tothe corresponding Eulerian level, but instead to a newLagrangian level which is equal to the allowable deviation.This guarantees that there is no horizontal discontinuities.The disadvantage of this method is, that the flow willin fact not be the ”real” Lagrangian flow but only anapproximation. Also since the new level is located at theactual boundary condition, it is highly likely that withina few time steps it will cross the boundary once moreand require an additional remapping. Thus increasing thetotal number of remappings considerably. The remappings

Figure 7. Flow of the LMCSL-LL scheme with dynamic remappingmethod II at individual levels, and tendency remapping to Lagrangianlevels. The four columns are respectively time step n, n + 1, n + 2 andn + 3.

from one Lagrangian level to another Lagrangian level is inessence a full remapping, and will not be any more desirablethan full remappings to Eulerian levels. This remapping willhowever be confined to a single level and not the full field.

Method III

The optimal method for remapping based on theconvergence and divergence of the Lagrangian flow, ishowever to demand that the thickness of a Lagrangian layermust not become too small or too large.

�1�p

k

< �p

k,L

< �2�p

k

(25)

where �1 =

23 could be used. In practice it has not been

neccessary to set an upper limit, i.e. �2 is set to a largenumber and only the lower constraint is enforced. Using

Figure 8. Flow of the LMCSL-LL scheme with dynamic remappingmethod III at individual levels, and tendency remapping to Lagrangianlevels. The four columns are respectively time step n, n + 1, n + 2 andn + 3.

this approach allows the Lagrangian levels to deviate muchmore than method I and II. This does however also allowmultiple layers to occupy the space where only one Eulerianlayer is present (and vice versa), i.e. greatly enhancingaccuracy in that area and decreasing the accuracy whereonly one Lagrangian layer spans multiple Eulerian layers.Therefore the minimum layer thickness should be chosenwith care.



Combined remapping

The fixed remapping method and the dynamic methods canalso be combined. When using dynamic remapping, it is notnecessary to remap the entire field to Eulerian levels usingthe LTS, but it can be done as well. A balance could possiblybe obtained between an almost fully Lagrangian flow andthe traditional quasi-Lagrangian flow.

5.3. Consistent vertical remapping

To further increase the consistency and the efficiencyof the scheme, a new vertical remapping routine hasbeen developed. The routine uses 1D cubic Lagrangeinterpolation. An equidistant Lagrangian polynomial, suchas the one used by Durran (2010), creates a third-orderpolynomial, matching � at the closest four grid points.In the vertical the grid is defined in pressure coordinates,which are irregularly spaced and changing through time. Acorresponding non-uniform cubic Lagrange polynomial canbe described by the following expression:

�

⇤

= � �1�2�6

�1�2�

i�2+↵�1�2

�1�4�7�

i�1

+

↵�1�6

�2�3�7�

i

�↵�2�6�3�4

�

i+1 (26)

with

�1 = 1 � ↵

�3 = 1 � �2

�4 = 1 � �1

�2 = �2 � ↵

�6 = ↵� �1

�7 = �2 � �1

where ↵ is the pressure distance between the grid point �i�2

and �⇤

, which is the ”departure” point. �1 is the distancebetween the grid point �

i�2 and �i�1, and �2 is the distance

between the grid point �i�2 and �

i

. These distances are allnormalized by dividing with the distance between the outergrid points �

i�2 and �

i+1. Equation (26) reduces to thetraditional formulation if the grid is equidistant, and is thusfully consistent with the bi-cubic horizontal interpolations,and the previously used cubic vertical interpolation foru, v, and T . The original vertical interpolation routine isformulated as a Newton polynomial thus giving the sameresult but without the possibility for re-using interpolationweights for multi-tracer efficiency. When formulated as aLagrange polynomial it also enables us to acquire massconservation using the general LMCSL approach describedin section 3.

bwk

=

w

kPK

l=1 wl

(27)

This method will generate over- and under-shoots as anytraditional higher-order interpolation, and should be usedin combination with a filter if the solution must be shapepreserving. This method does, however, mean that the actualremapping of tendencies is fast and efficient, in particularfor multiple tracers. A comparison between the efficiencyof the interpolations methods can be seen in 9.

In Figure 9 is shown four curves, which all matchesthe four grid values, �

i�2 = 8, �i�1 = 3, �

i

= 5, �i+1 = 2.

0.0 0.2 0.4 0.6 0.8 1.0�

0

2

4

6

8

10

�

LMCSL-NU

�1

= 1

3

�1

= 1

4

�1

= 1

5

�1

= 1

6

Figure 9. Example of cubic non-uniform representation. The solid (black)curve has �1 = 1

3 , the dot-dashed (red) has �1 = 14 , the dotted (blue) curve

has �1 = 15 , and the dashed (green) curve has �1 = 1

6 .

The solid (black) curve is the equidistant polynomial, ascreated by the traditional cubic interpolation. The remainingcurves show the polynomial when the grid is non-uniform.The dot-dashed (red) curve has �1 =

14 , the dotted (blue)

curve has �1 =

15 , and the dashed (green) curve has �1 =

16 .

The interpolation method will be referenced as LMCSL-NU(non-uniform).

6. 1D test

A simple one dimensional advection test has beenperformed to compare the new non-uniform LMCSLinterpolation with the traditional methods, i.e. SL andPPM2. The test is a solid body rotation on two non-uniformgrids. The two grids have a relatively large ratio betweenthe smallest and the largest cell§. In grid A the cells aresmoothly varying in size and in grid B the cells vary sharplybetween two sizes. The smoothly varying grid A is definedby

�x

i

= 1 + r sin

✓2⇡(i� 1)

N

◆(28)

and the sharply varying grid B is defined by,

�x

i

=

⇢0.5 i = even1.5 i = odd (29)

where �x

i

is the grid cell size of the i-th grid cell andN is the number of grid cells. r =

ratio�1ratio+1 is the ratio

factor. The ratio used in the test is 1:5. In test case A thevelocity is set to one in the entire domain, giving 100 stepsper revolution. In test case B the velocity is 1.25 since thedeparture points would otherwise coincide with the actualgrid points¶ (giving exact results).

Error analysis

To quantify the errors, the following common methods ofestimating errors is used.

§The cell size ratio in the Eulerian vertical grid is approximately 1:1.4 withonly 1:1.1 between neighboor cells. The Lagrangian grid can aquire muchlarger grid ratios, which is the reason that this test uses larger rations.¶We are forecasting cell averages and the arrival point is in the center ofeach cell.



l1-error

l1 =

PK

k=1 |�k �

k

|P

K

k=1 | k

|(30)

l2-error

l2 =

qPK

k=1

��

k

�

k

�2qP

K

k=1

�

k

�2 (31)

l

1

-error

l

1

=

max [|�k

�

k

|]max [|

k

|] (32)

where �k

is the simulation value and k

is the analytic ortrue value.

Table I and II show that both the PPM2 and LMCSL-NU are mass conserving and shape preserving if necessary.The PPM2 scheme returns slighlty better error scores, butthe LMCSL-NU method is less damping as can be seenin Table II, where the LMCSL-NU method combined withthe shape preserving filter, even after three revolutions, stillretains the initial maximum value. In section 15 it is alsoshown that the LMCSL-NU method is significantly moremulti-tracer efficient in real NWP applications.

7. Dynamic remapping

When using dynamic remapping, it is not possible to predictthe exact amount of remappings that is needed at eachtime step. Therefore while running the model, the totalnumber of dynamic remappings has been saved, so thedifference between the methods can be appreciated. Thedynamic remappings take additional time and will increasethe computational load. It is also in the very nature of thedynamic remappings, that the amount will change from timestep to time step. In an operational model it is unwanted,that the time used can not be determined precisely. It istherefore prefereable if the total amount of remappings isapproximately constant, so the time used per simulationis unchanged. Figure 11 illustrates the number of full

0 4 8 12 16 20 24hour

0

5

10

15

20

25

%of

tota

lle

vels

Dynamic remapping

LMCSL-D1

LMCSL-D2

LMCSL-D3

Figure 11. Total percentage of remapped levels per time step. The solid(red) curve is dynamic remapping method I, the dot-dashed (blue) curveis dynamic remapping method II, and the dotted (green) curve is dynamicremapping method III.

remappings, the individual dynamic remapping methodsrequires. The solid (red) curve is method I, the dot-dashed(blue) is method II, and the dotted (green) is the “ideal”method III. The x-axis is the time, the y-axis is thepercentage of the total number of vertical intersects in themodel, which is roughly 300.000. Method I requires almost

no remappings. Within the first two hours it stabilizes ataround 1.5%, and remains at that level after the initialincrease. Method II, which has the same constraints, butinstead remaps the Lagrangian levels to the allowed limit,shows a dramatical increase in the amount of remappingsneeded. It stabilizes, although not to quite as constant alevel, after approximately twelve hours just below 15%. It isexpected since the modified Lagrangian level is positionedexactly at the border, and it is very likely that an additionalremapping is required at the following time step. Method IIIshows a gradual increase, somewhat slower than method Iand method II, until it stabilizes at 2.5% after 20 hours.

0 2 4 6 8 10% of remappings

1

5

10

15

20

25

30

35

40

leve

l

Dynamic remapping - Vertical distribution

LMCSL-D1

LMCSL-D2

LMCSL-D3

Figure 12. Acumulated vertical distribution of remappings after one day.The solid (red) curve is dynamic remapping method I, the dot-dashed(blue) curve is dynamic remapping method II, and the dotted (green) curveis dynamic remapping method III.

The remappings are not distributed equally inthe vertical. Figure 12 shows the distribution of theaccumulated remappings after 24 hours. At that point allthree methods has stabilized. Method I and II are verysimilar even though the amount of remappings is a factorten different. In the upper part of the atmosphere, almostno remappings are needed, but in the middle layers theamount increases to a maximum between level 30 and35. At the surface the remappings decrease. Method IIreaches an equilibrium near the surface, where the numberof remappings stabilizes. This is again because the levelsthat has been remapped, stays at the allowed limit. MethodIII has a tiny maximum close to the top, where some layersconverge until they are remapped, and a large maxima close



0 10 20 30 40 50 60 70 80 90 100cell

0.0

0.2

0.4

0.6

0.8

1.0�

Test case A

0 10 20 30 40 50 60 70 80 90cell

0.0

0.2

0.4

0.6

0.8

1.0

�

Test case B

Analytic LMCSL-NU SL PPM2

Figure 10. One dimensional advection test on a highly non-uniform smoothly (upper) or sharply (lower) varying grid.

Table I. Error norms for test case A

Test Time steps l1 l2 l

1

�

mass

�

max

�

min

SL 100 0.1050 0.2004 0.3422 -1.3E-5 0.9427 0.0548PPM-2 100 0.0736 0.1621 0.3170 0.0 0.9473 0.0514PPM-2-SP 100 0.0654 0.1650 0.2972 0.0 0.9000 0.1000LMCSL 100 0.1054 0.2009 0.3434 0.0 0.9427 0.0546LMCSL-SP 100 0.0851 0.1958 0.3337 0.0 0.9000 0.1000SL 300 0.1441 0.2384 0.3662 3.4E-5 0.9706 0.0536PPM-2 300 0.0932 0.1860 0.3476 0.0 0.9433 0.0541PPM-2-SP 300 0.0810 0.1881 0.3250 0.0 0.9000 0.1000LMCSL 300 0.1445 0.2391 0.3674 0.0 0.9699 0.0541LMCSL-SP 300 0.1140 0.2294 0.3603 0.0 0.9000 0.1000

Table II. Error norms for test case B

Test Time steps l1 l2 l

1

�

mass

�

max

�

min

SL 80 0.1041 0.2013 0.3356 4.0E-1 0.9483 0.0491PPM-2 80 0.0903 0.1926 0.3902 0.0 0.9344 0.0623PPM-2-SP 80 0.0949 0.2042 0.3857 0.0 0.8958 0.1000LMCSL 80 0.1067 0.2062 0.3973 0.0 0.9485 0.0492LMCSL-SP 80 0.0866 0.2018 0.3865 0.0 0.9000 0.1000SL 240 0.1437 0.2391 0.3656 4.0E-1 0.9762 0.0527PPM-2 240 0.1164 0.2231 0.4069 0.0 0.9347 0.0734PPM-2-SP 240 0.1202 0.2281 0.3918 0.0 0.8808 0.1000LMCSL 240 0.1435 0.2425 0.4120 0.0 0.9828 0.0511LMCSL-SP 240 0.1145 0.2326 0.3999 0.0 0.9000 0.1000

to the surface where the layers are converging as well. In the

middle layers method III has almost no remappings since

the layers are allowed to move much more than with method

I and II (possibly several Eulerian levels).

8. Vertical numerical diffusion

Since the model is a fully operational NWP model, it isdifficult to assess the exact amount af unphysical numericalvertical diffusion, as it cannot be separated from the realdiffusion. The following test has therefore been constructedto quantify the vertical diffusion.



At the first time step three layers in the modelare initialized with a given value, the three layers arerespectively in the top, middle, and bottom of theatmosphere. Here the layers k = [5, 20, 35] have beenchosen, and the mass is initialized at 100 kg in everycell. Figure 13 shows the mean of the sum of the masspresent in each layer of the model, therefore giving us thevertical mass distibution at each time step. In the absense ofdiffusion, the mass would be constrained to the three initiallayers.

0 4 8 12 16 20 24hours

5

10

15

20

25

30

35

40

leve

l

LMCSL - default

0 4 8 12 16 20 24hours

5

10

15

20

25

30

35

40

leve

l

LMCSL - Fixed 1 hour

0 4 8 12 16 20 24hours

5

10

15

20

25

30

35

40

leve

l

LMCSL - Dynamic III

1 10 25 50 75 100% of emitted mass

Figure 13. Vertical diffusion after 24 hours forecast. Upper panel showsthe tradtional LMCSL scheme, middle panel shows the LMCSL-LL withfixed remapping every hour, and the lower panel shows the LMCSL-LLwith dynamic remapping 3.

The test has been run with three different settings -LMCSL, LMCSL-LL with dynamic remapping - methodIII, LMCSL-LL with fixed remapping every hour (6 steps).The duration is 24 hours of simulation corresponding to144 steps of 600 seconds. As expected the traditionalLMCSL scheme has the largest amount of vertical diffusion,but what is more interesting is the middle panel, wherethe effect of the fixed remapping can be seen. Theeffect is somewhat limited, which is surprising as the fullremappings are only one sixth of the traditional. But sincethe vertical remapping is quite accurate when the levels onlydiverge slightly as in the tradtional scheme, the remapping

0 4 8 12 16 20 24hour

80

85

90

95

100

%of

emis

sion

Vertical di�usion - Top

0 4 8 12 16 20 24hour

50

60

70

80

90

100

%of

emis

sion

Vertical di�usion - Middle

0 4 8 12 16 20 24hour

0

20

40

60

80

100

%of

emis

sion

Vertical di�usion - Bottom

LMCSL LMCSL-F1 LMCSL-D3

Figure 14. Vertical diffusion after 24 hours forecast. The lines indicatethe mass retained in the emission levels. The dashed (green) lines are thetradtional LMCSL scheme, the dot-dashed (black) lines are the LMCSL-LL with fixed remapping every hour, and the solid (red) lines are theLMCSL-LL with dynamic remapping 3.

becomes less so when the levels diverge more. This couldaccount for the reduced effect. The lower panel shows adrastic reducion of vertical diffusion, just as expected. Theupper and middle level experiences almost no diffusion,and the surface layers has slightly less diffusion as well.The surface layers do introduce a strong mixing, which isdirectly related to the real parameterized turbulent mixingin the boundary layer, but without the additional numericalmixing.

Figure 14 shows the mass retained in the initial levels(5, 20, and 35), again all schemes show some verticalmixing in the upper and middle layers, depending on themethod. In the lower layers the mixing is almost the samefor all methods, wich is fully consistent with the verticalmixing parameterized in the model.

9. Remapping efficiency

The increased amount of vertical interpolations (remap-pings) are the main factor in reducing the efficiency of theLMCSL-LL scheme. Therefore the new mass conserving



interpolation routine, LMCSL-NU, was created. Figure 15shows the amount of CPU-time used in a full remapping ofdifferent number of chemical species. The solid line (red) isthe LMCSL-NU method, the dot-dashed (blue) is the PPM2method, and the dotted (green) line is the non-conservingand non-monotone Nevilles algorithmk. It is evident thatthe PPM2-method is faster for a small number of fields,but already when four fields are remapped the new methodbecomes more efficient. The full Enviro-HIRLAM chemicalscheme currently uses 36 chemical species. The traditional

1 4 8 12 16 20 24 28 32 36remapped species

0.0

0.5

1.0

1.5

2.0

2.5

3.0

seco

nds

Remapping e�cency

PPM2

LMCSL-NU

Nevilles

Figure 15. Remapping efficiency. The x-axis is the number of remappedvariables, y-axis is the time used in seconds. The solid (red) line is theLMCSL-NU method, the daot-dashed (blue) line is the PPM2 method, andthe dotted (green) line is using Nevilles algorithm.

cubic interpolation, using Nevilles algorithm, has no multi-tracer efficiency and it cannot be used for mass conservinginterpolations. But since the LMCSL-NU scheme uses thenon-conserving interpolation weights, the same weights canbe used for all interpolations, also enhancing consistency.

10. NWP forecast

A thorough validation calls for a long term comparisonof forecast skill in simulations based on our new quasi-Lagrangian coordinate scheme with that in a referenceoperational model. Such validation is beyond the scopeof the present paper, and instead, to illustrate that thenew scheme performs realistically, we have comparedmeteorological fields in a single forecast. The referencemodel we compare with is HIRLAM (Unden et al. 2002)in a traditional NWP setup, which until recently has beenused operationally at a number of weather forecast centersin Europe. We have chosen the validation period 23rdto the 26th of October 1994 at 12 UTC. This synopticsituation is from the ETEX experiment period, whichoriginally was used for validating European dispersionmodels (Graziani et al. (1998) and Nodop et al. (1998)).As an example Figure 16 shows mean sea level pressureand temperature at 850 hPa in the verifying ERA-Interimre-analysis, and in 48 hour forecasts with the standardand the LMCSL-LL versions of HIRLAM valid on 25thOctober 1994 at 00 UTC. Similarly, Figure 17 shows the6 hourly accumulated precipitation in the two versions ofHIRLAM. In general LMCSL-LL is quite similar to thestandard version of HIRLAM, and we therefore concludethat the new dynamical core of LMCSL-LL provide

kNevilles algorithm was originally used for vertical interpolations of thenon-density variables, i.e. wind speed and temperature. It is of third orderas the LMCSL-NU interpolation.

”plausible” forecasts. As expected there are, however,some differences. In particular it is worth mentioning thesomewhat different distribtion of precipitation with e.g.a stronger development of the frontal zone over easternEuropa in the standard HIRLAM, and in general, forthis particular case, the standard HIRLAM shows strongerprecipitation. No attempts have been made to verify thesimulated precipitation against station data.

11. Conclusions

In this paper a new numerical scheme using a Lagrangianvertical coordinate has been introduced and tested against amore traditional quasi-Lagrangian numerical scheme withvertical remappings taking place every time step. Bothschemes are based on the LMCSL method which conservesmass locally at very little additional cost. The Lagrangiancoordinate scheme is implemented in an operational NWPmodel with the capability for online coupled chemistry. Aspart of the development a new vertical interpolation routinehas been developed and tested using a traditional rotationtest with a step function on a variable resolution.

The new scheme shows a decrease in unphysicalnumerical mixing when running in fully Lagrangian mode,where the levels are never remapped back to the Eulerianmodel levels. However, even a relativily small amountof full remappings, introduces a considerable amount ofnumerical diffusion in the vertical direction. The schemeis very multi-tracer efficient, as the additional prognosticvariables, such as chemical species, only increases theoverall computing cost by a minimum. The LMCSL-LLscheme is also shown to be very flexible, by enabling it tobe run both in traditional mode (LTS = 0) or with manyconfigurations for the remapping strategies including; fixed,dynamical, and combined remapping. The scheme producesrealistic forecasts as compared to a traditional NWP model.

The new interpolation routine LMCSL-NU is shown tobe very efficient and robust when used on very non-uniformgrids, while still returning a shape preserving and massconserving solution, with a very simple and local filter.

Acronyms

NWP Numerical weather predictionACT Atmospheric chemical transportSL Semi-LagrangianSISL Semi-implicit semi-LagrangianLMCSL Locally mass conserving semi-LagrangianLMCSL-LL LMCSL Lagrangian levelsLMCSL-NU LMCSL non-uniformLMCSL-M LMCSL monotonicLMCSL-MS LMCSL simple monotonicHIRLAM High resolution limited area modelCISL Cell integrated semi-LagrangianPPM2 Piecewise parabolic methodLTS Lagrangian time stepETEX European Tracer Experiment

Acknowledgement

The present study was a part of the research of the Centerfor Energy, En- vironment and Health (CEEH), financedby The Danish Strategic Research Program on SustainableEnergy under contract no 09-061417.



0� 10�E 20�E10�W

40�N

50�N

60�N9

9

0

9

9

5

1

0

0

0

1

0

0

5

1

0

1

0

1

0

1

0

1

0

1

5

1

0

1

5

1

0

1

5

1

0

1

5

1

0

2

0

1

0

2

0

1

0

2

0

Verifying reanalysis

0� 10�E 20�E10�W

40�N

50�N

60�N

9

9

0

9

9

5

1

0

0

0

1

0

0

5

1

0

1

0

1

0

1

0

1

0

1

5

1

0

1

5

1

0

1

5

1

0

2

0

HIRLAM default+48h

0� 10�E 20�E10�W

40�N

50�N

60�N

9

9

0

9

9

5

1

0

0

0

1

0

0

5

1

0

1

0

1

0

1

0

1

0

1

5

1

0

1

5

1

0

1

5

1

0

2

0

LMCSL-LL+48h

260 263 266 269 272 275 278 281 284 287 290850 mb temperature [C�]

Figure 16. Forecasts of mean sea level pressure and 850 hPa temperature compared with verifying ERA40 reanalysis. From left to right: ERA-Interimre-analysis on 25th October 1994 00UTC, 48 hour forecast with the standard HIRLAM, and 48 hour forecast with the LMCSL-LL (method III). Thecontour lines are the mean sea level pressure and the colored contours are the temperature at 850 mb.

0� 10�E 20�E10�W

40�N

50�N

60�N

9

9

0

9

9

5

1

0

0

0

1

0

0

5

1

0

1

0

1

0

1

0

1

0

1

5

1

0

1

5

1

0

1

5

1

0

2

0

HIRLAM default+48h

0� 10�E 20�E10�W

40�N

50�N

60�N

9

9

0

9

9

5

1

0

0

0

1

0

0

5

1

0

1

0

1

0

1

0

1

0

1

5

1

0

1

5

1

0

1

5

1

0

2

0

LMCSL-LL+48h

0.5 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.06 hour total precipitation [kg/m2]

Figure 17. Standard HIRLAM (left) and LMCSL-LL (right) forecasts of mean sea level pressure (same as Figure 16 and precipitation. The coloredcontours show 6 hourly accumulated precipitation up to 25th October 1994 00 UTC

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http://dx.doi.org/10.1002/qj.307

http://dx.doi.org/10.1002/qj.307

http://dx.doi.org/10.1016/j.jcp.2009.10.036

http://dx.doi.org/10.1016/j.jcp.2009.10.036

http://www.sciencedirect.com/science/article/pii/S004579300800056X

http://www.sciencedirect.com/science/article/pii/S004579300800056X

a locally mass-conserving semi-lagrangian regional nwp ...nate is understood that the pressure...

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