the utility of 6th-order, monotonic, numerical diffusion in the ...the ux and the sign of the...

The utility of 6th-order, monotonic, numerical diffusion in theAdvanced Research WRF Model

Jason C. Knievel,�

George H. Bryan, and Joshua P. HackerNational Center for Atmospheric Research, Boulder, Colorado, USA

1. Introduction

One of the great strengths of the WRF Model (Ad-vanced Research core) is its high effective resolution,which means that the model’s energy spectra begin todecay at shorter wavelengths than do the spectra of someother models (Skamarock 2004). The WRF Model’shigh effective resolution is achieved partly through thescale-selective diffusion implicit in the model’s advec-tion schemes (Skamarock 2004). Each of the model’sodd-ordered, up-wind biased advection schemes (e.g.,fifth) is equivalent to a centered scheme of the nexthigher order (e.g., sixth) plus a diffusive term (Hunds-dorfer et al. 1995; Wicker and Skamarock 2002). Thecoefficient in this diffusive term is proportional to thespeed of the advecting wind, so in light wind the diffu-sion is weak. It turns out that in some cases the diffusionis much too weak to filter poorly resolved kinematicalfeatures with wavelengths of 2–4 times the grid interval.These poorly resolved features can grow until they dom-inate fields of horizontal divergence and vertical velocityin the daytime boundary layer (Fig. 1).

To mitigate the problem, we added to the WRF Modelan explicit numerical diffusion scheme proposed by Xue(2000). The scheme is 6th-order, so its scale-selectivitypreserves the WRF Model’s high effective resolution. Inthis paper, we explain the diffusion scheme and why itis well suited to the WRF Model. Then we show resultsof simulations before and after the scheme was imple-mented.

2. Model and methods

2.1 WRF Model

The WRF Model we tested and modified is version2.0.3.1 of the Advanced Research core, released in De-cember 2004, which was the latest version available atthe time of writing. For our control simulation we usedthe unmodified model. To test the effects of additionalnumerical diffusion, we applied only the modificationsdescribed below (except for one additional, small change

�Corresponding author: Dr. Jason Knievel, NCAR, 3450 Mitchell

Lane, Boulder, CO, USA 80301; [email protected].

Divergence (contours) and terrain elevation (shaded)

Unmodified WRF Model

Domain: 3 Model level: 1

23-h forecast valid 2300 UTC 14 July 1998

Figure 1: Horizontal cross section from the control simulation by theunmodified WRF Model, valid 2300 UTC 14 July 1998 on domain3 of 4. Horizontal divergence at the lowest model level is contoured(positive in red and negative in blue) and terrain elevation (m AMSL)is shaded. The heavy green line marks the location of the vertical crosssection in Fig. 2. The solid black lines mark the perimeter of the GreatSalt Lake and the border between Utah and Nevada.

to the microphysics driver, which will be included in thenext official release of the model and does not directlybear on the results herein).

Details of the simulations are listed in Table 1. Weused four domains, with domains 2–4 nested inside thenext larger. Domain 1, the largest, encompassed a2940 km � 2520 km region of the eastern Pacific Oceanand western United States. Meteorological initial andboundary conditions are taken from the NCEP-NCARReanalysis (2.5

�grid interval). Most other initial and

boundary conditions (soil temperature, sea-surface tem-perature, etc.) are taken from the Eta Data AssimilationSystem (EDAS; 40-km grid interval). The skin tempera-ture of the Great Salt Lake is retrieved from data from theAdvanced Very High Resolution Radiometer (AVHRR).Our choice of meteorological data may seem surprisingly

3.15

WRF/MM5 Users' Workshop - June 2005

Table 1: Configurations of the WRF Model (Advanced Research core)that we tested. The control simulation, which is in bold text, was madewith the unmodified version 2.0.3.1 of the model. Multiple values inthe first two rows of the right column refer to domains 1–4.

Horiz. grid intervals (km) 30.0, 10.0, 3.3, 1.1Vertical levels 32, 32, 32, 32Temporal integration Runge-Kutta 3rd orderHorizontal advection 5th orderVertical advection 3rd orderCumulus convection Kain-Fritsch (Eta) on domains 1,2Microphysics Lin (6 class)Boundary layer Yonsei University (YSU)

Mellor-Yamada-JanjicMedium Range Forecast (MRF)none

Surface layer Monin-ObukhovJanjic Eta

Land surface NoahShortwave radiation DudhiaLongwave radiation Rapid radiative transfer model

coarse, but we wanted to provide to the model only thelarge-scale conditions so that locally driven circulationswould develop with little other mesoscale influence.

2.2 Numerical diffusion

Following the technique proposed by Xue (2000), wechose to apply explicit numerical diffusion by modify-ing the WRF Model’s time-dependent calculations, gen-erally represented by��

�� (1)

wherein the term on the far right is the 6th-order diffu-sive filter we added,

�is some predicted variable, � is

the sum of the terms already represented in the model,and � is a coefficient of diffusion. The coefficient canbe adjusted through the variable khdif in the namelistin order to specify the amount of diffusion in one timestep. Because diffusion rate is also a function of grid in-terval and time step, in order to maintain a constant rateof diffusion across all domains, the coefficient must beset lower for finer domains. Importantly, the coefficientis not a function of wind speed, in contrast to the im-plicit diffusion in the unmodified WRF Model. A goodreference for how the coefficient is related to the rate ofdiffusion is section 2.4.3 of Durran’s (1999) text.

In our modification of the WRF Model, we formulatedthe diffusion scheme in flux form:

�� ! � (2)

wherein �� "�#��$%��! $'& (3)

Xue (2000) calls � � a “diffusive flux.” Unfortunately,this 6th-order diffusion scheme suffers from Gibbs os-cillations, as do all such schemes of orders higher than2 (Hundsdorfer et al. 1995; Xue 2000). Gibbs oscilla-tions introduce new extrema to the simulated fields andcan intensify existing extrema through unphysical, up-gradient diffusion. To eliminate these undesirable prop-erties, we again followed the example of Xue (2000) andadded a simple flux limiter to the diffusion scheme. Thelimiter constrains diffusion to be down-gradient by re-setting the diffusive flux, � �(� to zero when the sign ofthe flux and the sign of the variable’s gradient are oppo-site (see Eq. 2). The addition of the limiter thus makesthe diffusion scheme monotonic.

Potential temperature (contours) and vertical velocity (shaded)


Domain: 3


Figure 2: Vertical cross section from the control simulation by the un-modified WRF Model, valid 2300 UTC 14 July 1998 on domain 3 of4. The location of the cross section is marked in Fig. 1. Potential tem-perature is contoured every 1 K and vertical velocity is shaded every5 cm s )!* (ascent in blue and descent in red). Black vectors, which areplaced at each horizontal grid point, indicate the flow in the plane ofthe cross section.

A side effect of the flux limiter is an effective reductionin diffusion rate. For example, although for our simula-tions we chose a coefficient of diffusion that produced anominal diffusion rate of 24% per time step, once the fluxlimiter was added, the effective rate was slightly less thanthat. The coefficient’s rather aggressive setting is some-what arbitrary; we have tested only a few other settingsand do not necessarily advocate this one, although it suitsour simulations.


2.3 Spectral analyses

In order to quantify the effect of explicit numerical diffu-sion on small features and phenomena in our simulations,we computed two-dimensional (2-D) spectra of verti-cal velocity, generally following the method of Errico(1985). First, the method forces vertical velocity on amodel level to be periodic in both directions. Next, a 2-D FFT (Fast Fourier Transform) algorithm decomposesthe field, and the resultant 2-D complex coefficients arebinned according to their approximate wavenumber tocompute the power.


Modified WRF Model



Figure 3: Same as Fig. 1 except for a simulation by the WRF Modelmodified to include explicit monotonic diffusion. The heavy green linemarks the location of the vertical cross section in Fig. 4.

3. Simulations

3.1 Control

The need for additional numerical diffusion in the WRFModel first became apparent when we examined fields ofdivergence and vertical velocity in simulations of terrain-forced circulations in the U. S. Great Basin (Figs. 1 and2). The fields are noisy in regions of low wind speed andwhere the boundary layer is well developed and approx-imately statically neutral, as is typical over land duringthe day. There is little or no noise where wind speed ishigh and in statically stable environments, such as oc-curs over cold water or at night. The wavelengths of thespurious patterns of divergence and vertical velocity areconsistently 2–4 times the grid interval. No domain isfree of the noise, although its amplitude varies inversely

Potential temperature (contours) and vertical velocity (shaded)

Modified WRF Model

Domain: 3


Figure 4: Same as Fig. 2 except for a simulation by the WRF Modelmodified to include explicit monotonic diffusion. The location of thecross section is marked in Fig. 3.

with the grid interval, as one might expect. Above theboundary layer, divergence and vertical velocity appearmore physically realistic (Fig. 2).

To get a better sense of the noise’s pervasiveness,we ran simulations with other model configurations (Ta-ble 1). The details and severity of the noise changed fromone configuration to the next, but some noise was alwayspresent. In particular, switching among boundary-layerschemes did not eliminate the noise, although the MRFscheme was slightly less noisy than the YSU scheme.

3.2 Modified

The addition of explicit numerical diffusion greatly mit-igates the grid-scale noise (cf. Figs. 1 and 3, and Figs. 2and 4), except at the edges of each domain, where the dif-fusion scheme’s 7-point stencil was not applied. Patternsin the diffused fields are not so obviously associated withthe model grid. Instead, the patterns appear to be influ-enced more by the terrain. For example, in Fig. 3 the con-tinuous, wavy arc of convergence (blue contours) imme-diately west and south of the Great Salt Lake is the lead-ing edge of the lake breeze (e.g., Rife et al. 2004). Otherbands of convergence tend to be collocated with tops ofhigh terrain, where up-slope winds meet (e.g., Whiteman2000). In Fig. 1 such phenomena are overwhelmed bygrid-scale noise. Indeed, the great extent to which thepatterns in Fig. 1 are due to small, insufficiently resolvedfeatures is dramatically apparent in Fig. 5, which shows


the difference between simulations before and after ex-plicit diffusion was added: Fig. 5 looks very much likeFig. 1.

The similarity between Figs. 1 and 5 raises the ques-tion of just how much of the information in the controlsimulation is removed by the monotonic diffusion. Spec-tra show that the scheme has little effect on scales largeenough to be well resolved (Fig. 6). Only features andphenomena smaller than + 6 times the grid interval arestrongly filtered. Without filtration (blue line in Fig. 6)the model’s spectrum is too flat at small scales.


Difference between unmodified and modified WRF Models



Figure 5: Same as Fig. 1 except the divergence is the difference be-tween simulations by the unmodified and modified WRF Models—thatis, without and with explicit monotonic diffusion, respectively.

4. Final remarks

Once we became familiar with the noise in oursimulations—in what fields it appeared, and where andwhen it was likely to occur—it was easy to find it inour colleagues’ simulations as well, some involving con-figurations and versions of the model that we did nottest. This anecdotally suggests that explicit numericaldiffusion is necessary for simulating many environmentsand should be considered a standard tool in the WRFModel. As mentioned above, environments of weakwind are especially problematic for the unmodified WRFModel, so model users who try to simulate terrain-driven,mesoscale circulations in weak synoptic forcing wouldgreatly benefit from the scheme we implemented.

Finally, we have not yet determined the origin of thegrid-scale noise that can dominate simulations that lack

wavelength (km)

po

we

r

Spectra of vertical motion

2300 UTC 14 July 1998

Modified WRF Model


Figure 6: Spectra of vertical velocity in simulations by the unmodifiedand modified WRF Models—that is, without and with explicit mono-tonic diffusion, respectively.

additional diffusion; this is an important topic of on-going research.

Acknowledgments. This work was funded primarilyby the U. S. Army Testing and Evaluation Command(ATEC). The National Center for Atmospheric Researchis sponsored by the National Science Foundation.

REFERENCES

Durran, D. R., 1999: Numerical Methods for Wave Equations in Geo-physical Fluid Dynamics. Springer, 465 pp.

Errico, R. M., 1985: Spectra computed from a limited area grid. Mon.Wea. Rev., 113, 1554–1562.

Hundsdorfer, W., B. Koren, M. van Loon, and J. G. Verwer, 1995: Apositive finite-difference advection scheme. J. Comput. Phys., 117, 35–46.

Rife, D. L., C. A. Davis, and Y. Liu, 2004: Predictability of low-level winds by mesoscale meteorological models. Mon. Wea. Rev., 132,2553–2569.

Skamarock, W. C., 2004: Evaluating mesoscale NWP models usingkinetic energy spectra. Mon. Wea. Rev., 132, 3019–3032.

Whiteman, C. D., 2000: Mountain Meteorology: Fundamentals andApplications. Oxford University Press, 355 pp.

Wicker, L. J., and W. C. Skamarock, 2002: Time-splitting methodsfor elastic models using forward time schemes. Mon. Wea. Rev., 130,2088–2097.

Xue, M., 2000: High-order monotonic numerical diffusion andsmoothing. Mon. Wea. Rev., 128, 2853–2864.


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