The Advanced Regional Prediction System (ARPS) – A multi ...

1 Center for Analysis and Prediction of Storms, University of Oklahoma, Norman OK 730192 School of Meteorology, University of Oklahoma, Norman OK 73019

The Advanced Regional Prediction System (ARPS) ± A multi-scalenonhydrostatic atmospheric simulation and prediction model.Part I: Model dynamics and veri®cation

M. Xue1;2, K. K. Droegemeier1;2, and V. Wong1

With 19 Figures

Received April 14, 2000Revised July 17, 2000

Summary

A completely new nonhydrostatic model system known asthe Advanced Regional Prediction System (ARPS) has beendeveloped in recent years at the Center for Analysis andPrediction of Storms (CAPS) at the University of Oklahoma.The ARPS is designed from the beginning to serve as aneffective tool for basic and applied research and as a systemsuitable for explicit prediction of convective storms as wellas weather systems at other scales. The ARPS includes itsown data ingest, quality control and objective analysispackages, a data assimilation system which includes single-Doppler velocity and thermodynamic retrieval algorithms,the forward prediction component, and a self-contained post-processing, diagnostic and veri®cation package.

The forward prediction component of the ARPS is a three-dimensional, nonhydrostatic compressible model formulatedin generalized terrain-following coordinates. Minimumapproximations are made to the original governing equa-tions. The split-explicit scheme is used to integrate thesound-wave containing equations, which allows the hori-zontal domain-decomposition strategy to be ef®cientlyimplemented for distributed-memory massively parallelcomputers. The model performs equally well on conven-tional shared-memory scalar and vector processors. Themodel employs advanced numerical techniques, includingmonotonic advection schemes for scalar transport andvariance-conserving fourth-order advection for other vari-ables. The model also includes state-of-the-art physicsparameterization schemes that are important for explicitprediction of convective storms as well as the prediction of¯ows at larger scales.

Unique to this system are the consistent code stylingmaintained for the entire model system and thorough internaldocumentation. Modern software engineering practices areemployed to ensure that the system is modular, extensibleand easy to use.

The system has been undergoing real-time prediction testsat the synoptic through storm scales in the past several yearsover the continental United States as well as in part of Asia,some of which included retrieved Doppler radar data andhydrometeor types in the initial condition.

As the ®rst of a two-part paper series, we describe here-in the dynamic and numerical framework of the model,together with the subgrid-scale turbulence and the PBLparameterization. The model dynamic and numerical frame-work is then veri®ed using idealized and realistic mountain¯ow cases and an idealized density current. Other physicsparameterization schemes will be described in Part II, whichis followed by veri®cation against observational data of thecoupled soil-vegetation model, surface layer ¯uxes and thePBL parameterization. Applications of the model to thesimulation of an observed supercell storm and to the pre-diction of a real case are also found in Part II. In the lattercase, a long-lasting squall line developed and propagated acrossthe eastern part of the United States following a historicalnumber of tornado outbreak in the state of Arkansas.

1. Introduction

Three-dimensional nonhydrostatic modeling ofatmospheric convection started in the mid-1970s

Meteorol. Atmos. Phys. 75, 161±193 (2000)

(e.g., Steiner, 1973; Miller and Pearce, 1974;Schlesinger, 1975; Tapp and White, 1976; Clark,1977; Klemp and Wilhelmson, 1978), followingthe success of earlier 2-D modeling studies thatused nonhydrostatic equations in either primitiveform (Lilly, 1962) or vorticity form (Orville,1968). These and other studies signi®cantlyadvanced our understanding of thunderstormdynamics (Lilly, 1979; Klemp, 1987) as well asother small-scale phenomena. However, modelingresearch on the storm-scale (de®ned here looselyas the scale at which nonhydrostatic dynamicsare important and attention is paid to individualstorm elements, e.g., updrafts and downdrafts)remained in the simulation mode for much ofthe last two decades. These simulationstypically used horizontally homogeneous initialconditions with arti®cial perturbations to initiateconvection.

Two major developments in the recent yearsprovided the impetus for moving from a mode ofconvective storm simulation to one of prediction.The ®rst is the deployment of about 160 Dopplerradars (Crum and Albert, 1993) in the U.S. thatprovides nearly continuous single-Doppler cover-age of spatial and temporal scales relevant tostorm prediction. The second concerns with tech-niques for retrieving unobserved quantities fromsingle-Doppler radar data to yield a consistent setof mass and wind ®elds appropriate for initializing

a storm-scale prediction model (e.g., Kapitza,1991; Liou et al., 1991; Sun et al., 1991; Qiu andXu., 1992; Shapiro et al., 1995; Sun and Crook,1994). Perhaps equally important for the realiza-tion of numerical weather prediction (NWP) onthe storm scale is the advent and accessibility ofincreasingly more powerful parallel-processingsupercomputers.

In 1989, the Center for Analysis and Predic-tion of Storms was established at the Universityof Oklahoma as one of the National ScienceFoundation's ®rst 11 Science and Technology(S&T) Centers. Its formal mission is to demon-strate the practicability of storm-scale numericalweather prediction and to develop, test, andvalidate a regional forecast system appropriatefor operational, commercial, and research appli-cations. Its ultimate vision is to make available afully functioning stormscale NWP-systemaround the turn of the century (Lilly, 1990;Droegemeier, 1990).

Central to achieving this goal is an entirely newthree-dimensional, nonhydrostatic model systemknown as the Advanced Regional PredictionSystem (ARPS). It includes a data ingest, qualitycontrol, and objective analysis package, a single-Doppler radar parameter retrieval and assimilationsystem, the prediction model itself, and a post-processing package. These components are illu-strated in Fig. 1.

Fig. 1. Principal elements of the ARPS model system. These include the ARPS data assimilation system (ARPSDAS), theforward prediction component and the post-processing tools used for product generation and forecast veri®cation. ARPSDASfurther includes the data ingest and analysis component known as the ARPS data analysis system (ADAS), Doppler radar dataretrieval algorithms and the 4-D variational data assimilation system

162 M. Xue et al.

In planning for its development, the ARPS wasrequired to meet a number of criteria. First, it hadto accommodate, through various assimilationstrategies, new data of higher temporal and spatialdensity (e.g., WSR-88D data) than had tradition-ally been available. Second, the model had toserve as an effective tool for studying the dyna-mics and predictability of storm-scale weather inboth idealized and more realistic settings. It mustalso handle atmospheric phenomena ranging fromregional scales down to micro-scales as interac-tions across this spectrum are known to haveprofound impacts on storm-scale phenomena.These needs required that the model have a¯exible and general dynamic framework andinclude comprehensive physical processes. Thesystem should also run ef®ciently on massivelyparallel computers. In short, it was our goal todevelop a model system that can be usedeffectively for both basic atmospheric researchand operational numerical weather prediction, onscales ranging from regional to micro-scales.

In Sect. 2 of this paper, we will describe thedynamic framework of the forward predictioncomponent of the ARPS system. We will describein Sect. 3 several options of subgrid-scaleturbulence parameterization together with a 1.5-order turbulent kinetic energy (TKE)-based pla-netary boundary layer (PBL) parameterizationscheme. Other physics parameterizations will bedetailed in Part II (Xue et al., 2000) and will bebrie¯y outlined in Sect. 4. The numerical treat-ment of various processes in the model ispresented in Sect. 5 with additional details foundin the appendices. Section 6 discusses the com-putational aspects of the model, and Sects. 7 and 8verify the dry dynamics of the model usingmountain ¯ows and a nonlinear density current. Asummary is found in Sect. 9.

2. Dynamic equations

2.1. Historical perspective

Three-dimensional nonhydrostatic models can bedivided into two broad categories: those contain-ing fast acoustic modes (Tapp and White, 1976;Klemp and Wilhelmson, 1978, KW hereafter) andthose that ®lter such modes via certain type ofanelastic approximation (Miller and Pearce, 1974;Schlesinger, 1975; Clark, 1977; Xue and Thorpe,

1991). For the former, commonly referred to ascompressible models, the acoustic waves must betreated in special ways to attain computationalef®ciency. Tapp and White (1976) used a semi-implicit integration scheme that is absolutelystable for linearized sound waves, while Klempand Wilhelmson (1978) employed a mode-split-ting technique where the acoustic waves andslow modes are integrated separately using differ-ent time steps. In the latter case, the verticalacoustic modes are usually treated implicitly toremove the time step limitation from these modesdue to the Courant-Fredrichs-Lewy (CFL) stabilitycondition.

In the anelastic (sound-proof) models, a prog-nostic equation for pressure (or alternativelydensity) is absent, and the pressure (or geopoten-tial height in pressure-based coordinates) has to bediagnosed from an elliptic equation derived fromthe equations of motion. In order to ®lter outacoustic modes, certain approximations have to bemade (see, e.g., discussion by Durran, 1989).

The mode-splitting technique has gained con-siderable popularity since KW because of itssimplicity and effectiveness (Tripoli and Cotton,1982; Chen, 1991; Tripoli, 1992; Dudhia, 1993;Hodur, 1997). An attractive feature of models usingthis approach is that all computations are local tothe grid points involved in the ®nite differencestencil, making their implementation on distrib-uted-memory parallel processor (PP) computersstraightforward through the use of domain decom-position strategies (Johnson et al., 1994; Droege-meier et al., 1995). Different from anelasticsystems, the compressible system of equationsdoes not have to make any approximation, makingit suitable to a wider range of applications.

The semi-implicit method used by Tapp andWhite (1976) for compressible systems has inrecent years been adopted by other models(Tanguay et al., 1990), and has been furtherextended to include linear gravity wave modes(Cullen, 1990) so as to remove its time steplimitation. Because of its absolute stability withrespect to modes treated implicitly, this method isoften combined with semi-Lagrangian advectionschemes (Tanguay et al., 1990; Golding, 1990) toachieve high computational ef®ciency. In practice,however, the ef®ciency of such schemes has to beconsidered together with solution accuracy. Forexample, it is known that implicit schemes distort

The Advanced Regional Prediction System (ARPS) ± A multi-scale nonhydrostatic atmospheric simulation 163

(slow down) gravity waves when used with largetime steps (Tapp and White, 1976). Semi-implicitsystems usually involve solving a global ellipticequation, making their ef®cient implementationon distributed-memory parallel computers lessstraightforward.

Based on the above considerations, we chooseto use a fully compressible system of equationsand solve them using the ` split-explicit'' timeintegration method.

2.2. The governing equations of ARPS

The governing equations of the ARPS includeconservation equations for momentum, heat,mass, water substance (water vapor, liquid andice), subgrid scale (SGS) turbulent kinetic energy(TKE), and the equation of state of moist air.Among the three state variables, i.e., temperature,pressure and density, prognostic equations for twoof them are needed and the third variable can bediagnosed from the equation of state.

For the temperature, modelers usually choosebetween temperature (e.g., Dudhia, 1993), andpotential temperature (e.g., KW). Some modelersfavor ice-liquid potential temperature (e.g., Tripoliand Cotton, 1981). In the ARPS, we choose topredict potential temperature and pressure thendiagnose density. The potential temperature ischosen because it is conservative for adiabaticprocesses. The ice-liquid potential temperature issupposed to be conserved even in the presence ofphase changes, but its de®nition involves approx-imations.

For the pressure equation, modelers again havethe choice of using pressure or Exner function asthe prognostic variable. Most existing compres-sible models predict the Exner function instead ofpressure (e.g., Klemp and Wilhelmson, 1978;Tapp and White, 1976), but we choose to predictpressure. In such a case, the pressure gradientforce (PGF) is written as in the original Navier-Stokes equations (e.g., Batchelor, 1967), so that afully conservative form of the momentum (notvelocity) equations can be formulated, both ana-lytically and numerically.

The ARPS governing equations are ®rst writtenin a Cartesian coordinate projected onto a planetangent to or intercepting the earth's surface.Using standard mathematical relations (Haltinerand Williams, 1980) for the transformation from a

local Cartesian space on the sphere to map pro-jection space, we obtain the following equationsof motion:

_u � ÿmpx�ÿ1 � � f � fm�vÿ ~f wÿ uwaÿ1 � Fu;

�1:1�_v � ÿmpy�

ÿ1 ÿ � f � fm�uÿ vwaÿ1 � Fv;

�1:2�_w � ÿpz�

ÿ1 ÿ g� ~f u� �u2 � v2�aÿ1 � Fw:

�1:3�In the above and in the equations to follow, the dotoperator denotes the total time derivative, e.g.,_u � du=dt, and subscripts t, x, y, z, �, � and �denote partial temporal or spatial derivative, e.g.,ux � @u=@x. In obtaining (1.1±3), no approxima-tion is made other than that the ellipticity of theearth is neglected and the atmosphere is assumedto be thin so that the radius is replaced by themean earth radius at the sea level, a. Note that thespatial derivative of map factor due to curvatureare retained in fm � umy ÿ vmx � u tan��=a, asare the Coriolis terms due to vertical motion(those involving ~f ). The de®nitions of othersymbols are found in Appendix A. Note that forthis system, only gravitational, pressure gradientand frictional forces (F terms) can change kineticenergy. All other terms cancel each other in thekinetic energy equation.

The equations of state for moist air (see Dutton,1986), mass continuity, heat energy conservation,and conservation of hydrometeor species are,respectively,

� � p�RdT�ÿ1�1ÿ qv� � qv�ÿ1��1� qv � qli�;�1:4�

_� � ÿ�fm2��u=m�x � �v=m�y� � wzg; �1:5�_� � _Q�Cp��ÿ1; �1:6�_q � Sq: �1:7�Here _Q denotes heat source, and Sq representssources due to moist processes.

2.3. The curvilinear coordinate system

The actual equations of the ARPS are written in acurvilinear coordinate system ��; �; �� de®ned by

� � ��x�; � � ��y�; and � � ��x; y; z�: �2�

164 M. Xue et al.

This coordinate system is a special case of thefully three-dimensional curvilinear system sinceconstant surfaces of � and � remain parallel tothose of constant x and y, respectively. The verticaltransformation allows grid stretching and ensuresthat the lower boundary conforms to the terrain.The horizontal transformation allows horizontalgrid stretching. Equation (2) represents a transfor-mation that maps a domain with stretched grid andirregular lower boundary to a regular rectangulardomain with equal grid space in each direction.We call the latter the computational domain.

The governing equations for ¯uid motion in afully 3-D curvilinear system can be found inThompson et al. (1985), Sharman et al. (1988) andShyy and Vu (1991). Following their work, we usethe Cartesian instead of the contravariant velocitycomponents as the basic dependent variables. Asshown in Sharman et al. (1988), the Cartesianvelocity components, u, v and w can be expressedas functions of the contravariant velocities Uc, Vc

and Wc and vice versa. For the transformationde®ned by (2), which is a special case of the fully3-D curvilinear transformation, we have

Uc � uJ3=��Gp� u=x�;V

c � vJ4=��Gp� v=y�;

and

Wc � �uJ1 � vJ2 � wx�y��=��Gp

; �3�where

J1 � ÿz�y�; J2 � ÿz�x�; J3 � z�y�; J4 � z�x�

and��Gp� z�x�y�: �4�

J1; J2; J3 and J4 are Jacobians of transformationand

��Gp

is the determinant of the Jacobian matrixof transformation from the ��; �; �� system to the�x; y; z� system. It is clear that Uc differs from uby a factor of x�, which is the grid stretchingfactor in the x-direction. The same is true in the ydirection. The formula for Wc is more compli-cated because this component is not orthogonalto the other velocity components.

The transformation relations for spatial deri-vatives from �x; y; z� to ��; �; �� coordinates are

�x � ��J3�� J1�� =��Gp

;

�y � ��J4�� J2��=��Gp

; and

�z � �x�y��=��Gp

: �5�

Most terrain-following coordinate models (e.g.,Clark, 1977; Pielke and Martin, 1981) de®ne thecoordinate transform therefore the transformationJacobians analytically. In the ARPS, the computa-tional grid is de®ned numerically and thereforecan be arbitrary. The Jacobians are calculatednumerically according to (4). This allows foradditional ¯exibility. In fact, the grid can be madetime dependent (Fiedler et al., 1998). The onlyrequirement for the grid generation is that thelowest grid level conforms to the terrain. Severalbuilt-in options for creating the computationalgrid with optional stretching are available in themodel. They allow for easy setup of, for example,quasi-uniform vertical levels at the lower andupper levels, and stretched levels in-between.One can also choose to ¯atten the coordinatesurfaces above a certain height, so that the errorassociated with calculating horizontal gradients(e.g., in horizontal PGF-terms) in a non-orthogo-nal grid is eliminated there. Figure 3 shows anexample of this generalized terrain-followingcoordinate in which the vertical grid is stretchedand the coordinate levels become ¯at at a givenheight.

Fig. 3. An illustration of ARPS computational grid basedon the coordinate transformation relation (2) with ahyperbolic-tangent stretching function in the vertical asdescribed in Xue et al. (1995). In this example, the gridintervals increases with height and the coordinate surfacesbecome ¯at above 7 km level. The formulation of equationsalso allows stretching in the horizontal directions


2.4. Final model equations

Following the practice of most non-hydrostaticatmospheric models (e.g., Clark, 1977; Dudhia,1993), we divide the atmospheric state variablesinto the base-state (reference state) and the devia-tion

' � �'�z� � '0:The base-state is intentionally chosen to beindependent of x and y so that explicit evaluationof its horizontal gradient in the ��; �; �� coordinateis avoided. This eliminates the usually largecancellation errors associated with such calcula-tions. The need to solve the perturbation equationsfor vertical acoustic waves implicitly is anotherreason for de®ning the reference state. As will beseen later, as long as we retain high-order per-turbation terms, the actual choice of the base statehas little effect on the ®nal solution.

The base state is required to satisfy the hydro-static relation:

�pz � ÿ��g; �6�where �� is the base-state density that contains theeffect of base-state water vapor.

For convenience of notation, we de®ne thefollowing:

�� Gp

; U� � ��Uc; V� � ��Vc and

W� � ��Wc: �7�The ®nal prognostic equations in the ARPS areobtained by transforming Eqs. (1.1±7) into thecurvilinear coordinate using relations given in theprevious section. In addition, there is an equationfor the sub-grid scale turbulent kinetic energy(TKE) E:

��u�t � m��ÿ1f�J3�p0 ÿ ��Div�� J1�p0 ÿ ��Div��g �ÿ ADV�u� � �� f � fm�vÿ ~f wÿ uwaÿ1 �

��Gp

Du; �8:1��v�t � m��ÿ1f�J4�p0 ÿ ��Div��

� �J2�p0 ÿ ��Div��g �ÿ ADV�v� ÿ �� f � fm�uÿ vwaÿ1 �

��Gp

Dv; �8:2�

��w�t � ��ÿ1�x�y��p0 ÿ ��Div�� g��ÿ1��p0� �p�ÿ1 ÿ �0��ÿ1� �ÿ ADV�w� � g��ÿ1��B0 � ��~f u

� �u2 � v2�aÿ1 ��Gp

Dw; �8:3��Gp

p0�t ÿ��Gp

��gw� �c2sfm2��

��Gp

Ucmÿ1��

��Gp

Vcmÿ1�� Gp

Wc�� g �ÿ fm�

��Gp

Ucp0� ��Gp

Vcp0� ��Gp

Wcp0�g�

��Gp

��c2s � _��ÿ1 � _AAÿ1�; �8:4�

��0�t � ��w��x � ÿADV��0� ��Gp

D� ��Gp

S�

�8:5��q�t �ÿ ADV�q� � ��Vqq=z��

��Gp

Dq ��Gp

Sq; �8:6��E�t �ÿ ADV�E�

� C � ��KmjDef j2 ÿ 2=3E Div�ÿ ��C"l

ÿ1E3=2 � 2��Gp

DE; �8:7�where the advection operator ADV�� is de®nedas

ADV�� m�U�� V�� W�� m2��U��mÿ1�� V��mÿ1�� W�� ÿ �

��Gp

Div� �9�and the density weighted divergence Div� is de-®ned as

Div� � r � ��~V�� 1=

��Gpfm2��U�mÿ1��V�mÿ1��W�� g:

�10�Div in Eq. (8.7) is de®nd as r � ~V .

2.5. Discussion of the equations

In vertical momentum Eq. (8.3), B0 includes thecontributions of water species and second-orderperturbation pressure and temperature to thebuoyancy:

B0 � q0v � �qv

ÿ q0v � qli

1� �qvÿ �

02��2ÿ 1ÿ

2 2

p02

�p2� �0p0

2 ��p:

�11�Retaining the second-order terms minimizes theimpact of approximations due to expansions

166 M. Xue et al.

around the reference state. Neglecting terms oforders higher than second order in (11) is the onlyapproximation made from equation set (1) to (8).Terms D in the equations denote subgrid scaleturbulence and computational mixing/numericaldiffusion, while most other terms are readilyrecognizable.

It should also be noted that in the horizontalPGF and other terms where the horizontal gradientof base-state variables is taken, we explicitly setthese terms to zero. By doing so, we avoidpotentially large cancellation error associated withcomputing horizontal ®nite differences in thetransformed coordinate. This problem becomesparticularly serious when the atmosphere isstrongly strati®ed in the vertical and the horizontalgrid spacing is much larger than the vertical(Janjic, 1977; Mesinger and Janjic, 1985). Byseparating the horizontally homogeneous base-state from the total state variables (which is nottypically done in hydrostatic models) and expli-citly setting their horizontal gradients to zero, thenumerical accuracy of the model is improved. Theuse of ¯attened coordinate surfaces at the upperlevels, as mentioned earlier, also helps reducesuch cancellation errors, particularly near thetropopause where the vertical change is strati®ca-tion is large.

In Eq (8.3), the hydrostatically balanced portionof the vertical pressure gradient is subtracted off,again to reduce cancellation error. The perturba-tion density �0 has to be diagnosed. To facilitatethe use of vertically implicit solver for acousticmodes (discussed further later), we expand �0 interms of other prognostic variables and retain all®rst-order terms as well as second-order terms in�0 and p0, as they appear in Eq. (11). This shouldgive suf®cient accuracy for almost all meteorolo-gical applications.

The terms involving �Div� in the momentumequations are arti®cial ` divergence damping''terms designed to attenuate acoustic waves, where��; �� and �� are the damping coef®cients in threedirections (Skamarock and Klemp, 1992). Byperforming a divergence operation on the momen-tum equations, one can obtain a 3-D divergenceequation of the form:

�Div��t � ��Div��xx � ��Div��yy

� ��Div��zz � � � � �12�

It is clear that these terms act to reduce small-scale mass divergence thereby damp acousticwaves. Different from Skamarock and Klemp(1992), we formulate the damping in terms ofmass weighted divergence instead of velocitydivergence. The inclusion of a divergence damp-ing is, however, not always needed, especiallywhen vertical acoustic waves are treated impli-citly with the forward biasing in the time averag-ing.

The pressure equation (8.4) is derived fromequation of state (1.4) and mass continuityequation (1.5). The last term on the RHS of theequation include contributions to pressure changefrom diabatic heating and changes in water vapor,liquid and ice water. A � 1� 0:61qv � qli. Ingeneral, such contributions are small and Dudhia(1993) argues that a model with a rigid lidbehaves more realistically (more like an atmo-sphere without an upper lid where air expandsisobarically) without these terms. The model hasthe option to neglect these terms.

Equations (8.5) and (8.6) are the conservationequations for potential temperature � and waterspecies (qv; qc; qr; qi; qs and qh). Terms S are thesources from microphysical, radiative and otherprocesses. Again explicit horizontal advection of ��and �qv is avoided. The second term on the RHS ofthe q equation represents hydrometer sedimenta-tion at a terminal velocity Vq, and is nonzero forrainwater, snow and hail or grapual.

It should be noted that the momentum andscalar conservation equations (8.1±7) have beenmultiplied by �� on both sides. Doing so yields aset of equations whose advection terms can bewritten in a ¯ux-divergence form for anelastic¯ows, and can be formulated to conserve thedensity-�� weighted ®rst and second moments ofthe advected quantities numerically, thereby con-trolling nonlinear computational instability.

Finally, since a minimum number of approx-imations were made in equation set (8), thesystem should maintain good energy conservationas does the original unapproximated set in (1).

3. Subgrid-scale and PBL turbulence

3.1. Subgrid-scale turbulence parameterization

In the ARPS, three subgrid-scale (SGS) closureoptions for turbulent mixing terms D in Eqs. (8.1±


6) are available: the ®rst-order Smagorinsky/Lillyscheme (Smagorinsky, 1963; Lilly, 1962); the1.5-order TKE-based scheme (Deardorff, 1980;Klemp and Wilhelmson, 1978; Moeng, 1984); andthe Germano dynamic closure scheme (Germanoet al., 1991; Wong, 1992; Wong and Lilly, 1994).We retain fully three-dimensional formulation atall scales and include the map factor, m, in theformulation.

According to Smagorinsky (1963) and Lilly(1962), the turbulent terms represented by D in themomentum equations (8.1±3) may be expressed interms of the Reynolds stress tensor �ij,

Dui� m��i1�x � ��i2�y � ��i3�z�; �13�

where index i�� 1; 2 or 3� represents the Carte-sian coordinates. The stress tensor �ij is related tothe deformation tensor Dij through

�ij � ��KmjDij; �14�where Kmj is the turbulent mixing coef®cient formomentum in the xj-direction and deformationtensor Dij is de®ned as

Dij � mimjmkf�ui=�mjmk��xi� �uj=�mimk��xj

g;�15�

where ui are velocity components and m1 � m2

� m and m3 � 1.The turbulent mixing for � and water variables

has a general form of

D� � m��H1�x � �H2�y� � �H3�z; �16�where Hj is the turbulent ¯ux of � in xj-direction,

Hj � ��KHjmj��xj; �17�

and KHj is the corresponding mixing coef®cient.In general, the same KH is used for heat, moistureand hydrometeor quantities and is related to Km

through the turbulent Prandtl number, Pr, i.e.,KH � Km=Pr. In the model, the above formulaeare expressed in curvilinear coordinates ��; �; ��.

a. The 1.5-order TKE-based turbulence closure

In the 1.5-order turbulence closure, the eddymixing coef®cient is related to a mixing length land a velocity scale measured by the SGSturbulent kinetic energy (TKE), E,

Kmj � 0:1E1=2lj: �18�

Here we make prevision for using different lengthscales in different directions.

For isotropic turbulence, the length scale is

l1 � l2 � l3 ��

min��; ls�

for unstable or

neutral case;for stable case;

8<:�19�

where � � ��x�y�z=m2�1=3and ls �

0:76E1=2Nÿ1 according to Moeng (1984).When the horizontal grid spacing is much

larger than vertical grid spacing, it becomesnecessary to use different horizontal length scale��h� than in the vertical ��v�. For this case ofanisotropic turbulence,

l1 � l2 � �h and l3

��v

min��v; ls�

for unstable or

neutral case;

for stable case:

�20�

8><>:In this case, the turbulent Prandtl number is deter-mined according to

Pr � max�1=3; �1� 2l3=�v�ÿ1�; �21�where the lower limit of 1/3 is effective when thevertical length scale l3 exceeds the vertical gridscale �v, which can occur when the TKE-basednon-local PBL parameterization scheme to bedescribed in Sect. 3.2 is used.

The time-dependent TKE is predicted by Eq.(8.7). The equation includes terms for buoyancyand shear production, and dissipation and diffu-sion of TKE. The ground surface heat andmoisture ¯uxes (to be discussed in Sect. 4) alsodirectly contribute the production of turbulence.The dissipation term is related to E and lengthscale l while the diffusion term has a similar formas that for other scalar variables. In the dissipationterm, we choose C" � 3:9 at the lowest modellevel and C" � 0:93 at the other levels afterDeardorff (1980) and Moeng (1984).

b. Smagorinsky-Lilly turbulence closure

The modi®ed Smagorinsky scheme (Smagor-insky, 1963; Lilly, 1962) relates Km to grid-scale¯ow deformation and static stability instead:

Kmj � �k�j�2�max�jDef j2 ÿ N2=Pr; 0��1=2; �22�

168 M. Xue et al.

where k � 0:21 after Deardorff (1972). �j is ameasure of the grid length scale. It is clear that Km

is non zero only when the Richardson numberRi � N2jDef jÿ2

is less than Pr. This criticalRichardson number often is de®ned to be a user-speci®ed value between 1/3 and 1. jDef j is themagnitude of the 3-D deformation and N is theBrunt-VaÈisaÈlaÈ frequency, calculated according toDurran and Klemp (1982) for moist air.

On a model grid with similar grid spacings inall three directions, the SGS turbulence is nearlyisotropic, so that

�j � ��x �y �z=m2�1=3for all j: �23�

When the grid aspect ratio ��x=�z� is large(e.g., for mesoscale and synoptic scale applica-tions), we use different length scales in thehorizontal and vertical, in the same way as we dowith the TKE turbulence option.

c. Germano dynamic closure scheme

This scheme is the same as the Smagorinsky-Lillyscheme except that the parameter k in Eq.(22)is dynamically determined based on local ¯owand varies with space and time. As such, theSGS representation is adjusted to match thestatistical structure of the smallest resolvableeddies. More details can be found in Germano etal. (1991), Wong (1992), and Wong and Lilly(1994). The non-terrain version of the Germanoscheme is currently available for the ARPS(Wong, 1994).

3.2. The non-local PBL parameterization

The turbulence closure schemes discussed in Sect.3.1 are designed to parameterize the local mixingdue to sub-grid scale turbulence. In a convectivelyunstable boundary layer, most of the verticalmixing is achieved by ` large'' boundary layereddies (Wyngaard and Brost, 1984). Unless thevertical as well as the horizontal resolutions ofthe model are on the order of 100 m or less so asto resolve most of the boundary layer eddies(100 m or less), additional parameterization isnecessary.

The treatment of convective boundary layerturbulence in the model is a combination of the3-D, 1.5-order Deardorff SGS turbulence schemediscussed in Sect. 3.1 and an ensemble turbulence

closure scheme of Sun and Chang (1986). Thevertical turbulent mixing length l3 in (20) isrelated to the (non-local) PBL depth instead of thelocal vertical grid spacing inside an unstable PBL.This relationship is based on the pro®le of peakvertical wavelength of vertical velocity derived byCaughey et al. (1979) from observational data;that is

l3 � l0f1:8zi�1ÿ exp�ÿ4z=zi�ÿ 0:0003exp�8z=zi��g; �24�

where z is the height above ground and zi the topof PBL. Constant l0 is chosen to be 0.25. In ourimplementation, zi is de®ned as the height atwhich a parcel lifted from the surface layer be-comes neutrally buoyant.

Under stable conditions or above the convectiveboundary layer, the length scale l reverts back tothat of the Deardorff scheme as in (19) or (20).The performance of this non-local TKE-basedscheme will be evaluated in Part II (Xue et al.,2001) together with the coupled soil-vegetationand the surface layer model.

4. The treatment of other physical processes

The state of the land surface has a direct impacton the sensible and latent heat exchange with theatmosphere. The time-dependent state of the landsurface is predicted by the surface energy andmoisture budget equations in a soil-vegetationmodel. The model used in the ARPS is based onNoilhan and Planton (1989), Pleim and Xiu (1995)and later improvements to their model. Surfacecharacteristics data sets with resolutions on theorder of 1 km have been derived from various datasources for use in the ARPS. The ARPS imple-mentation has the capability of de®ning multiplesoil types within each grid cell, so as to takeadvantage of the high-resolution data set.

For the precipitation processes, the ARPSincludes the Kessler (1969) two-category liquidwater (warm-rain) scheme and the modi®ed three-category ice scheme of Lin et al. (1983). Asimpli®ed ice parameterization scheme of Schultz(1995) is also available. When cumulus para-meterization is needed, the Kuo (1965; 1974) andKain-Fritsch (1990; 1993) schemes are available,with the latter being used for mesoscale applica-tions most of the time.


The treatment of shortwave radiation in theARPS is based on the models of Chou (1990;1992) and the long-wave radiation model on Chouand Suarez (1994). Enhancements to the cloud-radiation interaction in the presence of explicithydrometeor types is after Tao et al. (1996).

5. The numerical solution

5.1. Basic discretization

The continuous equations given in the previoussections are solved using ®nite differences on anArakawa C-grid (Arakawa and Lamb, 1977). TheC-grid represents the geostrophic adjustmentbetter than most other choices and allows for astraightforward and accurate treatment of theadvection-transport equations for the scalars. Withthis grid, all prognostic scalar variables are de-®ned at the center of the grid box while the normalvelocity components are de®ned on their respec-tive box faces. Other derived variables are eval-uated at locations that minimize spatial averagingin the difference operations (Fig. 2).

We de®ne the following standard average anddifference operators:

�'ns � �'�s� n�s=2� � '�sÿ n�s=2��=2;

�ns' � �'�s� n�s=2� ÿ '�sÿ n�s=2��=�n�s�;�25�

where ' is a dependent variable, s an independentvariable in space or time and n an integer.

Using the above notation, U�;V� and W� de-®ned in (7) are evaluated as follows:

U� � ��Uc;V� � ��Vc and W� � ��Wc:

�26�The contravariant vertical velocity, Wc, is eval-

uated according to

Wc � �u��J1

�

� v��J2

�

� w�x�y��Gp �

��ÿ1:

�27�Clark (1977) found that this form of discretizationis necessary for obtaining a correct kinetic energybudget in his anelastic model.

5.2. Time integration of the governingequations

The mode-splitting technique of KW is employedto integrate the dynamic equations (8.1±4).With this method, acoustically active terms, theterms on the LHS of those equations, areintegrated using a number of small time stepswithin a single large time step, and these termsare updated every small steps. The terms re-presenting slower modes, i.e., those on the RHS ofEqs. (8.1±4), are updated only once for all thesesmall steps.

The leapfrog scheme is used for the largetime step integration except when alternativeschemes such as the ¯ux-corrected transport

Fig. 2. A schematic depicting the staggering ofvariables on a grid box. The derived quantities arelocated so as to minimize spatial averaging in the®nite difference calculations

170 M. Xue et al.

scheme are used for scalar advection. In the smallsteps, u and v are integrated using forward-in-timescheme (with respective to PGF terms), and thew and p equations are integrated implicitly inthe vertical direction using the Crank-Nicolsonscheme.

Following Skamarock and Klemp (1994),ARPS also provides an option for treating theinternal gravity wave modes in the small timesteps. In this case, the �-equation is also integratedwithin the small steps, with only the verticaladvection of base-state �, i.e., the second term onthe LHS of Eq. (8.5), being updated every smallsteps. Correspondingly, the buoyancy term in thew equation is also evaluated in the small steps.Doing so removes the restriction resultingfrom the static stability on the large time step.All other scalar equations are integrated in largetime steps.

a. Small time step integration

The small step integration of Eqs. (8.1±5) in ®nitedifference form can be expressed as

��u�� ÿ u�

��

� ÿm��

�

��

�J3�p0 ÿ ��Div��

��

�J1�p0 ÿ ��Div��

�� f t

u �28:1�

��v�� ÿ v�

��

� ÿm��

�

��

�J4�p0 ÿ ��Div��

��

�J2�p0 ÿ ��Div��

�� f t

v �28:2�

��w�� ÿ w�

�� x�y� ��

��Div��

ÿ x�y��

��p

0�� 1ÿ ��p0��

� g��

� �p

� ��p0�

�� 1ÿ ��p0��h i

� g��=��0�h i�

� f tw; �28:3�

��Gp p0�� ÿ p0�

��

� ÿ�c2s m2 ��J3

�u=m� � ��J1u=m

��

��

��J4�v=m� � ��J2v=m

��

��

ÿ �c2s x�y��w�� 1ÿ ��w� �

� g��w�� 1ÿ ��w�� f tp; �28:4�

��0�� ÿ �0�

�� ÿ ��x�y�w��

�h i�� f t

�: �28:5�Here we also include the option for integrating

the potential temperature equation in the smallsteps. For each big time step, these equations areintegrated from t ÿ�t to t ��t with a number ofsmall time steps, with a step size of �� . Here,superscripts � and � �� denote current andfuture small step time levels, and t denotes termsupdated at the current time level in large steps. Wekeep � and cs constant in the small steps whenthey appear in the coef®cients even though theyare dependent on the fast-changing p0 and �0. Theterms related to slower modes (advection, diffu-sion, inertial oscillations, diabatic processes, etc.),i.e., the terms on the RHS of Eqs. (8.1±5), aregrouped in f t.

Weighted time averaging with coef®cient � isperformed on the vertical PGF and pressurebuoyancy terms in the w equation, Eq. (28.3),and on the vertical velocity divergence and base-state pressure advection terms in the p equation,Eq. (28.4). These are terms directly responsiblefor the vertically propagating acoustic waves; theywill impose a stringent limitation on �� if treatedexplicitly. This averaging couples the two equa-tions and makes the solution procedure implicit.At the same time, it removes the limitation on ��due to vertical acoustic modes as long as � > 0:5.Durran and Klemp (1983) showed that a � valuebetween 0.5 and 1.0 (effectively biasing thescheme towards the future time) offers additionalcomputational stability by damping the verticalacoustic modes. A value 0.6 is the default value inthe ARPS. The w and p equations are solved by®rst eliminating p0�� from the two equationsthen solving a linear tridiagonal system ofequations for w�� subject to top and bottomboundary conditions for w. Details can be found inAppendix B.


b. Terms related to slow modes

The ®nite difference form of the terms for slower(i.e., advection, diffusive and inertial) modesrepresented by f t in Eqs. (28) is as follows:

f tu �ÿ ADVU � ��v��v��mÿ �u��m��

h it

� ��f �v�� ÿ ��~f �w�

�� t

��Gp

Dtÿ�tu ; �29:1�

f tv �ÿ ADVV ÿ ��u��v��mÿ �u��m��

h it

ÿ ��f �u��

h it

��Gp

Dtÿ�tv ; �29:2�

f tw � ÿADVW � ��~f �u�

�� t

��B�t �

��Gp

Dtÿ�tw ;

�29:3�f tp � ÿADVPt; �29:4�

f tw � ÿADVT �

��Gp

Dtÿ�t� �

��Gp

St�: �29:5�

B in Eq. (29.3) represents the acoustically inactivebuoyancy terms, as in the second term on the RHSof Eq. (8). The mixing terms are lagged in time by�t for the linear stability consideration, while allother terms are calculated at time t. Finally, wepoint out that the discretized Coriolis terms, aswell as the terms involving differentiation of themap factor m, cancel each other in the globallyintegrated total energy equation, ensuring energyconservation.

In Eqs. (29), ADVU, ADVV, ADVW, ADVP andADVT are the advection terms for u, v, w, �0 andp0, respectively. Their continuous form is given by(9) but their discrete formulation depends on thechoice of advection scheme and the grid stagger-ing. We give the second- or fourth-order centeredformulation here for scalar �0 only. Those for u, v,w, and p0 can be found in Appendix C.

ADVT � � m U��0� � V��0

��

�W��0�

h i� �1ÿ �� m U��2��0

2�

� V��2��02�

� ��W��2��0

2��: �30�

When � � 1 the scheme is second-order andwhen � � 4=3 the scheme is the fourth-orderaccurate in space. As with most fourth-order

schemes the order of accuracy is true only forconstant ¯ows. When the ¯ow is not constant, thetruncation error is proportional to the gradient ofthe advective velocity, and the magnitude of erroris smaller than that of the fourth-order scheme ofWilhelmson and Chen (1982).

The advection terms are written in advectiveform, which is shown by Xue and Lin (2000) to benumerically equivalent to the ¯ux form consistingof a ¯ux term plus an anelastic correction. Thelater form is often used by other modelers (e.g.,Wilhelmson and Chen, 1982). Neglecting theeffect of compressibility, it can also be shown(Xue and Lin, 2000) that both the second-orderand fourth-order advection formulation in Eq. (30)are quadratically conserving, which is importantfor controlling nonlinear aliasing instability (Ara-kawa and Lamb, 1977) and for better representa-tion of the nonlinear energy cascade. Accordingto our knowledge, this quadratically conservingfourth-order formulation has not been used before.

For the scalars, two additional options areavailable. One is the multi-dimensional monotonic¯ux-corrected transport (FCT) scheme afterZalesak (1979), the other is the more ef®cientthough less accurate positive de®nite schemebased on leapfrog centered difference schemes(Lafore et al., 1998). Both schemes are suitablefor advecting positive de®nite variables, while theformer eliminates both undershoot and overshootassociated with conventional advection schemes.In the implementation of the ¯ux limiter, care hasbeen taken so that the extrema in the advectedscalar such as the potential temperature instead ofthe density weighted scalar are checked to preventovershoot and undershoot.

The discrete form of the mixing terms D in Eq.(29) uses second-order centered differencing andis straightforward based on their de®nitions inSect. 3.1.

c. Time integration of other scalar equations

The equations for water substances and TKE aresolved entirely on the big time step, and theirnumerical representation is given in a generalform for dependent variable q as

��qt��t ÿ qtÿ�t

2�t�ÿ ADVQt � �� Vqq=z�

�h it

��Gp

Dtÿ�tq �

��Gp

Stq; �31�

172 M. Xue et al.

where ADVQ has exactly the same functionalform as ADVT in Eq. (30) except when the FCTor the simple positive-de®nite advection schemeis used. The second term on the RHS is a ¯uxdivergence term, representing sedimentation of qat a terminal velocity Vq (positive downwards). Vq

is given by the microphysics parameterizationand is nonzero only for falling hydrometers. SinceVq can be large relative to w, split time steps basedon an upstream-forward advection scheme areused for this term inside each large time step.Even so, this process can take unproportionallylarge amount of total CPU time because the steptime size permitted can be very small when near-surface vertical grid spacing is very small. Avertical implicit treatment is being implementedfor this term and it should provide a betteref®ciency.

d. Special treatment of vertical mixing

Given that in the PBL the vertical mixing coef®-cients Kmv and KHv are based on the length scale lin Eq. (24), vertical turbulent mixing often resultsin a linear stability constraint more severe thanthat associated with advection, especially whenthe vertical resolution is high. To overcome thispotentially severe restriction on the large time stepsize, we apply the implicit Crank-Nicolson schemeto the vertical mixing so that the integration isabsolutely stable for these terms.

5.3. Boundary conditions

a. Lateral boundary conditions

Several types of boundary conditions can be usedin arbitrary combinations in the ARPS. At thelateral boundaries, they include rigid wall (mirror),periodic, zero-gradient, wave-radiating (open) andexternal (including one-way nested) conditions.Furthermore, several variations of the radiationlateral boundary condition are available. Twooptions are used most often. One is based on theOrlanski (1976) condition which applies a simplewave equation to the normal velocity component.Instead of using locally estimated phase speedsas proposed by Orlanski, we use the verticallyaveraged value of the outward-directed phasespeeds. Without the averaging, domain wide

pressure drift sometimes occurs in simulationswith a relatively small domain.

Another variation is originally proposed byKW. In this case, disturbances are assumed topropagate at the ¯ow speed plus a dominantinternal gravity wave speed; the latter is a user-speci®ed constant that is typically set to 30 to45 m sÿ1. Again, a simple wave equation is ap-plied to the normal velocity component only.Other variables on the boundary are obtained fromtheir respective prognostic equations, using up-stream advection when necessary.

One-way interactive self-nesting and nestingwithin other models are achieved by using theDavies-type (1983) lateral boundary conditionthat includes a boundary relaxation zone. Further-more, the ARPS offers a full implementation ofthe adaptive grid re®nement procedure of Ska-marock and Klemp (1993). This procedureprovides ARPS with unlimited level of two-wayinteractive nesting while allowing the nested gridsto be added and removed in response to the ¯owevolution during the model integration.

b. Vertical boundary conditions

At the lower and upper boundaries, rigid, zero-gradient and periodic boundary options are avail-able. For most applications, a free-slip mirrorcondition is applied at the lower boundary. Themirror condition is implemented in the computa-tional space; therefore, the contravariant verticalvelocity Wc � 0 at � � 0. This results in a ¯owthat follows the terrain surface at z � hm, wherehm is the terrain elevation. When surface frictionin the form of surface momentum ¯uxes isincluded, the lower-boundary condition is oftenreferred to as ` semi-slip''.

At the upper boundary, the wave-radiatingcondition of Klemp and Durran (1983) can beused in combination with a Rayleigh dampinglayer. When wave re¯ection is not anticipated,a rigid lid condition can be applied. Theimplementation is similar to Klemp and Duran,except that a cosine transform is used insteadof the full Fourier transform, thereby removingthe lateral periodicity requirement on w at thetop boundary. The ARPS implementation ofradiative upper-boundary condition is given inAppendix D.


5.4. Computational mixing

As in most numerical models, a certain amountof computational mixing or numerical smoothingis often needed to remove poorly resolved small-scale noise. This noise can originate from non-linear aliasing and numerical dispersion, frominitial analysis, or treatment of physical pro-cesses. In the ARPS, the computational mixing isincluded in all prognostic equations except forthe pressure equations, and has either a second-order �n � 2� or fourth-order �n � 4� form asgiven by��

Gp

D�n � �ÿ1�n=2�1Khn

@n��0�@�n

� @n��0�@�n

� �� Kvn

@n��0�@�n

�; �32�

where Khn and Kvn are the coef®cients of the n-thorder mixing in the horizontal and verticaldirections, respectively. High-order monotonicnumerical mixing/diffusion formulations of Xue(2000) are also available and the formulationensures global conservation of the mixed/diffusedvariables. It is important to note that unlike tur-bulent mixing, the computational mixing oper-ates along the model grid surfaces and acts on theperturbations from the base state instead of thetotal variables. This type of mixing does imposelimitations on the large time step size and theconstraint is a function of the magnitude ofmixing coef®cient.

6. Computational implementations

The ARPS computer code was developed under astringent set of rules and conventions. Uniformityof variable names is maintained across all sub-routines in the entire system. Readability, main-tainability and portability of the code have beenhigh priorities during the model developmentprocess. These virtues, together with extensiveinternal and external documentation (e.g., Xueet al., 1995), are perhaps unique to this codeamong atmospheric modeling systems. Thehighly modular design and the clearly de®nedmodule interfaces greatly ease the process ofcode modi®cation and the addition of newpackages. The uniform coding style throughoutthe model and the external documentation have

proven to be extremely bene®cial to both noviceand experienced users. The former makes theporting of the code to a variety of parallelplatforms straightforward (Droegemeier et al.,1995).

Currently unique to the ARPS, we maintain asingle version of the source code for all computerplatforms. Execution on distributed memory plat-forms are achieved by using MPI (MessagePassing Interface) message passing library. Thecalls to these routines are inserted into the modelin a pre-processing step by a small set of trans-lators written in C (Sathye et al., 1996). Given theuniform and consistent coding style followedthroughout the ARPS, the translators have to dealwith only a small subset of possible scenarios. Theversion of code prior to Version 5.0 is written inFORTRAN-77 for maximum portability. Conver-sion of the entire system into Fortran 90 under anew coding standard was recently completed withthe aid of a newly developed automatic codeconverter. This version makes use of, among otherthings, dynamic memory allocation and newFORTRAN intrinsic functions for additional¯exibility and better ef®ciency.

Signi®cant efforts have also been made in thecode optimization. This includes ®ne-tuning thecode structure for maximum vectorization and/orparallelization, and replacing all expensive powerand exponential functions with lookup tables. Thelatter is done without noticeable loss of solutionaccuracy. In the following sections, we presentresults of ARPS as applied to mountain ¯ow anddensity current problems.

7. Model veri®cations with mountain ¯ows

Analytic solutions of linear and nonlinear moun-tain waves in a constant ¯ow over idealized terrainhave been commonly used to verify the correct-ness and accuracy of numerical models (e.g.,Clark, 1977; Durran and Klemp, 1983; Xue andThorpe, 1991). Vertical momentum transport bymountain waves is an excellent measure of themodel's ability in handling the lower boundarydynamic forcing. Under certain circumstances,mountain forced waves can greatly amplify tocause wave breaking and formation of strong windson the lee slope. In this section, we compare thequasi-steady state solutions of the ARPS modelagainst analytical solutions for linear and non-

174 M. Xue et al.

linear mountain waves in both hydrostatic andnonhydrostatic regimes. The results validate thecoordinate transformation, lower and upperboundary conditions, as well as the time integra-tion procedure of the ARPS. We further test themodel's ability to simulate strong wave-breakingevents, such as the well documented 1972Boulder downslope windstorm (Lilly and Zipser,1972).

7.1. Veri®cation against analytic solutions

For non-rotational ¯ow forced by a small-amplitude 2-D mountain, the vertical displace-ment of a parcel, �, at a steady state is governedby a simple equation (Smith, 1979)

�xx � �zz � l2� � 0; �33�where l, also known as the Scorer parameter, isconstant for an isothermal, anelastic, and constant¯ow �l2 � g2��CpT0

�U2�ÿ1 ÿ �4R2T20 �ÿ1�, where g

is the gravitational acceleration, Cp is the speci®cheat of air at constant pressure, R the gas constantfor dry air, T0 the temperature of isothermalatmosphere, and �U the constant ¯ow speed]. Fora bell-shaped mountain, the solution to (33) canbe found using the Fourier transform methodsubject to lower-boundary condition ��x; 0� �h�x�, where h�x� is the mountain pro®le. Thesolution for � is proportional to the terrain heightand the sum of integrals over the horizontal wave-number, k, from 0 and l and from l to 1. Waveswith horizontal wave number less than l areevanescent in the vertical, while shorter waveshave vertical wave numbers equal to

��l2 ÿ k2p

.For a bell-shaped mountain, the dominant hor-izontal wave number is 1=a while the dominantvertical wave number is l. Furthermore the waveamplitude is inversely proportional to the squareroot of base-state density (see Smith, 1979). Thesolution can be evaluated numerically and used toverify the model.

The vertical ¯ux of horizontal momentumde®ned as

M ��1ÿ1

��u0w0dx; �34�

where M is constant with height for linearmountain waves in a uniform ¯ow (Eliassenand Palm, 1960). When the linear waves are

hydrostatic and irrotational, hydrostatic momen-tum ¯ux

M � ÿ�4�0N �Uh2

m; �35�

where �0 is the density and N the static stability atthe ground level. For both rotational and non-hydrostatic mountain waves, the vertical ¯ux issmaller than that of hydrostatic waves (Gill, 1982).

Long (1953) showed that for the special case ofBoussinesq and uniform ¯ow with constant staticstability, the vertical displacement � forced by a®nite-amplitude mountain satis®es an equationthat has the same form as (33). For such a ¯ow,the Scorer parameter l �l � N=�U� is also constant,and therefore the same Fourier transform proce-dure used for the linear case can again be used toobtain the solution for �. The main dif®culty hereis the enforcement of nonlinear lower boundarycondition ��x; z� � h�x�.

Instead of trying to ®nd the analytical solutionfor a pre-speci®ed mountain pro®le that satis®esthe nonlinear lower boundary condition, wefollow a procedure used by Durran and Klemp(1983) and determine a mountain pro®le so thatthe streamline given by the linear solution forcedby the original mountain follows this new pro®leat the lower boundary. For a bell-shaped mountainoriginally 570 m high, the resultant mountain hasa height of 503 m and the peak is shifted upstreamby about 400 m. In essence, the modi®ed moun-tain produces nonlinear responses that are equiva-lent to the linear responses produced by theoriginal taller mountain. The new mountainpro®le is used in our nonlinear experiment (seeTable 1) and the results will be compared with theanalytic solution obtained using the procedureoutlined above.

The ARPS is ®rst veri®ed against the 2-Dsolutions of linear mountain waves in both hydro-static and nonhydrostatic ¯ow regimes (as inSmith, 1979). In all experiments, the earth's rota-tion is neglected and an isothermal �T0 � 250 K�uniform upstream ¯ow ��U � 20 msÿ1� is speci-®ed. The experiments are impulsively started, i.e.,the mountain is introduced into the ¯ow at theinitial time. The Durran and Klemp (1983) radi-ation lateral boundary condition option is used forall control experiments, and the upper boundarycondition uses either Rayleigh damping or thewave permeable condition of Klemp and Durran


Table 1. List of mountain wave experiments

Experiment Parameters Linear hydrostatic(LH)

Linear nonhydrostatic(LNH)

Nonlinear nonhydrostatic(NLNH)

Boulder Windstorm(WSTORM)

hm (m) 1 1 503 Real terraina (m) 10000 2000 2000 N.A.�x�m� 2000 400 400 1000�z�m� 125 125 125 200L (domain width, km) 576 460 460 512H (domain depth, km) 24 24 24 28�t (s) 20 10 5 2.5�� (s) 5 1 1 2.5Rayleigh dampingcoef®cient (1/s)

0.0015 0.0015 N.A. N.A.

Height damper starts (km) 12 12 N.A. N.A.4th-order horizontalmixing coef®cient (1/s)

0 0 3� 10ÿ5 8� 10ÿ4

Fig. 4. Analytical solution of u0 and w0 (upper panel) and the model simulated solution at the ND time of 100 from experimentLH (lower panel), which is for linear hydrostatic mountain waves over a 1 m high bell-shaped mountain with a 10 km halfwidth. Note that the mountain pro®le in thick line has been ampli®ed by a factor of 500 for illustration purpose

176 M. Xue et al.

(1983), a small amount of horizontal spatialsmoothing (computational mixing) is applied onlyin the nonlinear run. The gravity wave modes areintegrated on the large time step. Divergencedamping is not used.

Three control experiments for idealized moun-tain waves are summarized in Table 1. For theparameters used here we have lÿ1 � 1 km. Inexperiment LH, a � 10 km� lÿ1, thus the ¯owis essentially hydrostatic. In experiments LNHand NLNH, a � 2 km � lÿ1, the ¯ow belongs tothe nonhydrostatic regime.

a. Linear mountain wave experiments

We present the model results at nondimensional(ND) times that are scaled by the advective timescale U0=a. Figure 4 shows the analytical (upperpanel) and model (lower panel) solutions of u0 andw0 for part of computational domain at U0t=a �100 (Note that the mountain depicted in the®gures has been ampli®ed by a factor of 500 forillustration purpose and it is done for all linearsolutions). The analytical ®elds were obtainedby numerically integrating the integral solutionusing mid-point method (Press et al., 1989). Ingeneral, the simulated waves are slightly weakerthan their analytical counterparts, and the errorincreases with height. The maximum relativeerror in w0 is about 5%, while that of u0 is about14%. The phases of the waves agree very well,however. Notice the amplitudes of the wavesincrease with height due to the effect of decreas-ing density.

Vertical pro®les of horizontal momentumtransport by gravity wave processes are plottedin Fig. 5 for experiment LH, together with thatfrom the analytical solution (bold line). Thesepro®les have been scaled by the analytical ¯ux forlinear hydrostatic waves given in (35). It can beseen that the analytical ¯ux is almost unity, whilethe simulated ¯uxes are about 0.97 at the surfaceand approach 0.96 at later times at the levelimmediately below the Rayleigh damping layer(12 km). This accuracy is at least as good as thosereported in the literature. For example, Durran andKlemp (1983) reported that the ¯ux at one verticalwavelength �z � 6:4 km� reached 94% of theanalytical value at an ND time of 60 for theircompressible model. A similar accuracy was alsoreported by Xue and Thorpe (1991). The improve-

ment in accuracy obtained here can be partlyattributed to higher vertical resolution.

We also performed an experiment (LHa) that isthe same as LH, except that the vertical grid isstretched from a minimum of 20 m at the surfacewhile keeping total number of levels the same.The stretching is based on a hyperbolic tangentfunction as described in Xue et al. (1995). Themomentum ¯uxes in Fig. 6 are even closer to

Fig. 5. Vertical pro®les of horizontal momentum atindicated ND times from experiment LH, along with thepro®le calculated from analytical solutions of u0 and w0

(thick line). All pro®les are normalized by the theoreticalvalue for linear irrotational hydrostatic waves in (35)

Fig. 6. Same as Fig. 3, except for experiment LHa, in whicha vertically stretched grid is used


unity (0.98) at the surface while the values atupper levels are slightly smaller, indicating thatthe solution accuracy is slightly sensitive to thevertical resolution. Another experiment (LHb)was conducted that used the wave-radiating topboundary condition without Rayleigh damping.In this case, the ¯ux pro®le (Fig. 7) is nearlyconstant, with values being of about 96% at thesurface and decreasing to 91% at the top by non-dimensional time 140. It shows that the radiationboundary condition is working well in this case.

Figure 8 shows the analytical and modelsimulated u0 and w0 ®elds for linear nonhydrostaticmountain waves from experiment LNH. Evidentin the solutions are the dispersive wave trainsdownstream of the mountain peak, especially atupper levels, distinguishing them from the hydro-static solutions obtained in previous experiments.The simulated wave pattern agrees quite well withtheory, with the amplitudes being slightly smaller(as in the previous cases). Figure 9a shows themodel simulated isentropes after �0 has beenampli®ed by 500 times for the purpose of illus-tration. These isentropes approximate parceltrajectories for an adiabatic, steady-state ¯ow. Itcan be seen that the lowest isentrope intercepts theterrain because of the linear boundary forcing. Inthese simulations, the pressure ®eld is found to bemost sensitive to contamination at the lateralboundaries (which use an open boundary condi-

tion) in a long simulation, and it is shown in Fig.9b that it remains well behaved by ND time 100.

The momentum ¯ux [scaled by the hydrostaticvalue given by (35)] from experiment LNH (Fig.10) is essentially constant at later times below theRayleigh damping layer, with a value of about0.76. This result is very close to the theoreticalprediction (Klemp and Durran, 1983) for linearnonhydrostatic mountain waves.

b. Nonlinear mountain waves

Because Long's solution requires the Boussinesqapproximation, the option for this approximationin the ARPS is turned on. It involves replacing ��by its constant surface value after �� and �p arespeci®ed. We also neglect the contribution by p0 tothe buoyancy as well as the vertical advection of�p in pressure equation. These simpli®cationsmake the system of equations analogous to theBoussinesq equations describing an incompres-sible ¯ow (the same approximations were made inXue et al., 1997). Finally, the atmosphere remainsisothermal and �U � 20 so that the Score's par-ameter has a value similar to that in our previousexperiments.

Figure 11 shows the analytical solution of u0and w0 (upper panel) for a 503 m high mountainobtained using the procedure described in Sect.7.1. The model solutions at ND time 100 are givenin the lower panel. Since the reference statedensity is constant, the wave amplitude no longerincreases with height; in fact, it decreases becausesigni®cant wave energy is dispersed downstream.The agreement between the two solutions is verygood, with the amplitudes in the numericalsolution being only slightly smaller.

The simulated isentropes and perturbationpressure are shown in Fig. 12. Unlike the previouslinear experiment (see Fig. 9), the isentrope atthe surface closely follows the terrain, whilethe waves at upper levels are weaker than those inFig. 9 for the lack of density scaling effect. Thepressure ®eld is again well behaved. Finally,Fig. 13 shows the vertical pro®les of momentum¯uxes. These ¯uxes have been scaled by that ofhydrostatic nonlinear mountain waves, the lattergiven by the formulation (35) for linear waves butwith hm � 570 m. The pro®le calculated from theanalytical u0 and w0 from Long's equation isshown by the thick line. The simulated vertical

Fig. 7. Same as Fig. 3, except for experiment LHb, in whichKlemp and Durran (1983) radiation boundary condition isapplied at the top boundary without a Rayleigh dampinglayer

178 M. Xue et al.

¯uxes overshoot at the early time due to theimpulsive startup but converge toward the analyt-ical value of about 0.76. This value is very closeto that in experiment LNH, indicating that bothlinear and nonlinear mountain waves in the non-hydrostatic regime with al � 2 transport momen-tum at a rate of about 76% of their hydrostaticcounterparts, a result that agrees with theory(Klemp and Durran, 1983). Furthermore, the factthat the ¯ux is nearly constant throughout thedepth of domain at later times indicates that theradiation top boundary condition works well even

for these ®nite amplitude waves (of course thewave amplitude has been signi®cantly reduced atupper levels due to downstream dispersion ofenergy).

7.2. Simulation of 1972 Boulder windstorm

A severe windstorm developed on the lee (east)side of the Front Range of the Rocky Mountainswas well observed and documented in Lilly andZipser (1972) and has been a subject of manysubsequent studies (e.g., Klemp and Lilly, 1975;

Fig. 8. Analytical solution of u0 and w0 (upper panel) and model simulated solution at the ND time of 100 from experimentLNH (lower panel), which is for linear nonhydrostatic mountain waves over a 1 meter high bell-shaped mountain with a 2 kmhalf width. As in Fig. 3, the mountain pro®le has been ampli®ed by a factor of 500


Peltier and Clark, 1979; Durran, 1986). Recently,2D simulations of this case with a bell-shapedmountain were conducted using 11 models(including ARPS) and the results intercompared(Doyle et al., 2000). Initialized with a upper-stream sounding taken at Grand Junction CO overan bell-shaped mountain that resembles the FrontRange, most models were able to simulate theupper-level wave breaking and intensi®cation of

downslope winds reasonably well, although sig-ni®cant differences exist among the solutions.

In this paper, we report the results of our simu-lation using a high-resolution real terrain pro®le.The terrain pro®le is derived from a 3 secondterrain database sub-sampled at 15 second inter-vals. The data were bilinearly interpolated to a1 km grid after which a 1-2-1 ®lter is applied onceto remove 2 grid interval terrain features. A500 km E-W cross-section through Boulder(40.027N) is taken and a 28 km deep domain isused. Radiation boundary conditions are used atthe top and lateral boundaries. The latter uses theKlemp and Wilhelmson (1978) formulation with aconstant phase speed of 50 msÿ1. The model isinitialized with the 1200 UTC 11 January 1972Grand Junction CO sounding (Fig. 14), whichextends up to a 28 km altitude. The sounding has acritical level �u � 0� at the 23 km level, thereforewaves are expected to be con®ned to below thislevel. The sounding also contains a relativelystable layer between 5 and 7 km levels, contribut-ing to the intensi®cation of downslope winds in aform of hydraulic jump ¯ow, according to Durranand Klemp (1986). Most previous simulationstudies of this case used signi®cantly smoothedsoundings with modi®ed wind pro®le at the upperlevels (e.g., Peltier and Clark, 1979; Durran andKlemp, 1983). Different from Doyle et al. (2000),

Fig. 9. Simulated isentropes, a and perturbation pressure, b at the ND time of 100 from experiment LNH, which is for linearnonhydrostatic mountain waves. The depiction of the potential temperature perturbation has been ampli®ed by a factor of 500,as has a mountain pro®le

Fig. 10. Same as Fig. 4, except for experiment LNH, whichis for linear nonhydrostatic mountain waves

180 M. Xue et al.

care is taken here to place the lowest level ofobserved sounding at the station height rather thanat the sea level so as to yield a correct distancebetween the mountain peak and the tropopause(and the stable layer). The grid resolution is 1 kmin the horizontal and 0.2 km in the vertical. Themodel ¯ow is abruptly started and the controlexperiment does not include surface friction.

Figure 15 shows the potential temperaturecontours and cross-mountain velocity ®elds at 3and 6 hours. The isentropes represent the ¯owtrajectories reasonably well outside the regions of

wave breaking. The most signi®cant features seenare the descent of mid-tropospherical isentropesalong the lee slope of the Front Range, accom-panied by strong surface winds of over 70 and80 msÿ1 at 3 and 6 hours, respectively. The maxi-mum surface wind reached 70 msÿ1 at 2 hours40 minutes and remained above 70 msÿ1 for therest of the simulation. The surface wind peaked94 msÿ1 at 4 hours 47 minutes in this simulation.The strong surface winds propagate downstreamwith the gust front, at which the ¯ow deceleratesabruptly and transitions into a subcritical ¯ow in

Fig. 11. Analytical solution of u0 and w0 (upper panel) and model simulated solution at the ND time of 100 from experimentNLNH (lower panel) which is for ®nite-amplitude (nonlinear) nonhydrostatic mountain waves over a 503 meter high mountainwhose pro®le satis®es ��z � h� � h, where � is the solution of Eq. (33) for a bell-shaped mountain with a 2 km half width.The mountain pro®le in thick line has its peak shifted upstream by about 400 m from the peak of original bell-shapedmountain


the form of hydraulic jump (Durran, 1986). Strongvertical motion is found at the front, signi®ed bythe nearly vertical isentropes. Above this strongsurface ¯ow and below tropopause, ¯ow reversal�u < 0� is seen shortly after 3 hours, resulting in¯ow overturning and strong mixing. Wave over-turning and breaking are also found above the

tropopause, where vertical wavelengths andamplitudes are smaller due to higher stability.The strongest upper-level wave activities arefound to be coupled with the strongest tropo-spheric forcing at the jump, whereas activitiesdirectly above the upper-tropospheric wave-break-ing region are weak. This well mixed region

Fig. 12. Simulated isentropes, a and perturbation pressure, b at the ND time of 100 from experiment NLNH, which is fornonlinear nonhydrostatic mountain waves

Fig. 13. Same as Fig. 4, except for experiment NLNH,which is for nonlinear nonhydrostatic mountain waves

Fig. 14. 1200 UTC, 11 January 1972 Grand Junction, COsounding used in the Boulder downslope windstormsimulations

182 M. Xue et al.

acts as a critical level that, in theory, reduces thevertical group velocity to zero and prevents up-ward transport of wave energy. The trapping ofwave energy at the lower levels tend to acceleratethe low-level ¯ow. Further upstream, waves ofsigni®cant amplitude are forced by lower (relativeto terrain height on the lee side) ridges nearx � 200 km, and these waves propagate verticallyinto the stratosphere forcing signi®cant wave

breaking as well (not shown). These waves aresuf®ciently far upstream of the Front Range anddo not appear to have signi®cantly affected theprimary wave system by 6 hours. Due to thepresence of very weak ¯ow at about the 23 kmlevel, nearly all wave activities are con®ned below23 km (not shown). The use of radiation topboundary condition does not appear critical to thelower-level mountain ¯ow.

Fig. 15. Isentropes (left panel) and u ®eld (right panel) at hour 3 and 6 of the 2-D simulation of downslope windstorm usingGrand Junction, CO soundings of 1200 UTC, 11 January 1972 and a high-resolution east-west terrain pro®le through Boulder,CO. Regions between � � 296K and 316K one the left and where u � 30msÿ1 are shaded. Only a portion of the 512� 28 kmintegration domain is shown


Overall, the simulated wave system at theearlier time resembles the observation depicted inFigs. 4 and 5 in Klemp and Lilly (1975). The peaksurface wind speed is larger than observed, almostcertainly due to the absence of surface friction.The results are also consistent with those ofprevious simulation studies, although most ofthem use idealized ridges. Our results also agreequalitatively with the simulation of COAMPSreported by Dolye et al. (2000) which applied asmoothed real terrain. The sounding used the inlatter, however, assumed height above sea levelinstead of ground level, therefore the tropopausein their case was about 1.5 km lower than reality.A simulation repeated using their soundingproduced a result closer to their solution. Finally,the extra ®ne-scale terrain features included in thecurrent experiment do not seem to signi®cantlyimpact the general behavior of the downslope¯ow, as is supported by experiments in whichsmall-scale features are ®ltered out or when a bellshaped ridge of similar scale and height is used(not shown). This can also be understood bynoting that the downslope ¯ow is mainly fed bythe upstream ¯ow from above the 4 km level.Had the terrain upstream of the Front Rangebeen replaced by air, the air would be too heavy(measured by the upstream Froude number) toclimb the mountain range. Another experimentthat included parameterized surface friction(through surface drag) resulted in a much weakerwave system, in which the downslope windsare limited to the lee slope (not shown). This resultis consistent with the ®nding of Richard et al.(1989).

7.3. Summary

We presented in this section a set of idealizedmountain wave experiments as well a realisticsimulation of a severe downslope windstorm. Forthe former, analytical solutions that cover linearand nonlinear waves in both hydrostatic andnonhydrostatic ¯ow regimes can be found. Quasi-steady state model solutions were comparedagainst these analytical solutions and excellentagreement was found. Experiments were con-ducted to examine solution sensitivity to verticalgrid stretching and the top boundary condition.These experiments, as well as the simulation of asevere downslope windstorm, demonstrated the

integrity of the dynamic and numerical frameworkof the model, in particular those aspects related tothe coordinate transformation, the treatment oflower-boundary forcing, and the top boundaryconditions.

8. Model validation with a nonlineardensity current

In this section, we examine the model's ability toaccurately handle highly nonlinear ¯ow withstrong interior gradients. A benchmark problemof a simple density current is chosen. Solutions forthis problem from a number of numerical models,including those of Carpenter et al. (1990) and Xueand Thorpe (1991), are documented in Strakaet al. (1993). Particular attention is paid to severaloptions of advection schemes in the ARPS andtheir impact on the solution accuracy.

8.1. The test problem

The test consists of a 2-D density current formedfrom a cold blob of air descending from anelevated level to the ground in a neutrally strati®edand initially static atmosphere. As the cold airreaches the ground, it spreads along the lowerboundary and develops rotors along the top of thecold pool boundary due to Kelvin-Helmholtzinstability (Fig. 16). In the spatial resolutionexperiments, the eddy-mixing coef®cient is keptthe same, so that the solutions may converge athigh resolutions.

The base-state atmosphere is calm and hasa constant potential temperature of 300 K. Anelliptic initial bubble is speci®ed in terms of tem-perature perturbation. It is centered at x � 0 kmand z � 3 km with a vertical half-axis of 2 km,a horizontal half-axis of 4 km and a minimumtemperature of ÿ15 K (see Straka et al., 1993).Free-slip wall conditions are used on all fourboundaries. The computational domain is 6.4 kmdeep and 25.6 km wide. Horizontal symmetry ofthe problem is exploited by centering the bubbleon the left boundary.

Since the amount of details that can exist in themodel solution are limited by the speci®ed and®xed eddy mixing coef®cient, it is possible toobtain a reference solution at a high resolutionbeyond which no noticeable improvement can beachieved. Such a reference solution was presented

184 M. Xue et al.

in Straka et al. (1993) using a compressible modelwith second-order advection at 25 m spatial re-solution. We present in Fig. 16 a similar referencesolution obtained using our model with fourth-order centered spatial difference, which is essen-tially identical to that obtained using second-order

scheme (not shown). Since this solution is veryclose to the reference solution in Straka et al.(1993, Fig. 2), we will use it as our benchmark.

8.2. The model results

We conducted a set of experiments using fouroptions of advection schemes at 400, 200 and100 m spatial resolutions (Table 2). The fouradvection options are: 1) second-order centered;2) fourth-order centered; the ¯ux-corrected trans-port (FCT) (Zalesak, 1979) with second-order (3),and fourth-order (4) higher-order scheme. For the®rst two options, the same advection schemeswere applied to both momentum and scalars,while for the latter two, momentum was advectedby the standard fourth-order centered scheme. TheFCT schemes preserve the monotonicity but donot require positive-de®niteness, they can there-fore be used to advect ®elds with both signs. As aspecial case, a positive ®eld will remain positivein the advective process. The result of using FCTin the model of Xue and Thorpe (1991) isdocumented in Straka et al. (1993). FollowingStraka et al. (1993), the experiments are run at400 m, 200 m and 100 m resolutions. At theseresolutions, the simulated density currents are,respectively, poorly resolved, reasonably resolvedand well resolved, measured in terms of the gridspacing as compared to the characteristic ¯owfeatures. The difference in the scheme perfor-mance, as will be shown, is more pronounced atlower resolutions.

Figure 17 shows the simulated �0 ®elds at 900 s,using four advection options at 400 m resolution.The corresponding solutions using 200 m and100 m resolutions are shown in Fig. 18 and Fig.19, respectively. The bottom panel of each ®gureis the reference solution averaged to the corre-sponding resolution. It is clear that undershootingin �0, as indicated by the minimum values, is

Fig. 16. �0 contours from the 25 m resolution referencesimulation, at a 0, b 600, and c 900 s. Contour interval is1� C. Only a portion of the 25:6� 6:4 km2 domain isshown. The ®gure shows that an oval shaped initial coldblob drops to the ground and spread along the ground in theform of density current. Kelvin-Helmholtz billows developalong the upper surface the cold pool

Table 2. List of density current experiments

Advection scheme Spatial resolution (m)

400 200 100

2nd-order-centered 2nd400 2nd200 2nd1004th-order-centered 4th400 4th200 4th1002nd-order FCT FCT2nd400 FCT2nd200 FCT2nd1004th-order FCT FCT4th400 FCT4th200 FCT4th100


occurring near the density current head in all butthe FCT solutions, with the problem being mostserious at the lowest resolution. The error isgenerally larger with second-order scheme thanwith fourth-order scheme. This undershoot, caus-ing the cold pool to be too cold, is believed to beresponsible for the faster propagation speed of thefront in these cases.

At all resolutions, the fourth-order schemesclearly outperform the second-order counterparts,in de®ning the frontal location and in simulatingthe shape and location of the billows (e.g.,compare Fig. 17a and Fig. 17b, Fig. 18a andFig. 18b). The FCT solutions are generally much

better than their non-monotonic counterparts.This is evident by comparing, e.g., Fig. 17c withFig. 17a, and Fig. 17d with Fig. 17b. The FCTscheme not only eliminates spurious oscillationsbut also resolves the ®ne-scale billow structuresbetter. At 100 m resolution, the differences in thesolutions are smaller but are still readily identi®-able, with the 4th-order and FCT options out-performing the others. Among all solutions,FCT4th100 in Fig. 19d compares best with thereference solution, agreeing with our expectation.The FCT scheme is about 3 times more expensivethan the conventional scheme of the same order,however.

Fig. 17. Fields of �0 at 900 s from the 400 m resolutionexperiments a 2nd400, b 4th400, c FCT2nd400, dFCT4th400 and e the 25 m reference run. The referencesolution has been averaged to the 400 m grid

Fig. 18. As in Fig. 13, but for the set of 200 m resolutionexperiments

186 M. Xue et al.

8.4. Summary

It has been documented in this section the be-havior of four advection options in the ARPS, asapplied to a density current for which a referencesolution is obtained at much higher resolution.The monotonic FCT scheme clearly outperformsthe regular centered difference schemes, espe-cially at relatively coarse resolutions. The fourth-order option exhibits clear improvement over thelower-order counterpart. The comparisons ofthese solutions with the grid-converged referencesolution obtained using ARPS as well as withthe benchmark solution in Straka et al. (1993)

establishes the reliability of the model in handlinghighly nonlinear and transient solutions.

9. Summary and Discussion

The design philosophy, the choice of equationsand their formulations, the numerical integrationprocedures, and the parameterizations of thesubgrid-scale and PBL turbulence processes inthe ARPS have been described in this paper.The dynamical and numerical framework ofthe model is veri®ed against known solutionsof mountain waves and an observed severedownslope windstorm. It is also veri®ed using agrid-converged solution of a nonlinear densitycurrent. The results of the latter also clearlydemonstrate the superiority of a high-ordermonotonic advection scheme over conventionalschemes that are commonly employed in atmo-spheric models.

We believe the use of the generalized coordi-nate transform with horizontal stretching, thetreatment of terms related to terrain-following co-ordinate for truncation error reduction, the formu-lation of conservative high-order advectionterms, the implementation of monotonic advectionfor scalars, the coupling of PBL with ` freeatmosphere'' turbulence, the coupling of soil-vegetation model, surface layer, PBL and atmo-spheric radiation, as well as the computationalimplementation of the system have their uniqueaspects compared to other regional atmosphericprediction models. The numerical frameworkand the computational paradigm that have beenestablished provide a solid foundation uponwhich future improvements can be rapidly imple-mented.

It should be pointed out that only the forwardtime integration components of the ARPS modelsystem have been described here. The completeforecast system includes real time data ingest,data analysis, retrieval and assimilation compo-nents (Brewster, 1996; Shapiro et al., 1996), the4D adjoint based data assimilation system (Wanget al., 1995), as well as a platform-independentpost-processing package. A complete descriptionof these components is outside the scope of thispaper.

More detailed description of the coupledsoil-vegetation model, the treatment of surfacelayer ¯uxes, the microphysics and cumulus

Fig. 19. As in Fig. 13, but for the set of 100 m resolutionexperiments


parameterizations, and the radiation scheme usedin the ARPS will be presented in Part II of thispaper series. Additional veri®cation experimentsand an application of the model to the simulationof a multi-scale event containing multipletornadic supercell storms and an intense long-lived squall line can also be found there. Thesource code and the online documentations of theARPS are available at http://www.caps.ou.edu/ARPS.

Appendix A. De®nition of symbols

Cp Speci®c heat of dry air at constantpressure (J kgÿ1 Kÿ1)

cs cs � � RT�1=2, the full acoustic wave

speed (m sÿ1)Cv Speci®c heat of dry air at constant

volume (J kgÿ1 Kÿ1)Dij Deformation tensors (sÿ1)

f ;~f Coriolis parameters. f � 2sin�� and~f � 2cos�� (sÿ1) where � is the earthlatitude;

g Acceleration due to gravity (m sÿ2)Hj Turbulence heat or moisture ¯uxes

(kg K mÿ2 sÿ1);J1; J2; J3; J4;

��Gp

Coordinate transformation Jacobians(ND);

�, RdCÿ1p

Km;Kh Turbulence mixing coef®cient formomentum and scalars, respectively(m2 sÿ1);

l Turbulent mixing length (m);L Latent heat of evaporation;m Map projection factor (ND);N Dry or moist Brunt-V�as�ail�a frequency,

depending on local static stability (sÿ1);p; �p; p0 Total, base-state and perturbation

pressure (Pascal);Pr Turbulent Prandtl number (ND);q Generic form of water vapor and other

hydrometeor species (kg kgÿ1);_Q Adiabatic heating rate (K sÿ1);qh Hail/grapaul mixing ratio (kg kgÿ1);qi Cloud ice mixing ratio (kg kgÿ1);qli Total liquid and ice water mixing ratio

(kg kgÿ1)qr Rain water vapor mixing ratio

(kg kgÿ1)qs Snow mixing ratio (kg kgÿ1)qv Water vapor mixing ratio (kg kgÿ1)qvs Saturation water vapor mixing ratio

(kg kgÿ1);Rd Gas constant for dry air (J kgÿ1 Kÿ1);Rv Gas constant for water vapor

(J kgÿ1 Kÿ1);T; �T ; T 0 Total, base-state and perturbation

temperature (K);

u, v, w Cartesian velocity components inx, y and z directions (m sÿ1);

Uc;Vc;Wc Contravariant velocity components in�; � and � directions (m sÿ1);

x,y,z Cartesian coordinates (m); Angular rotation rate of the earth (sÿ1); Rd=Rv;�;�; �0 Total, base-state and perturbation Exner

function. � � �p=p0�Rd=Cpand p0 � 105

Pascal;�e Equivalent potential temperature (K);�; ��; �0 Total, base-state and perturbation

potential temperature (K);��

��Gp

(kg mÿ3);�; ��; �0 total, base-state and perturbation density

(kg mÿ3);�ij Stress tensors (kg mÿ1 sÿ2)�; �; � Coordinates in the computational space

corresponding to x, y and z (m)

Appendix B. The vertically implicit solutionof the p and w equations

When coef®cient � is not zero, Eqs. (28.3) and (28.4)become simultaneous equations for two unknowns, w��

and p0�� and can not be solved independently.After regrouping the unknown terms, we rewrite the p-

equation (28.4) as

p0�� p0� ��g��w� ÿ c2s ��x�y�=

��Gp

��w�� Fp;

�B:1�where Fp includes other known terms:

Fp � ��Gp f t

p��1ÿ��Gp

g�w�ÿ c2s x�y��w

h i�n oÿ��c2

s m2��Gp �� J3

�u=m

� �� J4

�v=m

ÿ �h� ��J1u=m

��

� � ��J2v=m��

��

Substituting p0�� in (B.1) into w-equation (28.3) andregrouping again yields

w�� Fw ÿ ��2x�y�= ��

� �� g��w� ÿ c2s ��x�y�=

��Gp

��wh i��

ÿ ��2g=��c2s

��

� g��w�� ÿ c2

s ��x�y�=��Gp

��w�

� ��

; �B:2�

where the known terms on the RHS are grouped into Fw,which is

Fw � w� ��= ��

f tw ÿ x�y��p0 � �Fp��

ÿ ��x�y��Div��

� g ��0=��ÿ �p0 � �Fp�=��c2s ��

�h i��

188 M. Xue et al.

Equation (B.2) now has only one unknown, w�� , and thespatial averaging and differencing are all performed in thevertical direction. Expressing the equation in an explicit®nite difference form yields a set of linear algebraicequations for w�� at three adjacent vertical levels:

Akw��kÿ1 � Bkw��

k � Ckw��k�1 � Dk; �B:3�

where k is the index for vertical levels, and the knowncoef®cients Ak;Bk;Ck and Dk are

Ak � �Pk ÿ Nk��Mkÿ1 � Lkÿ1�;Bk � 1� Nk�Mk ÿMkÿ1 � Lk � Lkÿ1�

� Pk�Mk �Mkÿ1 � Lk ÿ Lkÿ1�;Ck � �Pk � Nk��Mk ÿ Lk�;Dk � Fwk;

where

Pk � ��2g=�2��c2s

��; Nk � ��2x�y�=��;Mk � g��=2; and Lk � ��c2

s=��Gp�:

Equation (B.3) forms a linear tridiagonal equation systemand is solved using the Thomas algorithm (e.g., Richtmyerand Morton, 1967) given upper and lower boundaryconditions on w. For a rigid top, w � 0. For the radiationtop boundary, the inverse Fourier transform of (D.4) inAppendix D is used as the condition. At the surface, w isobtained from the nonpermeable condition that requires the¯ow to be parallel to the ground. Finally, solution w�� issubstituted into Eq. (B.1) to obtain p0�� , thus complete onesmall time step integration cycle.

Appendix C. Second- and fourth-order centeredadvection formulation for u, v, w, and p0

The second- and fourth-order advection for � was given inEq. (30). We present here the formulations of ADVU, ADVV,ADVW, and ADVP for the advection of u, v, w and p0.

ADVU � � m U��u�

� V��u�

� ��W��u

�� 1ÿ�� m U�2�

�2�u2�

� V��2�u2�

� �� W��2�u

2��; �C:1�

ADVV � � m U��v� � V��v

��

�W��v�

� �� 1ÿ�� m U��2�v

2�

� V�2��2�v

2�� W��2�v

2��; �C:2�

ADVW � � m U��w�

� V��w�

� ��W��w

�� 1ÿ �� m U��2�w

2�

� V��2�w2�

� �� W�2�

�2�w2��; �C:3�

ADVP � � m��Gp �

Uc��p0�

��Gp �

Vc��p0�

!"

��Gp �

Wc��p0�#

� �1ÿ �� m��Gp �

Uc

�

�2�p02�

��Gp �

Vc

�

�2�p02�

0@ 1A24�

��Gp �

Wc

�

�2�p02�35: �C:4�

It is shown in Xue and Lin (2000) that the above advectionformulation, for both second and fourth-order cases, exactlyconserves the total kinetic energy and the total variance ofthe advected scalars if the mass continuity equationdifferenced using consistent second or fourth-order differ-ence scheme is exactly satis®ed. Under the same condition, italso conserves the global integral of the advected quantitiesthemselves (e.g., domain integrated momentum).

Appendix D. Implementation of radiation topboundary condition

The Klemp and Durran (1983) type wave-permeableradiation upper boundary condition is implemented in theARPS. The method is based on an analysis of linearhydrostatic mountain waves. By requiring the downwardenergy transport by hydrostatic gravity waves to be zero, thefollowing relationship between the Fourier transformedamplitudes of w and p0 at the top boundary can be obtained:

pnzÿ2 � N��

kwnzÿ1; �D:1�

where N is the Brunt-VaÈisaÈlaÈ frequency and k ��k2

x � k2y

qis

the horizontal wavenumber with kx � 2�x

sin kx�x2

ÿ �and

ky � 2�y

sinky�y

2

� �being the discretized approximations to

wavenumbers kx and ky in x and y directions, respectively. InEq. (D.1), pnzÿ2 is located one-half grid level below the topboundary, while wnzÿ1 is located at the boundary. In thederivation, it is assumed that the horizontal variation in thebase-state, and hence, in the coef®cients of the equation, issmall and can be neglected. Further, pnzÿ2 is approximated tobe the value at the w point.

Equation (B.1) in Appendix B can be rewritten for levelk � nzÿ 2 as

p0nzÿ2 � awnzÿ1 � bwnzÿ2 � c �D:2�

where

a � ��g��

2ÿ c2

s ��x�y��Gp

��

� �;

b � ��g��

2� c2

s ��x�y��Gp

��

� �;


and

c � Fp � p0�nzÿ2:

Note that we have dropped the superscript � �� forconvenience. Performing a double cosine Fourier transformon (D.2), and assuming that the coef®cients are slowlyvarying functions of x and y, Eq. (D.2) becomes

pnzÿ2 � awnzÿ1 � bwnzÿ2 � c: �D:3�eliminating pnzÿ2 from (D.1) and (D.3) yields

aÿ N��

k

� �wnzÿ1 � bwnzÿ2 � c � 0 �D:4�

which, after being transformed back into the physical space,serves as the top boundary condition required by (B.3). Thepressure at the top boundary is then obtained from Eq. (D.2).

Acknowledgements

The Center for Analysis and Prediction of Storms wassupported by the National Science Foundation and FederalAviation Administration through combined Grant andCooperative Agreement ATM92-20009. A special acknowl-edgment is in order for our colleagues who contributed invarious ways to the development and testing of the ARPS.We would like to thank H. Jin, D. Weber, Y. Liu, X. Song,D.-H. Wang, A. Sathye, D. Jahn, L. Zhao, G. Bassett, R.Carpenter, D. Hou, J. Zong, S. Weygandt, J. Levit, S.Lazarus, J. Zhang, Y. Richardson, Y. Tang, M. Zou, and S.-K.Park. Special thanks go to Dr. Alan Shapiro, who led a localusers group in the early days that tested various versions ofthe model, to Dr. Keith Brewster, who is the author ofADAS, and to Dr. Fred Carr, who is instrumental in thedevelopment of the data analysis system.

We have bene®ted greatly from collaborations andexchanges with numerous scientists, including Drs. W.Skamarock, A. Crook, J. Dudhia, G. Tripoli, W.-K. Tao, J.Straka, L. Wicker, J.S. Kain, C. Mattocks. Keith Brewsterand Dan Weber are thanked for reviewing the manuscript.

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Author's addresses: Dr. Ming Xue, School of Meteorology,University of Oklahoma, Sarkeys Energy Center, Suite 1310,100 East Boyd, Norman, OK 73019 (E-Mail: [email protected])


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