Research Article An MCV Nonhydrostatic Atmospheric Model ...downloads.hindawi.com/journals/amete/2016/4513823.pdf · e MCV scheme is adopted in this model to solve the governing equations

Research ArticleAn MCV Nonhydrostatic Atmospheric Model withHeight-Based Terrain following Coordinate Tests ofWaves over Steep Mountains

Xingliang Li12 Xueshun Shen23 Feng Xiao4 and Chungang Chen5

1National Meteorological Center China Meteorological Administration Beijing 10086 China2Center of Numerical Weather Prediction China Meteorological Administration Beijing 10086 China3State Key Laboratory of Severe Weather Chinese Academy of Meteorological Sciences Beijing 100086 China4Department of Energy Sciences Tokyo Institute of Technology Yokohama 226-8502 Japan5School of Human Settlement and Civil Engineering Xirsquoan Jiaotong University Xirsquoan Shaanxi 710049 China

Correspondence should be addressed to Chungang Chen cgchenmailxjtueducn

Received 20 November 2015 Accepted 4 February 2016

Academic Editor Xiao-Ming Hu

Copyright copy 2016 Xingliang Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A nonhydrostatic atmospheric model was tested with the mountain waves over various bell-shaped mountains The model isrecently proposed by using the MCV (multimoment constrained finite volume) schemes with the height-based terrain followingcoordinate representing the topography As discussed in our previous work themodel has some appealing features for atmosphericmodeling and can be expected as a practical framework of the dynamic cores which well balances the numerical accuracy andalgorithmic complexity The flows over the mountains of various half widths and heights were simulated with the model Thesemianalytic solutions to the mountain waves through the linear theory are used to check the performance of the MCV model Itis revealed that the present model can accurately reproduce various mountain waves including those generated by the mountainswith very steep inclination and is very promising for numerically simulating atmospheric flows over complex terrains

1 Introduction

Mountain weather processes such as lee waves rotors anddownslope windstorms which have great influence on theair quality over complex mountainous terrains [1 2] involvea wide range scales of air motions and present a challengeto numerical modeling With the rapid development ofcomputer hardware it is now possible for the atmosphericmodels to represent the complexity in topography with theincreasing horizontal resolutions and thus provide moreaccurate predictions for the mountain weather processesThe ability to simulate the atmospheric flows over complexmountainous areas becomes highly demanded for the non-hydrostatic atmospheric numerical models such as MM5[3] MC2 [4] COAMPS [5] LM [6] ARPS [7] WRF [8]and GRAPES [9] Though the significant advancements havebeen achieved during the past several decades adequate

simulations and predictions of the complex terrain-forcedweather processes for example mountain waves still remainan issue unsatisfactorily resolved Further efforts are stillrequired to develop more reliable dynamic cores with accu-rate representations of the topographic effects to improve thesimulation of atmospheric flows over complex terrains

Recently a new nonhydrostatic model was developed byusing the multimoment constrained finite volume (MCV)method [10] Different from the conventional finite volumemethod the unknowns (or the degree of freedom (DOFs))are defined as the values at the solution points distributedwithin each mesh cell In contrast to the direct multimomentmethod [11ndash18] where the moments are directly used asthe predicted unknowns the MCV formulation [19] updatesthe DOFs as the point values at the solution points whosetime evolution equations are derived by applying a setof constrained conditions imposing on different types of

Hindawi Publishing CorporationAdvances in MeteorologyVolume 2016 Article ID 4513823 12 pageshttpdxdoiorg10115520164513823

2 Advances in Meteorology

moments such as volume-integrated average (VIA) pointvalue (PV) and spatial derivative values (DV) and is thussimple efficient and easy to implement Being a high orderscheme more accurate numerical results can be obtainedin terms of the equivalent DOF resolution in comparisonwith the traditional finite volumemethod evenwith relativelycoarse grid spacing The rigorous numerical conservationin MCV model is exactly guaranteed by a constraint onthe VIA through a finite volume formulation of flux formBeing a new nodal-type high order conservative methodit is much beneficial to compute the metric and sourceterms in an MCV model which are always involved in thetreatments of spherical geometry in the horizontal directionand coordinate transformation in the vertical direction fortopographic effect The MCV model has some appealingfeatures for atmospheric modeling such as the rigorousnumerical conservation good computational efficiency andflexible configuration for solution points and thus can beexpected as a practical framework of the dynamic coreswhich well balances the numerical accuracy and algorithmiccomplexity [10 20] The competitive results of the widelyadopted benchmark tests can be referred to [10 20ndash24]

To deal with the bottom topography the height-basedterrain following coordinate is adopted in our MCV modelSince Phillipsrsquo pioneering work [25] the terrain followingcoordinates mainly classified into the pressure-based coor-dinate and the height-based coordinate have been widelyadopted to represent the underlying mountainous surface inatmospheric models During the past several decades theheight-based coordinate [26] has got an increasing popu-larity mainly due to its applicability for both hydrostaticand nonhydrostatic models and computational simplicityA variational grid generation technique [27] was adaptedto mountain wave simulation as well Modified version ofterrain following coordinate has been also proposed [28] tocircumvent to some extent the drawbacks of the coordinatein representing steep topography In this study a set ofbenchmark tests of mountain waves generated by a con-stant background flow over mountains with different steep-ness which essentially represent the complex mechanismsinvolved inmountain weather processes are examined by theMCV nonhydrostatic model using the height-based terrainfollowing coordinate in order to verify the performance of theMCV model in simulating the mountain weather processesand its potential for the further numerical investigations onthe flows in the atmosphere boundary layer and the air qualityover complex mountainous terrainThe numerical results areevaluated in comparison with the semianalytical solutionsobtained from the linear theory [29] as well as the numericalsolutions from other representative models

The remainder of this paper is organized as followsIn Section 2 the compressible nonhydrostatic atmosphericmodel using the MCV scheme and the height-based ter-rain following coordinate is briefly introduced Section 3describes the mountain tests and the semianalytical solutionsthrough the linear theory Section 4 discusses numericalresults of variousmountainwaves by theMCVmodel Finallya short conclusion is given in Section 5

2 2D MCV NonhydrostaticAtmospheric Model

In order to consider complex topography as the bottomboundary of the atmospheric model a height-based terrainfollowing coordinate is used in this study to map the physicaldomain (119909 119911) to the computational domain (119909 120577) through thetransformation

119911 (120577) = 119911119878 (119909) +120577

119911119879[119911119879minus 119911119878 (119909)] (1)

where 119911119878(119909) is the elevation of topography 119911

119879the altitude of

the model top and 120577 isin [0 119911119879]

Using a height-based terrain following coordinate 2Dcompressible and nonhydrostatic governing equations foratmospheric dynamics are written in flux form as

120597q120597119905+120597f120597119909

+120597g120597120577

= s (q) (2)

where

q =[[[[[[

[

radic1198661205881015840

radic119866120588119906

radic119866120588119908

radic119866 (120588120579)1015840

]]]]]]

]

f =[[[[[[

[

radic119866120588119906

radic1198661205881199062 + radic1198661199011015840

radic119866120588119908119906

radic119866120588120579119906

]]]]]]

]

g =[[[[[[

[

radic119866120588

radic119866120588119906 + radic119866119866131199011015840

radic119866120588119908 + 1199011015840

radic119866120588120579

]]]]]]

]

s =[[[[[

[

0

0

minusradic1198661205881015840119892

0

]]]]]

]

(3)

where 120588 is density (119906 119908) are velocity vector in the physicaldomain = 119889120577119889119905 is the vertical velocity in the transformedcoordinates 11986613 = 120597120577120597119909 andradic119866 = 120597119911120597120577 is the Jacobian oftransformation

Advances in Meteorology 3

The thermodynamic variables are split into a referencestate and the deviations to improve the accuracy of thenumerical model as

120588 (x 119905) = 120588 (119911) + 1205881015840 (x 119905)

119901 (x 119905) = 119901 (119911) + 1199011015840 (x 119905)

(120588120579) (x 119905) = (120588120579) (119911) + (120588120579)1015840 (x 119905)

(4)

where the reference pressure 119901(119911) and the density 120588(119911) satisfythe local hydrostatic balance x is the position vector 1199011015840 =1205760(120588120579)1015840 and 120576

0= 1205741198620(120588120579)120574minus1

The constants used in the simulations are specified asfollows Gravitational acceleration 119892 = 980616msminus2 idealgas constant for dry air 119877

119889= 287 Jkgminus1Kminus1 specific heat

at constant pressure 119888119901= 10045 Jkgminus1Kminus1 specific heat at

constant volume 119888V = 7175 Jkgminus1Kminus1 120574 = 119888

119901119888V = 14 refer-

ence pressure at the surface 1199010 = 105 Pa and constant 1198620 =

119877119889120574119901minus119877119889119888V0

The MCV scheme is adopted in this model to solve

the governing equations (2) The MCV scheme is a generalnumerical framework for developing high order numericalmodels to solve the hyperbolic systems A major featurewhich distinguishes MCV scheme from other conventionalnumerical schemes is the local high order spatial recon-struction For the sake of brevity we omit the details of thenumerical formulation of MCV nonhydrostatic model in thepresent paper The fourth-order MCV scheme and the 3rdTVDRunge-Kutta time scheme [30] are adopted in this studyA local Lax-Friedrich approximate Riemann solver [31] isused for computational efficiency The interested readers arereferred to [10] for details

The boundary conditions on the bottom surface are ofcrucial importance in atmospheric models especially forsimulations of the waves generated by complex topographyFor the test cases studied in this paper no-flux conditionis imposed along the bottom boundary and nonreflectingcondition is used for the lateral and the top boundaries

The no-flux boundary condition requires the velocityfield to satisfy the relation

u sdot n = 0 (5)

where n is the outward unit normal vector of the bottomsurface and u = (119906 119908)

119879 is velocity vector on the bottomboundary

The nonreflecting boundary conditions are realized by asponge layer along the lateral and top boundaries that relaxesthe numerical solution to the prescribed reference Thedamping terms are added to the momentum and potentialtemperature equations as

120597q120597119905

= [governing equation terms] minus 120591 (q minus q119887) (6)

where 120591 is the relaxation coefficients and q119887is the speci-

fied reference state More details about the strength of theRayleigh damping 120591 and the formulations of semidiscrete no-flux condition are described in [10]

Table 1 Configuration of mountains in different test cases

Test cases ℎ0(m) 119886 (m) 119897 (mminus1) 120572 (deg) 119886119897

1 LHM 1 10000 195 times 10minus3 0006 1952 LNHM 1 1000 1 times 10minus3 006 1

3 A3 100 100 2 times 10minus3 265 02

A4 100 50 2 times 10minus3 45 01

4 D1 500 500 1 times 10minus3 265 05D2 500 250 1 times 10minus3 45 025

3 Mountain Wave Test Cases

Satomura et al [32] suggested a set of mountain wavetests to evaluate the capability of atmospheric models inreproducing topographic effects The isolated bell-shapedbottom mountain to trigger the waves is specified as

119911119878 (119909) =ℎ0

1 + (119909 minus 1199090)21198862

(7)

where ℎ0is the maximum height of the mountain 119909

0is the

center of physical domain and 119886 is the half width of themountain

Six test cases with differentmountain height and horizon-tal width adopted in this study are given in Table 1 In thesecases themountain slopes (measured by averaged inclinationangles 120572) vary from 0006 to 45 degrees The cases of largeinclination angles indicate steep mountains and are verychallenging for the height-based terrain following coordinateThe initial hydrostatic conditions in these tests are specifiedin terms of Exner pressure Π = (119901119901

0)119877119889119888119901 and potential

temperature 120579 via the hydrostatic relation 119889Π119889119911 = minus119892(119888119901120579)

Setup of each case will be described in detail in Section 4The semianalytical solutions to mountain waves are

derived from the linear theory We first briefly introduce thelinear theory before discussing the numerical results

Using the linear theory [29] the small-amplitude 2Dmountain waves can be described by a single partial differ-ential equation as

120578119909119909 + 120578119911119911 + 1198972120578 = 0 (8)

where 120578 is the vertical displacement of a parcel at a steady stateand 119897 = 119873

0119880 is Scorer parameter which is constant for an

elastic isothermal and nonshear uniform flow1198730is Brunt-

Vaisala frequencyFor a bell-shaped mountain studied in this paper the

analytic solution of (8) can be obtained by using the Fouriertransform method The vertical displacement of a parcel120578(119909 119911) is obtained by

120578 (119909 119911) = (1205880

120588 (119911))

12

sdot Reint119897

0

ℎ (119896) exp [119894 (1198972 minus 1198962)12

119911] exp (119894119896119909) 119889119896

+ intinfin

119897

ℎ (119896) exp [minus (1198962 minus 1198972)12

119911] exp (119894119896119909) 119889119896

(9)


where 119896 is the wave number ℎ(119896) is Fourier transform of themountain shape 119894 = radicminus1 and Re indicates the real part ofa complex number

From (9) it is found that the solution for 120578(119909 119911) isproportional to the maximum height of mountain ℎ

0since

ℎ(119896) = ℎ0119886 exp(minus119896119886) The flow pattern of the mountain wave

depends on the dimensionless quantity 119886119897 When 119886119897 ≪ 1the wave disappears and a simple uniform flow is retrievedForced by the lower topographic surface the flow climbs upand goes down along themountain and the highest speed andlowest pressure occur at the top of the mountain When 119886119897 ≫1 the buoyancy-dominated hydrostatic flow over an isolatedridge develops and the disturbance energy is propagatingupward away from the mountain When 119886119897 = 1 the flow is inthe nonhydrostatic regime and the nonhydrostaticmountainwaves will appear It is also necessary to consider anothernondimensional parameter ℎ0119897 that is mountain heightscaled by Scorer parameter which suggests the applicabilityof linear theory Generally the small value of ℎ

0119897 (ℎ0119897 ≪ 1)

in test cases 1 to 3 meets the linear theory assumption sincethemountain is relatively low in comparison with the verticalstratification While ℎ

0119897 asymp 1 in test case 4 the nonlinear

effects have to be considered for a high mountainThe vertical flux of horizontal momentum is a measure of

the performance of a numerical model to simulate the waveenergy Following Eliassen and Palm [33] and Smith [29]the vertical flux of horizontal momentum (calledmomentumflux hereafter) is defined as

119872(119911) = int+infin

minusinfin

120588 (119911) 1199061015840(119909 119911) 119908

1015840(119909 119911) 119889119909 (10)

where 120588(119911) is the reference density and 1199061015840 and1199081015840 are the winddisturbances from the basic state

119872(119911) is constant for linear mountain waves in a constantmean wind [33] Furthermore when the linear mountainwaves are irrotational and hydrostatic the analytic momen-tum flux is given as

119872119867 (119911) = minus120587

4120588011987301198800ℎ

2

0 (11)

where 1205880and 1198800 are the reference density and horizontal

velocity values at the ground level

4 Numerical Results

41 Case 1 Linear Hydrostatic Mountain Waves The initialstate is an isothermal atmosphere with the constant tempera-ture of 119879 = 250Kmoving at a constant velocity119880 = 20msminus1The Brunt-Vaisala frequency119873

0 = 119892radic119888119901119879 in this case As aresult the reference Exner pressure is Π = exp(minus(119892119888

119901119879)119911)

The computational domain is [0 240000]m times [0 30000]mwith the grid spacing of Δ119909 = 1200m and Δ120577 = 240m TheMCVmodel is integrated up to 10 hours

According to the linear theory the parameter 119886119897 deter-mines the hydrostatically balanced mountain waves in thistest Figures 1(a) and 1(b) show the numerical and the semi-analytical solutions of vertical velocity component119908 at 119905 = 10

hours TheMCVmodel can accurately capture the mountainwaves triggered by the gentle slope except the numericalmaxima andminima contours in the vertical velocity field areslightly weaker than the analytic ones Compared with theexisting numerical results obtained by high order schemessuch as spectral element (SE) and discontinuous Galerkin(DG) the present results look quite similar to those in [34]

The momentum fluxes at different instants are plotted inFigure 1(c) Values normalized by119872119867 given in (11) are shownAs the linear hydrostatic mountain waves exist in this testthe vertical variation of the normalized momentum flux isusually used to examine the numerical dissipation of amodelAs shown in Figure 1(c) as the mountain wave develops intoa mature state the momentum flux gradually approaches avertically uniform distribution with a value close to unity ofthe analytical solution At 119905 = 10 (ie nondimensional time119880119905119886 = 72) the numerical flux is about 099 at the surface andapproaches 09645 at the height of double vertical wavelength(119911 = 2120587119897 asymp 64 km) Present results are competitive to thoseshown in [7 35] where the flux at 119911 = 64 km is 94 of itssteady value at nondimensional time of 60 in [35] and the fluxreaches 096 at later time at a height just below their Rayleighdamping layer (12 km) in [7] The results reveal that the highorder accuracy of MCVmodel is much beneficial to improvethe simulation of nondissipative hydrostaticmountain waves

42 Case 2 Linear Nonhydrostatic Mountain Waves Theinitial state of the atmosphere is specified as a flow ofuniform horizontal velocity 119880 = 10msminus1 with Brunt-Vaisalafrequency119873

0 = 001 sminus1The reference potential temperature

and Exner pressure are

120579 = 1205790exp(

1198732

0

119892119911) (12)

Π = 1 +1198922

119888119901120579011987320

(exp(minus11987320

119892119911) minus 1) (13)

respectively with 1205790= 280K The computational domain is

[0 144000]mtimes [0 30000]mwith a resolution of Δ119909 = 360mand Δ120577 = 300m The model runs for 5 hours The flow is inthe nonhydrostatic regime since the dimensionless parameter119886119897 = 1 The averaged inclination angle is about 006 degreeswhich is larger than that of case 1Thenonreflecting boundaryconditions in this case are imposed by placing the dampingterm in the computational domain 120577 ge 15 km for the topboundary and the thickness of 30 km for the lateral boundary

Figures 2(a) and 2(b) show the numerical and the semian-alytical solutions of vertical velocity 119908 after 5 hours (nondi-mensional time of 180) for linear nonhydrostatic mountainwaves It is observed that the linear nonhydrostatic mountainwaves are distinguished from the linear hydrostatic ones bythe dispersive character of wave trains behind the mountainpeak The simulated vertical velocity agrees well with theanalytical solution and the numerical result of other existinghigh order schemes for example DG3 model in [34]


100 120 16080 140X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

100 120 16080 140X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)

02

2 hours4 hours6 hours

8 hours10 hours

108 120 04 06Normalized momentum flux

0

2

4

6

8

10

12

Hei

ght (

km)

(c)

Figure 1 Numerical results of linear hydrostatic mountain waves Numerical solution (a) semianalytical solution (b) and normalizedmomentum flux (c) are shown The contour values vary from minus0005 to 0005 with an interval of 00005 Negative values are denoted bydashed lines

Similar to previous case the momentum flux profilesat 1 2 3 4 and 5 hours are plotted in Figure 2(c) It isnoted that the momentum flux profiles are normalized bythe analytic nonhydrostatic momentum flux 119872NH(119911) =

0457119872119867(119911) which is only valid for 119886119897 = 1 It can beseen that the model results have almost reached the steadystate at nondimensional time of 180 This situation is alsoobserved from the convergence of momentum flux at thenondimensional time of 72 108 and 144 As a matter of factthe flux at nondimensional time of 180 approaches 097 atthe height of 12 km The MCV nonhydrostatic model can

simulate the linear nonhydrostaticmountain waves quite wellas the linear theory predicts

43 Case 3 Simple Flows Two types of waves are checkedin case 3 which are denoted as A3 and A4 cases followingSatomura et al [32] In these cases the wave structuresdiminish and the topographic forcing is limited only to thelower atmosphere which is referred to as the simple flowpattern in this paperThe reference constantwind at the initialtime is 119880 = 10msminus1 with Brunt-Vaisala frequency 119873

0=

002 sminus1 for both cases The reference potential temperature


75 10560 90X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

75 10560 90X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)

02 108 120 04 06Normalized momentum flux

0

2

4

6

8

10

12

Hei

ght (

km)

1 hour2 hours3 hours

4 hours5 hours

(c)

Figure 2 Same as Figure 1 but for linear nonhydrostatic case

and the Exner pressure are determined the same as in case 2but with 120579

0= 250K

431 A3 Case The computational domain is [0 12000]m times

[0 8000]mwith a grid spacing of Δ119909 = 60m and Δ120577 = 60mThemodel runs for 20 minutes (nondimensional time of 120)until it reaches the steady state Compared with cases 1 and2 the mountain slope is steeper that is roughly 265 degreesin regard to averaged inclination anglesThe damping term isimposed in the computational domain 120577 ge 4 km for the topboundary and the thickness of 3 km for the lateral boundaryto assure the nonreflecting boundary conditions

Figures 3(a) and 3(b) depict the numerical and the semi-analytical solutions of vertical velocity 119908 at nondimensionaltime of 120The numerical results byMCVmodel are visuallyidentical to those by the linear theory In case of the equiv-alent DOF resolution that is Δ119909DOF = Δ1199093 = 20m thepresent results agree well with those in Satomura et al [32]The dimensionless parameter 119886119897 = 02 indicates that a simpleirrotational flowwill appear in the simulation It ismanifestedby the streamlines shown in Figure 3(c) where the steadyflows climb up to the peak of mountain and then go downand the largest speed and the lowest pressure are observedat the top of the mountain The momentum flux scaled by119872119867 given in (11) is presented in Figure 3(d) It should be


55 6 65 75X (km)

0

05

1

15

2Z

(km

)

(a)

55 6 65 75X (km)

0

05

1

15

2

Z (k

m)

(b)

0

05

1

15

Z (k

m)

55 6 65 75X (km)

(c)

0

05

1

15

2

25

3

35

4H

eigh

t (km

)


20 min(d)

Figure 3 Numerical results of case A3 Numerical solution (a) semianalytic solution (b) streamlines from 0 to half vertical wavelengthheight over an isolated mountain (c) and normalized momentum flux (d) are shown Contour interval is 025msminus1 All negative values aredenoted by dashed lines

pointed out here that we use ℎ0= 27m instead of the real

mountain height to calculate 119872119867to make the normalized

values of momentum flux distributed around unity for betterillustration The values of normalized flux in our modelremain between 02 and 12 Similar treatment is also adoptedin following test cases The profile below the nonreflectingabsorbing layer at 4 km shows small fluctuations which arealso observed in [32 Figure 9(a)]


[0 2000]mwith a grid spacing of Δ119909 = 15m and Δ120577 = 15m

Themodel is integrated for 10minutes (nondimensional timeof 120) The nonreflecting boundary conditions in this caseare imposed by placing damping term in the computationaldomain 120577 ge 1 km for the top boundary and |119909 minus 15 km| ge12 km for the lateral boundary

Figures 4(a) and 4(b) show that the numerical andthe semianalytical solutions of vertical velocity 119908are atnondimensional time of 120 which agree quite well Ourresults also agree well with those in [32] when using theequivalent DOF resolution that is Δ119909DOF = 5m Theconcavity of vertical velocity near the ground is observed in


14 15 16 1713X (km)

0

01

02

03

04

05Z

(km

)

(a)

14 15 16 1713X (km)

0

01

02

03

04

05

Z (k

m)

(b)

12 14 16 18 21X (km)

0

05

1

15

Z

(c)

0

02

04

06

08

1H

eigh

t (km

)


10 min(d)

Figure 4 Same as Figure 3 but for case A4 Contour interval is 10msminus1

the present results which also appears in MRINPD-NHMmodel with horizontally explicit and vertically implicit (HE-VI) method and TSOmodel in [32] Compared with A3 casethe dimensionless parameter 119886119897 is smaller and equals 01 Asa result the wave structure predicted by the linear theory isthe simple pattern flow and confirmed by streamlines shownin Figure 4(c)

Themountain slope in this case is steeper than theA3 casedue to a narrower half mountain width 119886 and the averagedinclination angle is about 45 degrees This case verifies theperformance of the proposed numerical model to simulatethe waves generated by steep mountains Furthermore thenormalized flux profile scaled by formulation (11) with ℎ

0=

21m is shown in Figure 4(d) It is observed that the fluxprofile below the nonreflecting absorbing layer at 1 km isnearly unity which is better than those in Satomura et al [32]

44 Case 4 Nonlinear Mountain Waves Similar to case 3two types of mountain waves are tested in case 4 and denotedbyD1 andD2 cases in Satomura et al [32]The initial constantmean flow is119880 = 10msminus1 with Brunt-Vaisala frequency1198730 =001 sminus1 for both cases The reference potential temperatureand the Exner pressure are computed the same as in case2 The model runs for 100 minutes In case 4 the parameterℎ0119897 asymp 1 and the mountain is specified to be high enough to


0

2

4

6

8

10Z

(km

)

10 12 14 16 18 20 228X (km)

(a)

0

2

4

6

8

10

Z (k

m)

10 12 14 16 18 20 228X (km)

(b)

0

1

2

4

6

Z (k

m)

5 15 20 2510X (km)

(c)

0

2

4

6

8

10

12H

eigh

t (km

)

02 04 06 08 1 120Normalized momentum flux

100 min(d)

Figure 5 Same as Figure 3 but for case D1 Contour interval is 02msminus1

take the nonlinear effects into account [29] which is differentfrom other test cases in this study

441 D1 Case The computational domain is [0 18000]m times

[0 30000]m with a grid spacing of Δ119909 = 150m and Δ120577 =150m Similar with A3 case the averaged inclination angleis about 265 degrees The damping term is placed in thecomputational domain 120577 ge 10 km for the top boundary and|119909| ge 10 km for the lateral boundary

Figures 5(a) and 5(b) give the numerical and the semi-analytical solutions of vertical velocity 119908 at nondimensionaltime of 120Thepattern of vertical velocity simulated byMCVmodel compares quite well with that of the linear theory

Compared with the results shown in [32] our results agreewell with those of TSOmodelwhen using the equivalentDOFresolution that is Δ119909DOF = 50m It is noticed that thereis slight difference in the amplitudes between our result andthe analytical solution It may owe to the nonlinear effects ofthe high mountain Different from case 3 the dimensionlessparameter 119886119897 equals 05 so that the waves will propagateupward The streamlines are plotted in Figure 5(c) and it isobserved that the mountain waves propagate upward behindthe mountain peak The normalized flux profile scaled byformulation (11) with ℎ

0= 250m is presented in Figure 5(d)

It can be seen that the flux profile below 12 km is nearlyunity whereas the flux fluctuation is visible under the height


10 12 14 16 18 20 228X (km)

0

2

4

6

8

10Z

(km

)

(a)

10 12 14 16 18 20 228X (km)

0

2

4

6

8

10

Z (k

m)

(b)

5 15 20 2510X (km)

0

1

2

4

6

Z (k

m)

(c)100 min


0

2

4

6

8

10

12H

eigh

t (km

)

(d)

Figure 6 Same as Figure 3 but for Case D2 Contour interval is 05msminus1 in this case

of 8 km Although the mountain is high enough to takeinto account the nonlinear effects the flux profile from thenumerical model is close to that of the linear theory exceptthe fluctuation structure

442 D2 Case In this case all mountain parameters are thesame as in the D1 case except the half width of mountain 119886 =250m which leads to the increase of the averaged inclinationangle of mountain to about 45 degrees The dimensionlessparameter 119886119897 = 025 and in this case the vertically propa-gating wave will not be noticeable The boundary conditionsare imposed the same as D1 case

The numerical and semianalytical solutions of verticalvelocity 119908 at nondimensional time of 240 are plotted inFigures 6(a) and 6(b) It is seen that the wave propagates withthe similar structure compared with that by the linear theoryHowever the vertical velocities are larger than the analyticalsolution due to the significant nonlinear effects of the highmountain which are also observed in [32]Thedimensionlessparameter 119886119897 equals 025 and is smaller than D1 case whichmeans the upward propagatingwaves are relativelyweakThisis confirmed by the numerical results shown in Figure 6(c)for the streamlines The normalized flux profile scaled byformulation (11) with ℎ

0= 180m is shown in Figure 6(d)


The values of normalized flux by the MCV model arenearly unity and fluctuate with the height The flux profilestructure at nondimensional time of 240 is similar to thoseof MRINPD-NHM model (HE-VI) and TSO model in [32]when using the equivalent DOF resolution Δ119909DOF = 50m

5 Conclusions

The mountain waves triggered by a constant flow over abell-shaped mountain are investigated by MCV compress-ible nonhydrostatic atmospheric model with the applicationof the height-based terrain following coordinate A set ofmountain wave test cases in which the mountain slopesrange from 0006 to 45 degrees are systematically checkedComparison with the semianalytical solution from the lin-ear theory reveals that the present model can accuratelyreproduce various mountain waves in the linear regimeThe numerical results are also compared with other existingmodels including high order models using SE and DGschemes as well as other operational and research modelswhich verifies the numerical accuracy of the present modelin representing complex topography of different steepness

Verified in this work theMCVmodel can accurately cap-ture various mountain-triggered waves by using the height-based terrain following coordinate which thus provides a toolfor better understanding the mountainous weather processesand the related air pollution in the boundary layers overmountainous terrains

Being nodal-type high order scheme the MCV methodhas significant advantage in treating geometrical metricterms It can be concluded from the present study thatthe present numerical framework which adopts the MCVscheme and the height-based terrain following coordinate ispromising for further developing 3D nonhydrostatic atmo-spheric model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study is supported by National Key Technology RampDProgram of China (Grant no 2012BAC22B01) National Nat-ural Science Foundation fund for Creative Research Group(Grant no 41221064) Natural Science Foundation of China(Grant nos 11372242 41375108 and 41522504) and in partbyGrants-in-Aid for Scientific Research Japan Society for thePromotion of Science (Grant no 24560187)

References

[1] F K Chow S F J De Wekker and B J Snyder Moun-tain Weather Research and Forecasting Recent Progress andCurrent Challenges Springer Atmospheric Sciences SpringerDordrecht The Netherlands 2013

[2] X-MHu Z QMaW L Lin et al ldquoImpact of the Loess Plateauon the atmospheric boundary layer structure and air quality

in the North China Plain a case studyrdquo Science of the TotalEnvironment vol 499 pp 228ndash237 2014

[3] J Dudhia ldquoA nonhydrostatic version of the Penn State-NCARmesoscale model validation tests and simulation of an Atlanticcyclone and cold frontrdquoMonthlyWeather Review vol 121 no 5pp 1493ndash1513 1993

[4] R Benoit M Desgagne P Pellerin S Pellerin Y Chartierand S Desjardins ldquoThe Canadian MC2 asemi-Lagrangiansemi-implicit wideband atmospheric model suited for finescaleprocess studies and simulationrdquo Monthly Weather Review vol125 no 10 pp 2382ndash2415 1997

[5] R M Hodur ldquoThe naval research laboratoryrsquos coupled oceanatmospheremesoscale prediction system (COAMPS)rdquoMonthlyWeather Review vol 125 no 7 pp 1414ndash1430 1997

[6] G Doms and U Schattler ldquoThe nonhydrostatic limited-areamodel LM (lokal modell) of DWD Part I Scientific documen-tationrdquo Deutscher Wetterdienst Rep LM F90 135 DeutscherWetterdienst (DWD) Offenbach Germany 1999

[7] M Xue K K Droegemeier and V Wong ldquoThe AdvancedRegional Prediction System (ARPS)mdasha multi-scale nonhy-drostatic atmospheric simulation and prediction model PartI model dynamics and verificationrdquo Meteorology and Atmo-spheric Physics vol 75 no 3-4 pp 161ndash193 2000

[8] W C Skamarock and J B Klemp ldquoA time-split nonhydrostaticatmospheric model for weather research and forecasting appli-cationsrdquo Journal of Computational Physics vol 227 no 7 pp3465ndash3485 2008

[9] D Chen J Xue X Yang et al ldquoNew generation of multi-scaleNWP system (GRAPES) general scientific designrdquo ChineseScience Bulletin vol 53 no 22 pp 3433ndash3445 2008

[10] X L Li C G Chen X S Shen and F Xiao ldquoA multi-moment constrained finite-volume model for nonhydrostaticatmospheric dynamicsrdquo Monthly Weather Review vol 141 no4 pp 1216ndash1240 2013

[11] T Yabe and T Aoki ldquoA universal solver for hyperbolic equationsby cubic-polynomial interpolation I One-dimensional solverrdquoComputer Physics Communications vol 66 no 2-3 pp 219ndash2321991

[12] T Yabe F Xiao and T Utsumi ldquoThe constrained interpolationprofile method for multiphase analysisrdquo Journal of Computa-tional Physics vol 169 no 2 pp 556ndash593 2001

[13] F Xiao ldquoUnified formulation for compressible and incom-pressible flows by using multi-integrated moments I One-dimensional inviscid compressible flowrdquo Journal of Computa-tional Physics vol 195 no 2 pp 629ndash654 2004

[14] F Xiao R Akoh and S Ii ldquoUnified formulation for com-pressible and incompressible flows by using multi-integratedmoments II Multi-dimensional version for compressible andincompressible flowsrdquo Journal of Computational Physics vol213 no 1 pp 31ndash56 2006

[15] F Xiao and S Ii ldquoCIPMulti-moment finite volume methodwith arbitrary order of accuracyrdquo in Proceedings of the 12thComputational Engineering Conference vol 12 pp 409ndash415Japan Society for Computational Engineering and ScienceTokyo Japan May 2007 httparxivorgabs12076822

[16] S Ii and F Xiao ldquoCIPmulti-moment finite volume method forEuler equations a semi-Lagrangian characteristic formulationrdquoJournal of Computational Physics vol 222 no 2 pp 849ndash8712007

[17] C G Chen and F Xiao ldquoShallow water model on sphericalcubic grid by multi-moment finite volume methodrdquo Journal ofComputational Physics vol 227 pp 5019ndash5044 2008


[18] X Li D Chen X Peng K Takahashi and F Xiao ldquoAmultimoment finite-volume shallow-water model on the Yin-Yang overset spherical gridrdquoMonthly Weather Review vol 136no 8 pp 3066ndash3086 2008

[19] S Ii and F Xiao ldquoHigh order multi-moment constrainedfinite volume method Part I basic formulationrdquo Journal ofComputational Physics vol 228 no 10 pp 3669ndash3707 2009

[20] C Chen X Li X Shen and F Xiao ldquoGlobal shallow watermodels based on multi-moment constrained finite volumemethod and three quasi-uniform spherical gridsrdquo Journal ofComputational Physics vol 271 pp 191ndash223 2014

[21] S Ii and F Xiao ldquoA global shallow water model using highorder multi-moment constrained finite volume method andicosahedral gridrdquo Journal of Computational Physics vol 229 no5 pp 1774ndash1796 2010

[22] C G Chen J Z Bin and F Xiao ldquoA global multimomentconstrained finite-volume scheme for advection transport onthe hexagonal geodesic gridrdquoMonthlyWeather Review vol 140no 3 pp 941ndash955 2012

[23] C G Chen J Z Bin F Xiao X L Li and X S Shen ldquoA globalshallow-water model on an icosahedral-hexagonal grid by amulti-moment constrained finite-volume schemerdquo QuarterlyJournal of the Royal Meteorological Society vol 140 no 679 pp639ndash650 2014

[24] X L Li X S Shen X D Peng F Xiao Z R Zhuang and CG Chen ldquoAn accurate multimoment constrained finite volumetransport model on Yin-Yang gridsrdquo Advances in AtmosphericSciences vol 30 no 5 pp 1320ndash1330 2013

[25] N A Phillips ldquoA coordinate system having some specialadvantages for numerical forecastingrdquo Journal of Meteorologyvol 14 no 2 pp 184ndash185 1957

[26] T Gal-Chen and R C J Somerville ldquoOn the use of a coordinatetransformation for the solution of the Navier-Stokes equationsrdquoJournal of Computational Physics vol 17 pp 209ndash228 1975

[27] T Satomura ldquoCompressible flow simulations on numericallygenerated gridsrdquo Journal of the Meteorological Society of Japanvol 67 pp 473ndash482 1989

[28] C Schar D Leuenberger O Fuhrer D Luthi and C GirardldquoA new terrain-following vertical coordinate formulation foratmospheric prediction modelsrdquoMonthly Weather Review vol130 no 10 pp 2459ndash2480 2002

[29] R B Smith ldquoThe influence of mountains on the atmosphererdquoAdvances in Geophysics vol 21 pp 87ndash230 1979

[30] C-W Shu ldquoTotal-variation-diminishing time discretizationsrdquoSIAM Journal on Scientific and Statistical Computing vol 9 no6 pp 1073ndash1084 1988

[31] C-W Shu and S Osher ldquoEfficient implementation of essentiallynonoscillatory shock-capturing schemesrdquo Journal of Computa-tional Physics vol 77 no 2 pp 439ndash471 1988

[32] T Satomura T Iwasaki K Saito C Muroi and K TsubokildquoAccuracy of terrain following coordinates over isolated moun-tain steep mountain model intercomparison project (St-MIP)rdquoAnnuals of Disaster Prevention Research Institute 46B KyotoUniversity Kyoto Japan 2003

[33] A Eliassen and E Palm ldquoOn the transfer of energy in stationarymountain wavesrdquo Geofysiske Publikationer vol 22 pp 1ndash231960

[34] F X Giraldo and M Restelli ldquoA study of spectral elementand discontinuous Galerkin methods for the Navier-Stokesequations in nonhydrostatic mesoscale atmospheric modelingequation sets and test casesrdquo Journal of Computational Physicsvol 227 no 8 pp 3849ndash3877 2008

[35] D R Durran and J B Klemp ldquoA compressible model for thesimulation of moist mountain wavesrdquoMonthlyWeather Reviewvol 111 no 12 pp 2341ndash2361 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of


EarthquakesJournal of


Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining


Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of


Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering


GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of


OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in


MineralogyInternational Journal of


MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Geological ResearchJournal of


Geology Advances in


moments such as volume-integrated average (VIA) pointvalue (PV) and spatial derivative values (DV) and is thussimple efficient and easy to implement Being a high orderscheme more accurate numerical results can be obtainedin terms of the equivalent DOF resolution in comparisonwith the traditional finite volumemethod evenwith relativelycoarse grid spacing The rigorous numerical conservationin MCV model is exactly guaranteed by a constraint onthe VIA through a finite volume formulation of flux formBeing a new nodal-type high order conservative methodit is much beneficial to compute the metric and sourceterms in an MCV model which are always involved in thetreatments of spherical geometry in the horizontal directionand coordinate transformation in the vertical direction fortopographic effect The MCV model has some appealingfeatures for atmospheric modeling such as the rigorousnumerical conservation good computational efficiency andflexible configuration for solution points and thus can beexpected as a practical framework of the dynamic coreswhich well balances the numerical accuracy and algorithmiccomplexity [10 20] The competitive results of the widelyadopted benchmark tests can be referred to [10 20ndash24]

To deal with the bottom topography the height-basedterrain following coordinate is adopted in our MCV modelSince Phillipsrsquo pioneering work [25] the terrain followingcoordinates mainly classified into the pressure-based coor-dinate and the height-based coordinate have been widelyadopted to represent the underlying mountainous surface inatmospheric models During the past several decades theheight-based coordinate [26] has got an increasing popu-larity mainly due to its applicability for both hydrostaticand nonhydrostatic models and computational simplicityA variational grid generation technique [27] was adaptedto mountain wave simulation as well Modified version ofterrain following coordinate has been also proposed [28] tocircumvent to some extent the drawbacks of the coordinatein representing steep topography In this study a set ofbenchmark tests of mountain waves generated by a con-stant background flow over mountains with different steep-ness which essentially represent the complex mechanismsinvolved inmountain weather processes are examined by theMCV nonhydrostatic model using the height-based terrainfollowing coordinate in order to verify the performance of theMCV model in simulating the mountain weather processesand its potential for the further numerical investigations onthe flows in the atmosphere boundary layer and the air qualityover complex mountainous terrainThe numerical results areevaluated in comparison with the semianalytical solutionsobtained from the linear theory [29] as well as the numericalsolutions from other representative models

The remainder of this paper is organized as followsIn Section 2 the compressible nonhydrostatic atmosphericmodel using the MCV scheme and the height-based ter-rain following coordinate is briefly introduced Section 3describes the mountain tests and the semianalytical solutionsthrough the linear theory Section 4 discusses numericalresults of variousmountainwaves by theMCVmodel Finallya short conclusion is given in Section 5

2 2D MCV NonhydrostaticAtmospheric Model

In order to consider complex topography as the bottomboundary of the atmospheric model a height-based terrainfollowing coordinate is used in this study to map the physicaldomain (119909 119911) to the computational domain (119909 120577) through thetransformation

119911 (120577) = 119911119878 (119909) +120577

119911119879[119911119879minus 119911119878 (119909)] (1)

where 119911119878(119909) is the elevation of topography 119911

119879the altitude of

the model top and 120577 isin [0 119911119879]

Using a height-based terrain following coordinate 2Dcompressible and nonhydrostatic governing equations foratmospheric dynamics are written in flux form as

120597q120597119905+120597f120597119909

+120597g120597120577

= s (q) (2)

where

q =[[[[[[

[

radic1198661205881015840

radic119866120588119906

radic119866120588119908

radic119866 (120588120579)1015840

]]]]]]

]

f =[[[[[[

[

radic119866120588119906

radic1198661205881199062 + radic1198661199011015840

radic119866120588119908119906

radic119866120588120579119906

]]]]]]

]

g =[[[[[[

[

radic119866120588

radic119866120588119906 + radic119866119866131199011015840

radic119866120588119908 + 1199011015840

radic119866120588120579

]]]]]]

]

s =[[[[[

[

0

0

minusradic1198661205881015840119892

0

]]]]]

]

(3)

where 120588 is density (119906 119908) are velocity vector in the physicaldomain = 119889120577119889119905 is the vertical velocity in the transformedcoordinates 11986613 = 120597120577120597119909 andradic119866 = 120597119911120597120577 is the Jacobian oftransformation



120588 (x 119905) = 120588 (119911) + 1205881015840 (x 119905)

119901 (x 119905) = 119901 (119911) + 1199011015840 (x 119905)

(120588120579) (x 119905) = (120588120579) (119911) + (120588120579)1015840 (x 119905)

(4)


0= 1205741198620(120588120579)120574minus1





119901119888V = 14 refer-


119877119889120574119901minus119877119889119888V0





u sdot n = 0 (5)




120597q120597119905







3 A3 100 100 2 times 10minus3 265 02

A4 100 50 2 times 10minus3 45 01




119911119878 (119909) =ℎ0

1 + (119909 minus 1199090)21198862

(7)


0is the



0)119877119889119888119901 and potential





120578119909119909 + 120578119911119911 + 1198972120578 = 0 (8)






120578 (119909 119911) = (1205880

120588 (119911))

12

sdot Reint119897

0

ℎ (119896) exp [119894 (1198972 minus 1198962)12

119911] exp (119894119896119909) 119889119896

+ intinfin

119897

ℎ (119896) exp [minus (1198962 minus 1198972)12

119911] exp (119894119896119909) 119889119896

(9)




0since



0119897 (ℎ0119897 ≪ 1)





119872(119911) = int+infin

minusinfin

120588 (119911) 1199061015840(119909 119911) 119908

1015840(119909 119911) 119889119909 (10)



119872119867 (119911) = minus120587

4120588011987301198800ℎ

2

0 (11)



4 Numerical Results



119901119879)119911)








120579 = 1205790exp(

1198732

0

119892119911) (12)

Π = 1 +1198922

119888119901120579011987320

(exp(minus11987320

119892119911) minus 1) (13)





100 120 16080 140X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

100 120 16080 140X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)

02


8 hours10 hours


0

2

4

6

8

10

12

Hei

ght (

km)

(c)






0=



75 10560 90X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

75 10560 90X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)


0

2

4

6

8

10

12

Hei

ght (

km)


4 hours5 hours

(c)



0= 250K





55 6 65 75X (km)

0

05

1

15

2Z

(km

)

(a)

55 6 65 75X (km)

0

05

1

15

2

Z (k

m)

(b)

0

05

1

15

Z (k

m)

55 6 65 75X (km)

(c)

0

05

1

15

2

25

3

35

4H

eigh

t (km

)


20 min(d)










14 15 16 1713X (km)

0

01

02

03

04

05Z

(km

)

(a)

14 15 16 1713X (km)

0

01

02

03

04

05

Z (k

m)

(b)

12 14 16 18 21X (km)

0

05

1

15

Z

(c)

0

02

04

06

08

1H

eigh

t (km

)


10 min(d)




0=




0

2

4

6

8

10Z

(km

)

10 12 14 16 18 20 228X (km)

(a)

0

2

4

6

8

10

Z (k

m)

10 12 14 16 18 20 228X (km)

(b)

0

1

2

4

6

Z (k

m)

5 15 20 2510X (km)

(c)

0

2

4

6

8

10

12H

eigh

t (km

)


100 min(d)










10 12 14 16 18 20 228X (km)

0

2

4

6

8

10Z

(km

)

(a)

10 12 14 16 18 20 228X (km)

0

2

4

6

8

10

Z (k

m)

(b)

5 15 20 2510X (km)

0

1

2

4

6

Z (k

m)

(c)100 min


0

2

4

6

8

10

12H

eigh

t (km

)

(d)








5 Conclusions






Acknowledgments


References















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



120588 (x 119905) = 120588 (119911) + 1205881015840 (x 119905)

119901 (x 119905) = 119901 (119911) + 1199011015840 (x 119905)

(120588120579) (x 119905) = (120588120579) (119911) + (120588120579)1015840 (x 119905)

(4)


0= 1205741198620(120588120579)120574minus1





119901119888V = 14 refer-


119877119889120574119901minus119877119889119888V0





u sdot n = 0 (5)




120597q120597119905







3 A3 100 100 2 times 10minus3 265 02

A4 100 50 2 times 10minus3 45 01




119911119878 (119909) =ℎ0

1 + (119909 minus 1199090)21198862

(7)


0is the



0)119877119889119888119901 and potential





120578119909119909 + 120578119911119911 + 1198972120578 = 0 (8)






120578 (119909 119911) = (1205880

120588 (119911))

12

sdot Reint119897

0

ℎ (119896) exp [119894 (1198972 minus 1198962)12

119911] exp (119894119896119909) 119889119896

+ intinfin

119897

ℎ (119896) exp [minus (1198962 minus 1198972)12

119911] exp (119894119896119909) 119889119896

(9)




0since



0119897 (ℎ0119897 ≪ 1)





119872(119911) = int+infin

minusinfin

120588 (119911) 1199061015840(119909 119911) 119908

1015840(119909 119911) 119889119909 (10)



119872119867 (119911) = minus120587

4120588011987301198800ℎ

2

0 (11)



4 Numerical Results



119901119879)119911)








120579 = 1205790exp(

1198732

0

119892119911) (12)

Π = 1 +1198922

119888119901120579011987320

(exp(minus11987320

119892119911) minus 1) (13)





100 120 16080 140X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

100 120 16080 140X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)

02


8 hours10 hours


0

2

4

6

8

10

12

Hei

ght (

km)

(c)






0=



75 10560 90X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

75 10560 90X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)


0

2

4

6

8

10

12

Hei

ght (

km)


4 hours5 hours

(c)



0= 250K





55 6 65 75X (km)

0

05

1

15

2Z

(km

)

(a)

55 6 65 75X (km)

0

05

1

15

2

Z (k

m)

(b)

0

05

1

15

Z (k

m)

55 6 65 75X (km)

(c)

0

05

1

15

2

25

3

35

4H

eigh

t (km

)


20 min(d)










14 15 16 1713X (km)

0

01

02

03

04

05Z

(km

)

(a)

14 15 16 1713X (km)

0

01

02

03

04

05

Z (k

m)

(b)

12 14 16 18 21X (km)

0

05

1

15

Z

(c)

0

02

04

06

08

1H

eigh

t (km

)


10 min(d)




0=




0

2

4

6

8

10Z

(km

)

10 12 14 16 18 20 228X (km)

(a)

0

2

4

6

8

10

Z (k

m)

10 12 14 16 18 20 228X (km)

(b)

0

1

2

4

6

Z (k

m)

5 15 20 2510X (km)

(c)

0

2

4

6

8

10

12H

eigh

t (km

)


100 min(d)










10 12 14 16 18 20 228X (km)

0

2

4

6

8

10Z

(km

)

(a)

10 12 14 16 18 20 228X (km)

0

2

4

6

8

10

Z (k

m)

(b)

5 15 20 2510X (km)

0

1

2

4

6

Z (k

m)

(c)100 min


0

2

4

6

8

10

12H

eigh

t (km

)

(d)








5 Conclusions






Acknowledgments


References















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in




0since



0119897 (ℎ0119897 ≪ 1)





119872(119911) = int+infin

minusinfin

120588 (119911) 1199061015840(119909 119911) 119908

1015840(119909 119911) 119889119909 (10)



119872119867 (119911) = minus120587

4120588011987301198800ℎ

2

0 (11)



4 Numerical Results



119901119879)119911)








120579 = 1205790exp(

1198732

0

119892119911) (12)

Π = 1 +1198922

119888119901120579011987320

(exp(minus11987320

119892119911) minus 1) (13)





100 120 16080 140X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

100 120 16080 140X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)

02


8 hours10 hours


0

2

4

6

8

10

12

Hei

ght (

km)

(c)






0=



75 10560 90X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

75 10560 90X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)


0

2

4

6

8

10

12

Hei

ght (

km)


4 hours5 hours

(c)



0= 250K





55 6 65 75X (km)

0

05

1

15

2Z

(km

)

(a)

55 6 65 75X (km)

0

05

1

15

2

Z (k

m)

(b)

0

05

1

15

Z (k

m)

55 6 65 75X (km)

(c)

0

05

1

15

2

25

3

35

4H

eigh

t (km

)


20 min(d)










14 15 16 1713X (km)

0

01

02

03

04

05Z

(km

)

(a)

14 15 16 1713X (km)

0

01

02

03

04

05

Z (k

m)

(b)

12 14 16 18 21X (km)

0

05

1

15

Z

(c)

0

02

04

06

08

1H

eigh

t (km

)


10 min(d)




0=




0

2

4

6

8

10Z

(km

)

10 12 14 16 18 20 228X (km)

(a)

0

2

4

6

8

10

Z (k

m)

10 12 14 16 18 20 228X (km)

(b)

0

1

2

4

6

Z (k

m)

5 15 20 2510X (km)

(c)

0

2

4

6

8

10

12H

eigh

t (km

)


100 min(d)










10 12 14 16 18 20 228X (km)

0

2

4

6

8

10Z

(km

)

(a)

10 12 14 16 18 20 228X (km)

0

2

4

6

8

10

Z (k

m)

(b)

5 15 20 2510X (km)

0

1

2

4

6

Z (k

m)

(c)100 min


0

2

4

6

8

10

12H

eigh

t (km

)

(d)








5 Conclusions






Acknowledgments


References















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


100 120 16080 140X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

100 120 16080 140X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)

02


8 hours10 hours


0

2

4

6

8

10

12

Hei

ght (

km)

(c)






0=



75 10560 90X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

75 10560 90X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)


0

2

4

6

8

10

12

Hei

ght (

km)


4 hours5 hours

(c)



0= 250K





55 6 65 75X (km)

0

05

1

15

2Z

(km

)

(a)

55 6 65 75X (km)

0

05

1

15

2

Z (k

m)

(b)

0

05

1

15

Z (k

m)

55 6 65 75X (km)

(c)

0

05

1

15

2

25

3

35

4H

eigh

t (km

)


20 min(d)










14 15 16 1713X (km)

0

01

02

03

04

05Z

(km

)

(a)

14 15 16 1713X (km)

0

01

02

03

04

05

Z (k

m)

(b)

12 14 16 18 21X (km)

0

05

1

15

Z

(c)

0

02

04

06

08

1H

eigh

t (km

)


10 min(d)




0=




0

2

4

6

8

10Z

(km

)

10 12 14 16 18 20 228X (km)

(a)

0

2

4

6

8

10

Z (k

m)

10 12 14 16 18 20 228X (km)

(b)

0

1

2

4

6

Z (k

m)

5 15 20 2510X (km)

(c)

0

2

4

6

8

10

12H

eigh

t (km

)


100 min(d)










10 12 14 16 18 20 228X (km)

0

2

4

6

8

10Z

(km

)

(a)

10 12 14 16 18 20 228X (km)

0

2

4

6

8

10

Z (k

m)

(b)

5 15 20 2510X (km)

0

1

2

4

6

Z (k

m)

(c)100 min


0

2

4

6

8

10

12H

eigh

t (km

)

(d)








5 Conclusions






Acknowledgments


References















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


75 10560 90X (km)

0

2

4

6

8

10

12Z

(km

)

(a)

75 10560 90X (km)

0

2

4

6

8

10

12

Z (k

m)

(b)


0

2

4

6

8

10

12

Hei

ght (

km)


4 hours5 hours

(c)



0= 250K





55 6 65 75X (km)

0

05

1

15

2Z

(km

)

(a)

55 6 65 75X (km)

0

05

1

15

2

Z (k

m)

(b)

0

05

1

15

Z (k

m)

55 6 65 75X (km)

(c)

0

05

1

15

2

25

3

35

4H

eigh

t (km

)


20 min(d)










14 15 16 1713X (km)

0

01

02

03

04

05Z

(km

)

(a)

14 15 16 1713X (km)

0

01

02

03

04

05

Z (k

m)

(b)

12 14 16 18 21X (km)

0

05

1

15

Z

(c)

0

02

04

06

08

1H

eigh

t (km

)


10 min(d)




0=




0

2

4

6

8

10Z

(km

)

10 12 14 16 18 20 228X (km)

(a)

0

2

4

6

8

10

Z (k

m)

10 12 14 16 18 20 228X (km)

(b)

0

1

2

4

6

Z (k

m)

5 15 20 2510X (km)

(c)

0

2

4

6

8

10

12H

eigh

t (km

)


100 min(d)










10 12 14 16 18 20 228X (km)

0

2

4

6

8

10Z

(km

)

(a)

10 12 14 16 18 20 228X (km)

0

2

4

6

8

10

Z (k

m)

(b)

5 15 20 2510X (km)

0

1

2

4

6

Z (k

m)

(c)100 min


0

2

4

6

8

10

12H

eigh

t (km

)

(d)








5 Conclusions






Acknowledgments


References















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


55 6 65 75X (km)

0

05

1

15

2Z

(km

)

(a)

55 6 65 75X (km)

0

05

1

15

2

Z (k

m)

(b)

0

05

1

15

Z (k

m)

55 6 65 75X (km)

(c)

0

05

1

15

2

25

3

35

4H

eigh

t (km

)


20 min(d)










14 15 16 1713X (km)

0

01

02

03

04

05Z

(km

)

(a)

14 15 16 1713X (km)

0

01

02

03

04

05

Z (k

m)

(b)

12 14 16 18 21X (km)

0

05

1

15

Z

(c)

0

02

04

06

08

1H

eigh

t (km

)


10 min(d)




0=




0

2

4

6

8

10Z

(km

)

10 12 14 16 18 20 228X (km)

(a)

0

2

4

6

8

10

Z (k

m)

10 12 14 16 18 20 228X (km)

(b)

0

1

2

4

6

Z (k

m)

5 15 20 2510X (km)

(c)

0

2

4

6

8

10

12H

eigh

t (km

)


100 min(d)










10 12 14 16 18 20 228X (km)

0

2

4

6

8

10Z

(km

)

(a)

10 12 14 16 18 20 228X (km)

0

2

4

6

8

10

Z (k

m)

(b)

5 15 20 2510X (km)

0

1

2

4

6

Z (k

m)

(c)100 min


0

2

4

6

8

10

12H

eigh

t (km

)

(d)








5 Conclusions






Acknowledgments


References















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


14 15 16 1713X (km)

0

01

02

03

04

05Z

(km

)

(a)

14 15 16 1713X (km)

0

01

02

03

04

05

Z (k

m)

(b)

12 14 16 18 21X (km)

0

05

1

15

Z

(c)

0

02

04

06

08

1H

eigh

t (km

)


10 min(d)




0=




0

2

4

6

8

10Z

(km

)

10 12 14 16 18 20 228X (km)

(a)

0

2

4

6

8

10

Z (k

m)

10 12 14 16 18 20 228X (km)

(b)

0

1

2

4

6

Z (k

m)

5 15 20 2510X (km)

(c)

0

2

4

6

8

10

12H

eigh

t (km

)


100 min(d)










10 12 14 16 18 20 228X (km)

0

2

4

6

8

10Z

(km

)

(a)

10 12 14 16 18 20 228X (km)

0

2

4

6

8

10

Z (k

m)

(b)

5 15 20 2510X (km)

0

1

2

4

6

Z (k

m)

(c)100 min


0

2

4

6

8

10

12H

eigh

t (km

)

(d)








5 Conclusions






Acknowledgments


References















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


0

2

4

6

8

10Z

(km

)

10 12 14 16 18 20 228X (km)

(a)

0

2

4

6

8

10

Z (k

m)

10 12 14 16 18 20 228X (km)

(b)

0

1

2

4

6

Z (k

m)

5 15 20 2510X (km)

(c)

0

2

4

6

8

10

12H

eigh

t (km

)


100 min(d)










10 12 14 16 18 20 228X (km)

0

2

4

6

8

10Z

(km

)

(a)

10 12 14 16 18 20 228X (km)

0

2

4

6

8

10

Z (k

m)

(b)

5 15 20 2510X (km)

0

1

2

4

6

Z (k

m)

(c)100 min


0

2

4

6

8

10

12H

eigh

t (km

)

(d)








5 Conclusions






Acknowledgments


References















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


10 12 14 16 18 20 228X (km)

0

2

4

6

8

10Z

(km

)

(a)

10 12 14 16 18 20 228X (km)

0

2

4

6

8

10

Z (k

m)

(b)

5 15 20 2510X (km)

0

1

2

4

6

Z (k

m)

(c)100 min


0

2

4

6

8

10

12H

eigh

t (km

)

(d)








5 Conclusions






Acknowledgments


References















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



5 Conclusions






Acknowledgments


References















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in





























Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in










Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in

Documents

Research Article An MCV Nonhydrostatic Atmospheric Model ...downloads.hindawi.com/journals/amete/2016/4513823.pdf · e MCV scheme is adopted in this model to solve the governing equations