Improving variational mass-consistent models of hydrodynamic flows via boundaryconditions

Marco A. Nunez

Departamento de Fısica, Universidad Autonoma Metropolitana Iztapalapa, A. P. 55-534, C.P. 09340, D. F.,Mexico. e-mail: manp@xanum.uam.mx

Abstract

Variational mass-consistent models have been used by mesoscale meteorological community to modeling thewind field in a bounded region Ω with boundary Γ. According to the Sasaki’s proposal, the problem isreduced to the solution of an elliptic equation for a multiplier Lagrange λ subject to the Dirichlet BoundaryCondition (DBC) λ = 0 for flow-through boundaries. In this work we study the effect of this boundarycondition on the field vn obtained with numerical estimations λn of λ. It is shown that the use of theDBC in any of the open boundaries yields a field vn that satisfies poorly the constraint ∇ · vn = 0 in theclosed region Ω∪Γ. To avoid the use of the DBC, we study some approximations of the boundary conditionv · n = vT · n on the whole boundary Γ, where vT denotes the true field. Formal and numerical resultsshow that the field U0 = v0 + w0k obtained by direct integration of equation ∇ ·U0 = 0, yields a Neumannboundary condition for λ that increases in several orders of magnitude the accuracy with which a numericalfield vn satisfies the constraint ∇ · vn = 0 in Ω ∪ Γ.

Keywords: Variational mass-consistent models, air pollution, data assimilation, oceanography, particle imagevelocimetry.

1. Introduction

The estimation of a hydrodynamic velocity from data measured in a finite domain, is a basic problem inseveral areas of fluid mechanics, such as meteorology [1,2], oceanography [3] and experimental fluid mechanics[4]. There is a wide class of solutions that go from a simple interpolation to the numerical solution ofprimitive-equation models, which require an important computational effort to set up the boundary andinitial conditions on the relevant hydrodynamic variables. Mass-consistent models is a class of diagnosticmodels developed in meteorology that is intermediate in sophistication between interpolated and primitive-equation models and attempts to satisfy the continuity equation [2,5−7]. Several studies give evidencethat mass-consistent models are suitable for predicting the atmospheric dispersion of hazardous materialsin emergency response scenarios and, among these models, variational mass-consistent models (VMCM’s)appear to be best since they introduce a fewer number of arbitrary parameters [5−11]. VMCM’s havebeen applied to modeling the transport, diffusion and dispersion of atmospheric pollutants and as input ofprognostic models [12−21], a review of these models is given in Refs. [6−8]. The simplicity of VMCM’shas motivated the development of new computational algorithms [22-28] and applications for air qualitymodeling and climatological studies [29-33] over the last decades. The VMCM’s reduce the estimation ofthe velocity field to the solution of an elliptic equation subject to certain boundary conditions. The mainaim of this work is to study how these models can be improved by means of the boundary conditions usedto solving the elliptic problem.

An approach to estimate a hydrodynamic velocity field from data measured approximately at a giventime in a bounded region Ω, is the interpolation of the data to get an initial field v0 which, in general, doesnot satisfy physical constraints. The variational method proposed by Sasaki [12] to adjust v0 consists in

1

computing the field v that minimizes the functional

J(v) =∫

Ω(v − v0) · S(v − v0)dΩ (1.1)

and satisfies some physical constraints, where S is a given positive-definite symmetric matrix. Following themeteorological literature [5−21], we assume that the density is approximately constant in the domain Ω insuch a way that the continuity equation takes the form

∇ · v = 0 in Ω (1.2)

The minimization of functional (1.1) subject to the constraint (1.2) yields the field

v = v0 + S−1∇λ (1.3)

where λ is solution of the elliptic equation

−∇ · S−1∇λ = ∇ · v0 (1.4)

To get a unique solution of this equation, two boundary conditions are commonly used, the Dirichlet Bound-ary Condition (DBC) or “natural boundary condition” λ = 0 for open or ”flow-through” boundaries andthe Neumann boundary condition (NBC) ∂λ/∂n = 0 for closed or “no-flow-through” boundaries [6−8,30].In Ref. [34,35] it was shown that the last boundary condition is consistent when S is the unit matrix andwe have v0 · n = 0 on the terrain, however these values are not always used in meteorology.

In principle, if vT is the true velocity field and Γ denotes the boundary of the domain Ω, the bestboundary condition (BC) to compute a mass-consistent field v (1.3) may be the condition

v · n = vT · n on the whole boundary Γ . (1.5)

However, the term vT · n is not always known on Γ. In meteorology and oceanography the BC (1.5) isknown partially since vT ·n = 0 holds on the terrain, this condition has been used by VMCM’s that employconformal coordinates [6]. In general, if ΓN denotes the part of Γ where vT · n is known, the BC (1.5) isreplaced by

v · n = vT · n on ΓN . (1.6)

An argument based on the orthogonal decomposition of L2(Ω), the set of vectors fields whose componentsare square integrable in Ω, shows that if the functional (1.1) is minimized subject to the conditions (1.2),(1.6), then the minimizer v is given by Eq. (1.3) and λ must satisfy the DBC

λ = 0 on the complementary boundary ΓD ≡ Γ \ ΓN , (1.7)

see, e.g., section 6 of Ref. [35]. In the same reference (section 3) it was shown that the DBC (1.6) generatesdiscontinuities in λ and v when one of the elements of a constant and diagonal matrix S takes smallor large values. In meteorology the DBC (1.4) is commonly used on the lateral boundaries or top of Ω[6−28,30,34,35]. The results reported in sections 3−5 of this work, show that the use of DBC (1.7) in anyof the open boundaries of Ω to solving numerically the equation (1.4), yields a field v (1.3) that satisfiespoorly (1.1) in the closed region Ω ∪ Γ.

In section 6 we study some approximations of vT · n to avoid the use of DBC (1.7). Since the verticalvelocity is not routinely measured by meteorological networks, the field v0 has the form v0 = u0i + v0j.

2

This yields the simplest mass-consistent model U0 = v0 +w0k where the vertical velocity w0 is obtained byintegration of equation ∇ ·U0 = 0 subject to v · n = 0 on the topography. Analytic and numerical resultsreported in section 6 show that the BC

v · n = U0 · n on Γ (1.8)

improves significantly the regularity of λ and the accuracy with which numerical approximations vn of v(1.3), satisfy the constraint (1.1) in the closed region Ω = Ω ∪ Γ. The estimation of U0 is simple but manyauthors have preferred to compute the field v (1.3) because U0 is sensitive to data errors while the field vis less sensitive to the same errors and, mainly, v can exhibit real physical properties with a suitable matrixS [6−8]. However, if v0 = u0i + v0j is the true horizontal field, then U0 could be the true field. Thus, weshall consider that a mass-consistent field v (1.3) is physically consistent when U0 can be obtained from v.In this work we show the relation

limS33→0

v = U0 (1.9)

holds for several boundary conditions and a more general matrix S = Sij than that used in meteorology[6−8]. A summary of the results of sections 1−6 was reported in Ref. [36] where we give a contravariantformulation of mass consistent models that use the BC (1.8) for a complex terrain. Section 7 is devoted tosome concluding remarks and proofs of the main results are given in the Appendix.

2. Boundary value problem 1 and the limit S3 → 0

Hereafter the following notation will be used. Partial derivatives are denoted by ∂xf = ∂f/∂x, xyz willbe a cartesian system with unit vectors i ≡ x1, j ≡ x2, k ≡ x3, r = xi + yj + zk = xixi (repeated indicesindicate summation); Ω denotes an open, bounded and connected region in R3 with Lipschitz boundary Γ, nis the outward unit normal to Γ and Ω = Ω∪Γ. L2(Ω) is the space of square integrable functions on Ω withthe inner product 〈f, g〉 =

∫Ω fgdΩ and the norm ‖f‖ = 〈f | f〉1/2, H1(Ω) = f ∈ L2(Ω) : ∂xif ∈ L2(Ω); w

is the vector with components wi and w · u = wiui.

We begin with a formulation of the minimization of functional J(v) (1.1) subject to the constraints (1.2),(1.6). The interpolation methods used to estimate the initial field v0 guarantee that it belongs to spaceH(div) ≡

w ∈ L2(Ω) : ∇ ·w ∈ L2(Ω)

with L2(Ω) =w : wi ∈ L2(Ω)

. The field v that minimizes J(v)

and satisfies (1.2), (1.6), belongs to the set

V∗ = w ∈ H(div) : ∇ ·w = 0 in Ω , w · n = vT · n on ΓN .

According to the standard literature [5−33] we assume that the region ΓD ≡ Γ \ ΓN is at least a subsetof the flow-through boundaries with a non-zero area. The matrix S = Sij used in meteorology, is usuallyconstant and diagonal, but we can consider a more general matrix with elements Sij in the set C(0)(Ω) = f :f is continuous on Ω. In the appendix we give a proof of the following result

Proposition 1: Let S = Sij be symmetric positive-definite on Ω with elements Sij in C(0)(Ω). If v0 belongsto H(div), then there exits a unique field v in V∗ that minimizes the functional J(v) (1.1). Furthermore,there is a unique λ in the space H1

D(Ω) ≡ φ ∈ H1(Ω) : φ = 0 on ΓD

, with which the relation (1.3) holds

true in Ω.

The relation (1.3) reduces the problem of calculating v to solving the Boundary Value Problem (BVP)

Lλ = ∇ · v0 in ΩLλ =

(vT−v0

) · n on ΓN

λ = 0 on ΓD

(2.1)

3

where we set L ≡ −∇ · S−1∇ and L ≡ n · S−1∇. The main method used in meteorology to solve problem(2.1) is the finite difference method [6−8]. The Ritz method (which includes the finite element method) isa good choice, since it replaces BVP (2.1) by its weak form where only the DBC λ = 0 is considered sincethe NBC is an inherent property of λ [37]. In this work we shall study some versions of the BVP (2.1)considered in the meteorological literature [5−35].

In meteorology the initial field v0 has the form

v0 = u0i + v0j

and the usual domain is Ω = (x, y) ∈ Ωxy, h (x, y) < z < zM, Ωxy ≡ (0, xM )× (0, yM ), where h (x, y) is theterrain elevation on (x, y, z = 0). The open boundaries are the planes Γx = x = 0, xM, Γy = y = 0, yM,ΓzM = z = zM, and Γh = z = h (x, y) is the terrain. We have vT · n = 0 on Γh, but vT · n is unknownon the open boundary Γ \Γh, so that the most common BVP considered in meteorology [6−8] has the form

Lλ = ∇ · v0 in ΩLλ = −v0 · n on ΓN = Γz=h(x,y)

λ = 0 on ΓD = Γ \ Γz=h(x,y) .

(2.2)

To simplify the study of the effects of DBC λ = 0 we shall consider a flat topography h (x, y) = 0. Thisleads to the Boundary Value Problem 1 (BVP1)

Lλ = ∇ · v0 in Ω∂zλ = 0 on ΓN = Γz=0

λ = 0 on ΓD = Γ \ Γz=0 .

(2.3)

The simplest mass-consistent model obtained from v0 = u0i + v0j is U0 = v0 + w0k where the verticalvelocity

w0 = −∫ z

0∇ · v0 ds . (2.4)

is obtained by integration of ∇ ·U0 = 0 with U0 ·n = 0 on z = 0. In this work we shall see that, for severalboundary conditions, the field v (1.3) obeys the consistency relation (1.9), whose precise meaning is clarifiedbelow, when the matrix S has the form

S =

S1(x, y) S12(x, y) 0S12(x, y) S2(x, y) 0

0 0 S3

, S3 = constant. (2.5)

We shall establish Eq. (1.9) in terms of approximations of v obtained with the Ritz method and theeigenfunctions of the operator L which has the form L = Lxy + S−1

3 (−∂2z ) with

Lxy = −∇xy · S−1r ∇xy with Sr =

(S1 S12

S12 S2

), ∇xy =

(∂x

∂y

).

The eigenvalue problemLxyφij = Eijφij in Ωxy with φij = 0 on ∂Ωxy,

has a set of eigenvalues Eij with finite multiplicity, and the (orthonormalized) eigenfunctions φij constitute abasis of the space L2 (Ωxy) endowed with the inner product 〈f, g〉xy =

∫Ωxy

fgdΩxy [37]. The one-dimensionalproblem (−∂2

z

)φk = Ekφk with ∂zφk (0) = φk (zM ) = 0 ,

4

has the eigenvalues Ek = ω2k with ωk = (2k − 1)π/2zM for k = 1, 2, . . ., the normalized eigenfunctions

φk =√

2/zM cosωkx constitute a basis of L2 (0, zM ) endowed with the inner product 〈f, g〉z =∫ zM

0 fgdz.Thus, the eigenvalue problem corresponding to the operator L subject to the conditions of BVP1, is

Lφijk = Eijkφijk with φijk

⌋ΓD

= 0 , ∂zφijk

⌋z=0

= 0 ,

and has the solution φijk = φijφk, Eijk = Eij + Ek.

Hereafter, we set F ≡ ∇ · v0. According to the Ritz method, BVP1 is replaced by the problem

Lλmnl = Fmnl with Fmnl =mn∑

ij

Fijkφijφk, Fijk =< F, φijφk > . (2.6a)

The solution is

λmnl = S3

mnl∑

ijk

cijkφijφk with cijk = Fijk/ (Ek + S3Eij) , (2.6b)

and the horizontal components of the field vmnl = v0 + S−1∇λmnl = umnli + vmnlj + wmnlk are(

umnl

vmnl

)=

(u0

v0

)+ S3 S−1

r

mnl∑

ijk

cijk

(∂xφij

∂yφij

)φk.

This yieldslim

S3→0

∣∣umnl − u0∣∣ = lim

S3→0

∣∣vmnl − v0∣∣ = 0 in Ω . (2.7)

The series λmnl is not suitable to compute limS3→0 wmnl. Instead we replace BVP1 by the problem

Lλmn = Fmn with Fmn =mn∑

ij

Fij(z)φij , Fij =< F, φij >xy , λmn =mn∑

ij

λij(z)φij .

which leads to the one-dimensional problems

∂2zλij − S3Eijλij = −S3Fij with ∂zλij (0) = λij (zM ) = 0 (2.8)

whose solution is

λij(z) = −W−1S3

[ϕ0(z)

∫ zM

zFij(s)ϕ1(s)ds + ϕ1(z)

∫ z

0Fij(s)ϕ0(s)ds

]

with ϕ0 = cosh(rz), ϕ1 = sinh(rz − rzM ), r =√

S3Eij , W = r cosh(rzM ). Hence the field

vmn = v0 + S−1∇λmn

has the vertical velocity

wmn = S−13 ∂zλmn =

mn∑

ij

S−13 ∂zλij(z) φij

withS−1

3 ∂zλij(z) = −W−1

[∂zϕ0(z)

∫ zM

zFij(s)ϕ1(s)ds + ∂zϕ1(z)

∫ z

0Fij(s)ϕ0(s)ds

]. (2.9)

The horizontal components of vmn obey relations like (2.7). In terms of wmn and the following Fourier seriesof the vertical velocity w0 (2.4),

w0mn =

mn∑

ij

w0ijφij , w0

ij =⟨φij , w

0⟩xy

, (2.10)

5

we have the following relation proved in the Appendix

limS3→0

wmn = w0mn . (2.11)

We can summarize with Result 1: For S given by (2.5) the relation limS3→0 vmn = U0mn holds in Ω with

U0mn =

mn∑

ij

< U0ij , φij >xy φij .

3. BVP1 with a constant and diagonal matrix S

The most common matrix S used in meteorology is diagonal with constant elements Si [5−30]. In thiscase the operator Lxy = S−1

1

(−∂2x

)+ S−1

2

(−∂2y

)has the eigenvalues and eigenfunctions

Eij = S−11 Ei + S−1

2 Ej with Ei = ω2i , ωi = iπ/xM , for i = 1, 2, . . . ,

φij = φi(x)φj(y) with φi =√

2/xM sinωix

where Ej and φj have similar expressions. We have

Fmnl =mnl∑

ijk

Fijk φiφjφk , Fijkl =⟨F, φiφjφk

⟩, (3.1)

and the Ritz approximation λmnl (2.6b) yields a field vmnl = v0 + S−1∇λmnl that satisfies

∇ · vmnl = ∇ · v0 +∇ · S−1∇λmnl = F − Fmnl . (3.2)

Thus, the accuracy with which vmnl satisfies the constraint (1.2) is independent of S = δijSi and dependsonly of the convergence of the series Fmnl (3.1) towards F . This assertion is confirmed by the numericalresults reported in section 4 of Ref. [35] with S3 from 10−10 to 1010. The independence of S is not surprisingsince Proposition 1 asserts that the exact λ yields a field v = v0 + S−1∇λ that satisfies Eq. (1.2) for anygiven matrix S.

In section 3 of Ref. [35] we show that one of the components of v = v0 + S−1∇λ becomes discontinuousas one of the matrix elements Si tends to 0 or ∞. In this work we study more carefully the case S3 → 0 inorder to get characterize the convergence of v towards U0. A basic result is that DBC (1.7) decreases thegoodness of numerical approximations such as vmnl to conserve the mass and also generates a discontinuityin v as S3 → 0. To establish the discontinuity property let us replace BVP1 by the problem

Lλnl = Fnl with λnl =nl∑

jk

λjk(x)φjφk , Fnl =nl∑

jk

Fjk(x)φjφk ,

where Fjk is the Fourier coefficient of F with the basis φjφk. Hence we get the problem

−S−11 ∂2

xλjk + S−13 Ejkλjk = Fjk with λjk = 0 at x = 0, xM , (3.3a)

and Ejk = Ek + S3S−12 Ej > 0. The solution is

λjk(x) = −W−1S1

[ϕ0(x)

∫ xM

xFjk(s)ϕ1(s)ds + ϕ1(x)

∫ x

0Fjk(s)ϕ0(s)ds

](3.3b)

with ϕ0 = sinh(rx), ϕ1 = sinh(rx − rxM ), r =√

S1Ejk/S3, W = r sinh(rxM ). The horizontal componentsof vnl = v0 + S−1∇λnl satisfy relations like (2.7) and the vertical velocity wnl has the following behavior,proved in the Appendix.

6

Result 2. The limiting velocity limS3→0 wnl is discontinuous at Γx because of DBC λ = 0 on Γx.

In a similar manner the series λml =∑ml

ik λik(y)φiφk yields a field vml = v0 + S−1∇λml whose verticalcomponent wnl becomes discontinuous at Γy as S3 → 0. These results are independent of S1 and S2. Asummary of additional limiting properties of v (1.3) is given in Table I of Ref. [35].

4. Examples

Following the meteorological literature, the next examples consider a diagonal matrix S with constantelements Si. We begin with a version of BVP1 in the xz plane.

Example 1. For v0 = xi in Ω = (0, xM ) × (0, zM ) we have F = 1, w0 = −z. (a) To verify Eq. (2.11)consider the Ritz approximation

Lλm = Fm with λm =m∑

i=1

λiφi , Fm =m∑

i=1

Fiφi, Fi =< F, φi >x .

Equation (2.9) yields

S−13 ∂zλi(z) = −Fi

[zsinh (rz)

zr

1− cosh(rz − rzM )

cosh (rzM )+

cosh(rz − rzM )cosh (rzM )

zsinh (rz)

zr

]

with r =√

S3Ei. Hence we get limS3→0 S−13 ∂zλi = −Fiz for i = 1, . . . , m and, therefore,

limS3→0

wm = −zFm

where −zFm = w0m is the Fourier series of w0 = −z in the basis φi, this is the relation (2.11). (b) Since

φi∞i=1 is a basis of L2(0, xM ), Fm converges to F = 1 in the L2(0, xM )−norm as m increases but thisconvergence is not uniform since Fm = 0 holds at x = 0, xM , for any m and a similar result occurs with theconvergence of w0

m = limS3→0 wm towards w0. Thus Fm and w0m will exhibit the Gibbs phenomenon as m

increases (see [35, Fig. 2]). This problem is due to the DBC λ = 0 on Γx and generates a discontinuity onΓx as S3 → 0 (Result 2). To verify Result 2 consider the problem

Lλl = Fl with λl =l∑

k=1

λk(x)φk(z) , Fl =l∑

k=1

Fk cosωkz , Fk = 〈F, φk〉z .

Hence we get the equation

∂2xλk −EkS1S

−13 λk = −S1Fk with λk = 0 at x = 0, xM .

The solution is λk = −S3FkE−1k ψk with

ψk = sinh (rx)1− cosh(rx− rxM )

sinh (rxM )+ sinh(rx− rxM )

cosh(rx)− 1sinh (rxM )

,

r =√

S1Ek/S3 [Eq. (3.3b)]. We have ψkcx=0,xM= 0 for any S3 > 0 and limS3→0 ψk = −1 for 0 < x < xM ,

therefore, the vertical velocity wl =∑l

k=1 S−13 λk∂zφk becomes discontinuous at Γx as S3 → 0.

Let us consider three-dimensional examples of problem (2.6) in a parallelepiped Ω with sides xM = yM =20, zM = 5 (km), and the initial field v0 = u0i given by

u0 = β e(y) f(z) g(x) (4.1)

7

where β is a constant obtained from the average velocity⟨u0

⟩= |Ω|−1 ∫

Ω u0dΩ = 10 ms−1, |Ω| = xMyMzM .We have F = βef∂xg, Fmnl = βenflgm, where en, fl, gm, are Fourier series of e, f and ∂xg, respectively,namely,

en =n∑

j=1

ejφj , fl =l∑

k=1

fkφj , gm =m∑

i=1

giφi . (4.2)

The accuracy with which the field vmnl = v0 + S−1∇λmnl satisfies ∇ · vmnl = 0 is given by

∇ · vmnl = F − Fmnl = β (e f ∂xg − en fl gm) . (4.3)

This shows that if gm, en, fl do not converge to e, f , ∂xg, on ΓD, we have

∇ · vmnl 6= 0 on ΓD = Γ \ Γz=0 , for any m,n, l. (4.4)

Thus, ∇ · vmnl (x, y, z) can increase as the point (x, y, z) tends to ΓD. This expectation is confirmed by thenumerical results reported below. Instead of computing ∇ · vmnl point–by-point, we report the flux

I (Γi,vmnl) =∫

Γi

vmnl · nds =∫

Ωi

∇ · vmnldΩ (4.5)

in three regions Ω1 ⊂ Ω2 ⊂ Ω of the form Ωi = (ai, bi)2 × (ci, di) with boundary Γi. Calculations were done

with a 15−digit precision PC.

We begin with a three-dimensional version of Example 1, v0 = βxi, F = βef∂xg with e = f = ∂xg = 1.The divergence ∇·vmnl is determined by the convergence of the Fourier series en, fl, gm, toward the constantfunction 1. Since these series do not converge to 1 in a vicinity of y = 0, yM , z = zM , x = 0, xM , then∇ · vmnl(x, y, z) becomes worse as (x, y, z) tends to the flow-through boundary ΓD, where λmnl satisfiesthe DBC λmnl = 0. This expectation is confirmed by the results reported in Table I, which show thatthe flux I (Γi,vmnl) increases rapidly as Γi goes from Γ1 to Γ. The meaning of values for v(2)

mnl, v(3)mnl,

is explained below. Despite its simplicity, this example shows that DBC (1.7) produces bad results in avicinity of the open boundaries Γx, Γy, ΓzM . Similar results are reported in Tables II, III, IV, for u0 = βxz,βz coswx, βze−z/H coswx, respectively. In Table IV I (Γi,vmnl) goes from −4× 10−5 to −3. These resultsare supported by two-dimensional finite-element calculations reported in Ref. [28] for a complex terrain.

TABLE I. Flux I (Γi,vm=n=l=50) in km3s−1 for v0 = iβx, Ωi = (ai, bi)2 × (ci, di) km3.

Ωi (8, 12)2×(2, 3) (4, 16)2×(1,4) (0, 20)2×(0, 5)

vmnl 5× 10−7 3× 10−5 4× 10−2

v(2)mnl 9× 10−8 6× 10−6 8× 10−3

v(3)mnl zero zero zero

TABLE II. Flux I (Γi,vm=n=l=50) in km3s−1 for v0 = iβxz, S3 = 1 was used for v(3)mnl.

Ωi (8, 12)2×(2, 3) (4, 16)2×(1,4) (0, 20)2×(0, 5)

vmnl 6× 10−7 3× 10−5 5× 10−2

v(2)mnl 2× 10−7 1× 10−5 2× 10−2

v(3)mnl zero zero zero

TABLE III. Flux I (Γi,vm=n=l=50) in km3s−1 for v0 = iβz cosωx, ω = π/2xM , S3 = 1.

8

Ωi (8, 12)2×(2, 3) (4, 16)2×(1,4) (0, 20)2×(0, 5)

vmnl −4× 10−7 −3× 10−5 −4× 10−2

v(2)mnl −2× 10−7 −1× 10−5 −1× 10−2

v(3)mnl −2× 10−8 −6× 10−7 zero

TABLE IV. Flux I (Γi,vm=n=l=50) in km3s−1 for v0 = iβze−z/H cosωx, H = 10 km, ω = π/2xM .

Ωi (8, 12)2×(2, 3) (4, 16)2×(1,4) (0, 20)2×(0, 5)

vmnl −4× 10−5 −3× 10−3 −3v(2)

mnl −2× 10−5 −1× 10−3 −1v(3)

mnl −6× 10−7 −3× 10−5 −3× 10−14

To check the reliability of a field vmnl to conserve the mass we use the fact that the constraint ∇ ·v = 0is valid under the assumption that the density ρ is almost constant in the closed region Ω. Hence the massand mass-flow rate through the open boundaries of Ω are M ∼ ρ |Ω|, dM/dt ∼ −ρ

∮Γ v ·nds, and the fraction

of mass that flows through region Ω in a time tc is

∆M (Γ,v, tc)M

∼ − tc|Ω|

∮

Γv · nds . (4.6)

If a phenomenon under study has a characteristic time tc, vmnl may be considered mass-consistent if it yields|∆M/M | ≤ 10−2, although air-quality studies require a very small ratio |∆M/M | [30,31]. For instance,consider the transport and diffusion of atmospheric pollutants released at the origin of region Ω used in theexamples above. Since the time required by a fluid particle to go from x = 0 to xM is t∗ ∼ xM/

⟨u0

⟩= 33

min, a suitable estimation of pollutants concentration requires |∆M/M | ≤ 10−2 for tc = 33 min. TableV reports the ratio ∆M/M for the field vmnl of Table IV with tc = 1 s, 1, 10 min, we see that vmnl ismass-consistent for tc ≤ 10 s, for tc ≥ 1min the field vmnl has no physical sense.

TABLE V. Fraction of mass (4.6) for fields of Table IV.

tc 1s 1 min 10 min

vmnl 2× 10−3 10−1 1v(2)

mnl 5× 10−4 3× 10−2 3× 10−1

v(3)mnl −2× 10−17 10−15 10−14

5. Boundary value problem 2

In this section we consider the BC v · n = vT · n on the lateral boundaries Γx, Γy. According toProposition 1, this implies the replacement of the BC λ = 0 in BVP1 by the condition Lλ =

(vT − v0

) · non Γx ∪ Γy. Thus the new BVP in a region Ω with flat topography is

Lλ = ∇ · v0 in Ω

Lλ =

0(vT − v0

) · non Γz=0

on Γx ∪ Γy

λ = 0 on ΓD = ΓzM .

(5.1)

In practice vT · n is unknown on Γx ∪ Γy, but v0 · n may be a reasonable approximation. This leads to theBoundary Value Problem 2 (BVP2)

Lλ(2) = ∇ · v0 in ΩLλ(2) = 0 on ΓN = Γz=0 ∪ Γx ∪ Γy

λ(2) = 0 on ΓD = ΓzM .

(5.2)

9

To study the corresponding velocity field v(2) = v0 + S−1∇λ(2) with a matrix S given by (2.5), we use theeigenfunctions of L = Lxy + S−1

3 (−∂2z ). The eigenvalue problem for Lxy is

Lxyφ(2)ij = E

(2)ij φ

(2)ij with n · S−1

r ∇xyφ(2)ij = 0 on ∂Ωxy . (5.3)

The lowest eigenvalue is E(2)ij=0 = 0 and its eigenfunction is the constant function φ

(2)ij=0 = 1/

√xMyM .

The remaining eigenvalues are positive and the (orthonormalized) eigenfunctions φ(2)ij constitute a basis of

L2 (Ωxy) [37].

We begin with a formulation of relation (1.9) by considering the problem

Lλ(2)mn = F (2)

mn with λ(2)mn =

mn∑

ij

λ(2)ij φ

(2)ij , F (2)

mn =mn∑

ij

F(2)ij φ

(2)ij , F

(2)ij =< F, φ

(2)ij >xy . (5.4)

The horizontal components of v(2)mn = v0 + S−1∇λ

(2)mn satisfy relations like (2.7) and the vertical one w

(2)mn

converges to the Fourier series w0,2mn =

∑mnij w0,2

ij φ(2)ij of w0 (2.4) as S3 → 0. This leads to the following result

whose proof is similar to that of Result 1. Result 3: For S given by (2.5) the relation limS3→0 v(2)mn = U0,2

mn

holds in Ω with

U0,2mn =

mn∑

ij

< U0ij , φ

(2)ij >xy φ

(2)ij .

Let us see how the replacement of λ = 0 by Lλ(2) = 0 on Γx ∪ Γy improves the field v(2) given by BVP2when S is constant and diagonal. In this case the NBC (5.2) is equivalent to the conditions

∂xλ(2) |Γx= ∂yλ(2) |Γy= ∂zλ

(2) |z=0= 0 . (5.5)

Using these conditions and λ(2) = 0 at z = zM to compute the eigenvalues and eigenfunctions of L, we get

E(2)ijk = E

(2)ij + S−1

3 Ek with E(2)ij = S−1

1 E(2)i + S−1

2 E(2)j (5.6a)

φ(2)ijk = φ

(2)i φ

(2)j φk ,

whereE

(2)i = ω2

i , ωi = iπ/xM , for i, j = 0, 1, . . . ,φ

(2)i=0 = 1/

√xM , φ

(2)i≥1 =

√2/xM cosωix ,

(5.6b)

the pair E(2)j , φ

(2)j (y) is similar to E

(2)i , φ

(2)i , and Ek, φk, are those used to solving BVP1. If BVP2 is replaced

by the problem

Lλ(2)nl = F

(2)nl with λ

(2)nl =

nl∑

jk

λ(2)jk (x) φ

(2)j φk , F

(2)nl =

nl∑

jk

F(2)jk (x) φ

(2)j φk ,

where F(2)jk is the coefficient of F with the basis φ

(2)j φk, we get the equation

−S−11 ∂2

xλ(2)jk + E

(2)jk λ

(2)jk = F

(2)jk with ∂xλ

(2)jk = 0 at x = 0, xM ,

and E(2)jk > 0. The solution is

λ(2)jk (x) = S1

[ϕ0(x)

r sinh(rxM )

∫ xM

xFjk(s)ϕ1(s)ds +

ϕ1(x)r sinh(rxM )

)∫ x

0Fjk(s)ϕ0(s)ds

](5.7)

10

with ϕ0 = cosh(rx), ϕ1 = cosh(rx − rxM ), r = (S1E(2)jk /S3)1/2, W = −r sinh(rxM ). The velocity field

v(2)nl = v0 + S−1∇λ

(2)nl has horizontal components that satisfy relations like (2.7) but the main feature is the

correct behavior of the vertical velocity w(2)nl as S3 → 0, whereas the velocity wnl given by BVP1 becomes

discontinuous (Result 2). The result in question, proved in the Appendix, is

Result 4: The limiting vertical velocity limS3→0 w(2)nl is continuous at Γx.

In a similar way, if λ(2) is estimated by means of the series λ(2)ml =

∑mlik λik(y)φ(2)

i φk, the field v(2)ml =

v0 + S−1∇λ(2)ml has a vertical component w

(2)ml that is continuous at Γy as S3 → 0. These results are

independent of S1 and S2.

In a general case, where S is constant and diagonal, BVP2 is replaced by the complete Ritz approximation

Lλ(2)mnl = F

(2)mnl with λ

(2)mnl =

mnl∑

ijk

λ(2)ijkφ

(2)ijk , F

(2)mnl =

mnl∑

ijk

F(2)ijk φ

(2)ijk, F

(2)ijk =< F, φ

(2)ijk > , (5.8)

where φ(2)ijk = φ

(2)i φ

(2)j φk is an eigenfunction (5.6b) of L. We have λ

(2)ijk = F

(2)ijk /E

(2)ijk and the corresponding

field is v(2)mnl = v0 + S−1∇λ

(2)mnl. Let us see some examples.

Example 1 (continued). (a) To verify Result 3 consider the version Lλ(2)m = F

(2)m of problem (5.4). The

series F(2)m =

∑mi=0 F

(2)i φ

(2)i = 1 coincides with F = 1 since the basis φ

(2)i includes the constant function

φ(2)i=0 = 1/

√xM because of the NBC (5.5) at x = 0, xM . Hence λ

(2)m = λ

(2)0 (z) and problem (5.4) takes the

form−S−1

3 λ(2)0 = 1 with λ

(2)0 |z=0= 0 , λ

(2)0 |z=zM = 0 .

The solution λ(2)0 = S3

(z2M − z2

)/2 yields v(2) = u0i+w0k with w0 = −z, thus v(2) = U0 confirming Result

3. (b) The DBC λ = 0 at z = zM in BVP2 produce bad numerical results when we consider the completeRitz approximation (5.8). In this case we have to use the series Fl =

∑lk=0 〈1, φk〉z φk of F = 1. Since every

φk vanishes at z = zM , Fl exhibits the Gibbs phenomenon at z = zM as l → ∞ (see [35, Fig. 1]). Hencethe field v(2)

l = v0 + S−1∇λ(2)l satisfies ∇ · v(2)

l = F − Fl 6= 0 at z = zM for all l and we can expect a poorquality of v(2)

l in a vicinity of z = zM . This is confirmed by the three-dimensional results of Table I wherewe see that the flux I(Γi,v

(2)mnl) is only slightly lower than that given by BVP1.

Three-dimensional solutions of problem (5.8) were used to compute the results reported in Tables I−Vfor v(2)

mnl. We see that the flux I(Γi,v(2)mnl) is similar to that I(Γi,vmnl) from BVP1 and increases as Γi → Γ,

e.g., in Table IV I(Γi,v(2)mnl) goes from −2 × 10−5 to −1. Similar results were reported in Ref. [28] for a

two-dimensional version of BVP2 with a complex terrain. In Table V the ratio ∆M/M obtained via BVP1and BVP2 is similar as well.

Example 2. A variation of BVP1 and BVP2, to modeling the wind field and the transport of atmosphericpollutants, was worked in Refs. [23,26]. It consists in solving the equation Lλ = ∇ · v0 with λ = 0 on thelateral boundaries Γx, Γy, Lλ =

(vT − v0

) ·n on the terrain and top, S is constant and diagonal. To see theeffect of the DBC consider a region Ω with flat terrain and v0 = u0i + v0j. According to Proposition 1, λ issolution of the problem

Lλ = ∇ · v0 with λ = 0 on Γx ∪ Γy, ∂zλ = 0 at z = 0, zM . (5.9)

To solve this problem we use the basis functions φi, φj , of BVP1 and the solutions of problem

−∂2zφ

(2)k = E

(2)k φ

(2)k with ∂zφ

(2)k = 0 at z = 0, zM . (5.10)

11

The pair E(2)k , φ

(2)k , is similar to that given by Eq. (5.6b). A Ritz approximation is obtained from equation

Lλmnl = Fmnl with λmnl =mn∑

i,j=1

l∑

k=0

λijkφiφjφ(2)k ,

where Fmnl is the series

Fmnl =mn∑

i,j=1

l∑

k=0

Fijkφiφjφ(2)k , Fijk =< F, φiφjφ

(2)k > .

The divergence of the field vmnl = v0 + S−1∇λmnl is ∇ · vmnl = F − Fmnl. As a particular case considerthe field v0 = iβx of Table I, which yields F = β and Fmnl = βengmfl where en, gm, f

(2)l , are Fourier

series of the constant function 1 with the basis φj , φi, φ(2)k , respectively. Since the basis φ

(2)k includes the

constant function φ(2)k=0 = 1/

√zM we have f

(2)l = 1 and, therefore, the divergence ∇ · vmnl is determined

by the convergence of the series en, gm, toward the constant function 1. As occurs with BVP1, the DBCon Γx, Γy, generates a bad convergence of Fmnl. This is confirmed by the flux I (Γi,vmnl) computed in theregions Ωi of Table I with m = n = 50. The values I = 4× 10−7, 2× 10−5, 3× 10−2 (km3s−1) for Γ1, Γ2, Γ,respectively, are slightly lower than those given by BVP1, in agreement with the use of the NBC at z = zM .However, the global effect of the DBC on the lateral boundaries is as bad as occurs in BVP1 and BVP2.

6. Approximations of condition v · n = vT · n on Γ

The best boundary condition on a mass-consistent field v (1.3) is

v · n = vT · n on the whole boundary Γ (6.1)

of the region of interest Ω. The use of this boundary condition requires the following formulation of Propo-sition 1, which is proved in the Appendix.

Proposition 2. Let S, v0, be as in Proposition 1, and let V∗N = w ∈ H(div) : ∇·w = 0 in Ω, w ·n = vT ·non Γ. Then there is a unique field v in V∗N that minimizes the functional J(v) (1.1) and there exists λ inH1(Ω) with which the relation (1.3) holds true.

Using (1.3), (1.2), (6.1), we get the problem

Lλ = ∇ · v0 in Ω (6.2a)

Lλ =(vT − v0

) · n on Γ . (6.2b)

Multiplying Eq. (6.2a) by φ in H1(Ω) and integrating by parts we get the weak form of BVP (6.2), namely,λ has to satisfy

a (λ, φ) =⟨∇ · v0 | φ⟩

Ω+

⟨(vT−v0

) · n | φ⟩ΓN

for all φ ∈ H1(Ω) (6.3a)

where we set a (λ, φ) ≡ ⟨S−1∇λ | ∇φ

⟩Ω. In particular, for φ = 1, we have a (λ, 1) = 0 and Eq. (6.3a) takes

the form ⟨∇ · v0 | 1⟩Ω

+⟨(

vT−v0) · n | 1⟩

Γ= 0 . (6.3b)

Since the relation⟨∇ · v0 | 1⟩

Ω− ⟨

v0 · n | 1⟩Γ

= 0 holds for any field v0 in H(div), the Eq. (6.3b) takes theform

∮Γ vT · nds = 0, which is, by hypothesis, true. This guarantees that there exists a unique λ ∈ H1(Ω)

(module an additive constant) that satisfies the weak form (6.3a) of BVP (6.2) [37]. We see that the BC(6.1) is equivalent to the NBC (6.2b) on the whole boundary Γ, avoiding the use of DBC λ = 0. Using

12

equations similar to (6.3a), (6.3b), one sees that a field v∗ can be used to approximate vT ·n on Γ only whenit satisfies the relation ∮

Γv∗ · nds = 0 . (6.4)

A simple choice to replace the DBC λ = 0 at z = zM in BVP2 (5.2), is ∂zλ = 0, this condition is equivalentto estimate vT · n with v0 · n, but in general v0 does not satisfy Eq. (6.4).

Since the term vT · n is, in general, unknown on the flow-through boundary Γx ∪ Γy ∪ ΓzM , it has to beestimated. Since the field U0 satisfies (6.4) and it can be used to define the BC

v · n = U0 · n on Γ . (6.5a)

This leads to the BVP

Lλ = ∇ · v0 in Ω (6.5b)

Lλ = w0k · n on Γ . (6.5c)

The BC (6.5a) does not imply the field v = v0 + S−1∇λ will be equal to U0 in Ω, but it guarantees thatthe field U0 is a limiting case of v, as is shown below. In this work and Ref.[35] (Table I) we saw that thecondition λ = 0 generates discontinuities in v when the elements of S are small or large. The results reportedbelow support the expectation that these discontinuities disappear with the NBC (6.5c), and the numericalestimations of v satisfy the physical constraint (1.2) in the closed region Ω with much more accuracy thanthe fields obtained with BVP1 and BVP2.

Let us consider the field v0 = u0i + v0j in a region Ω with flat topography. The BVP (6.5) leads to theBoundary Value Problem 3 (BVP3)

Lλ(3) = F (6.6a)

∂xλ(3)⌋

Γx

= ∂yλ(3)

⌋Γy

= ∂zλ(3)

⌋z=0

= 0, ∂zλ(3)

⌋z=zM

= S3w0M (x, y) . (6.6b)

where we set w0M (x, y) ≡ w0 (x, y, zM ) [Eq. (2.4)]. The desired field is v(3) = v0 + S−1∇λ(3). To get a

formulation of Eq. (1.9) with S given by Eq. (2.5), we replace Eq. (6.6a) by the Ritz approximation

Lλ(3)mn = F (2)

mn (6.7a)

where F(2)mn is given by Eq. (5.4) but now the series

λ(3)mn =

mn∑

ij=0

λ(3)ij (z)φ(2)

ij is subject to ∂zλ(3)mn

⌋z=0

= 0, ∂zλ(3)mn

⌋z=zM

= S3w0Mmn , (6.7b)

with

w0,2Mmn =

mn∑

ij=0

w0,2Mijφ

(2)ij and w0,2

Mij =< w0M , φ

(2)ij >xy , (6.8)

where the φ(2)ij ’s are solutions of the eigenvalue problem (5.3). To get an homogeneous boundary condition

at z = zM , λ(3)ij is replaced by a new unknown ψij given by

λ(3)ij = S3ψij + S3qij with qij = cz2/2, c = w0,2

Mij/zM . (6.9)

Replacing (6.9) into (6.7b), (6.7a), we get the problem

Aijψij = −S3F(2)ij −Aijqij with ∂zψij = 0 at z = 0, zM , (6.10)

13

where we set Aij = ∂2z − S3E

(2)ij . The solution of this problem leads to the following result proved in the

Appendix.

Result 5. Let U0,2mn be the field of Result 3. For S given by (2.5) the field v(3)

mn = v0 + S−1∇λ(3)mn satisfies

limS3→0

v(3)mn = U0,2

mn . (6.11)

To study the numerical effects due to the BC (6.7b) at zM we consider a diagonal and constant matrixS and the complete Ritz approximation of BVP3

Lλ(3)mnl = F

(3)mnl with λ

(3)mnl =

mn∑

ij=0

λ(3)ijl (z)φ(2)

ij , (6.12)

where F(3)mnl is the Fourier series

F(3)mnl =

m,n∑

i,j=0

F(3)ijl φ

(2)i φ

(2)j , F

(3)ijl =

l∑

k=0

< F, φ(2)i φ

(2)j φ

(2)k > φ

(2)k , (6.13)

and φ(2)k is solution of problem (5.10). To compute λ

(3)mnl we set λ

(3)ijl = S3ψijl + S3qij where ψijl is the Ritz

approximation of problem (6.10) in the space generated by the set φ(2)k l

k=0, namely,

Aijψijl = −S3F(3)ijl − Aijl with Aijl =

l∑

k=0

⟨Aijqij , φ

(2)k

⟩zφ

(2)k , ψijl =

l∑

k=0

ψijkφ(2)k . (6.14)

This leads to the relationAijλ

(3)ijl = −S3F

(3)ijl − S2

3E(2)ij

c

2(z2 − z2

l

)

where we set z2l =

∑lk=0 < z2, φ

(2)k >z φ

(2)k and, consequently, λ

(3)mnl satisfies

Lλ(3)mnl = F

(3)mnl +

S3

2zM

(z2 − z2

l

) mn∑

ij=0

w0,2MijE

(2)ij φ

(2)ij . (6.15)

Hence the divergence of the field v(3)mnl = v0 + S−1∇λ

(3)mnl is given by

∇ · v(3)mnl = F − F

(3)mnl −

S3

2zM

(z2 − z2

l

)Lxyw

0,2Mmn. (6.16)

From Eqs. (6.8), (5.3), we get

Lxyw0,2Mmn =

mn∑

ij=0

w0,2MijE

(2)ij φ

(2)ij .

The series w0,2Mmn converges to w0

M in the L2 (Ωxy)−norm but the convergence of Lxyw0,2Mmn towards Lw0

M

may not be good. The z2l −series converges uniformly to z2 on [0, zM ), except in a vicinity of zM where

we have ∂zz2 6= 0 whereas ∂zz

2l = 0 at z = zM . The bad convergence of z2 − z2

l and Lxyw0,2Mmn may be

compensated by small values of S3. This result suggests that small values of S3 may also improve v(3)mnl.

Thus, for small S3 the goodness of v(3)mnl depends basically of the convergence of F

(3)mnl towards F .

To illustrate the above results consider a three-dimensional version of Example 1 v0 = xi which yieldsF = 1, U0 = v0+kw0 with w0 = −z. (a) Hence we get w0

M = −zM and BVP3 takes the form Lλ(3) = 1 withthe boundary conditions (6.6b) and ∂zλ

(3) = −S3zM at z = zM . The exact solution λ(3) = −S3z2/2 + λ0,

where λ0 is an arbitrary constant, yields v(3) = U0, in agreement with Result 5. (b) If BVP3 is replaced

14

by the approximated problem (6.7) we get the same pair λ(3),v(3) because the series F(3)mn coincides with

F = 1. (c) If we consider the complete Ritz approximation (6.12) we get F(3)mnl = F = 1, since the constant

eigenfunctions φ(2)i=0, φ

(2)j=0, φ

(2)k=0 [Eq. (5.6b)], belong to the basis because of the boundary conditions (6.6b)

on Γx, Γy, and (5.10) at z = 0, zM . In BVP2 the DBC at z = zM generates a bad convergence of theseries F

(2)mnl in a vicinity of z = zM and the poor divergence ∇ · v(2)

mnl observed in Table I. In Eq. (6.16)we have w0,2

Mmn = w0M = −zM since the basis φ

(2)ij includes the constant function φ

(2)i=0φ

(2)j=0 which yields

Lxyw0,2Mmn = 0. Thus we get ∇ · v(3)

mnl = 0 in the closed region Ω for all m, n, l, in contrast with the resultsin Table I for fields vmnl, v(2)

mnl, obtained with boundary conditions used in the literature [6−21,23-28,30].

From Eq. (6.16) we get the flux

I(Γi,v(3)mnl) = ∆Fi − (S3/2zM ) ∆z2

i Lwi (6.17)

reported in Tables I-IV with S3 = 1, where we set

∆Fi =∫

Ωi

(F − F(3)mnl)dΩi, ∆z2

i =∫ di

ci

(z2 − z2

l

)dz ,

Lwi =∫ bi

ai

∫ bi

ai

Lxyw0,2Mmndxdy .

The flux I(Γi,v(3)mnl) in Table II for u0 = βxz is also 0 because we have Lwi = 0 and the series F

(3)mnl = F

(3)l

(the series of z in the basis φ(2)k ) converges rapidly. Tables I-V show that the field v(3)

mnl: (i) is better thanvmnl, v(2)

mnl, in the inner regions Ω1, Ω2 and, (ii) is almost solenoidal in a vicinity of the boundary Γ. Thuswe can say that the NBC in (6.5a) can provide reliable mass-consistent models in the closed region Ω.

Table VI reports integrals in Eq. (6.17) for u0 = β(cosωx)ze−z/H . We see that ∆z2i is reasonably

small and only Lwi has a “large” value. This result can be attributed to the bad convergence of the seriesLxyw

0,2Mmn. Nevertheless the product Ai = (S3/2zM )∆z2

i Lwi is as small as ∆Fi with S3 = 1, zM = 5 km,so that if we choose S3 < 1, the goodness of v(3)

mnl inside Ω is determined by ∆Fi. The numerical resultsreported in [25,35] suggest that for S3 ≤ 10−4, we get v(3) ≈ U0 and in this case we have Ai ¿ ∆Fi. Thefact that each integral in the flux I on the outer boundary Γ is practically equal to 0, can be attributed tothe NBC (6.5c), confirming its goodness to obtain reliable mass-consistent velocity fields in Ω.

TABLE VI. Integrals of the flux (6.17) reported in Table IV, with S3 = 1, zM = 5 km.

∆Fi Ai ∆z2i Lwi

Γ1 −5× 10−6 4× 10−6 −4× 10−5 4× 10−2

Γ2 −2× 10−4 2× 10−4 −2× 10−4 4× 10−1

Γ −6× 10−14 zero zero −5× 10−17

Example 3. In mesoscale meteorology the equations of motion are simplified decomposing the velocity vT

and other variables into the form vT = v0(z) + v′(x, y, z, t) where

v0(z) = |Ωxy|−1 ∆t−1

∫ t+∆t

t

∫

Ωxy

vT (x, y, z) dΩxy

and v′ is a subgrid-scale perturbation [2]. This suggests the use of v0 = iu0 + jv0 to get the field

v0h(z) = |Ωxy|−1

∫

Ωxy

[i u0 (x, y, z) + j v0 (x, y, z)

]dΩxy ≡ iu0

h + jv0h

15

which satisfies (6.4) and, therefore, can be used to estimate vT ·n. In the case of a region with flat topographywe have to solve equation (6.2a) with the boundary conditions

∂xλ(3)⌋

x=0,xM

= (u0h − u0), ∂yλ

(3)⌋

y=0,yM

= (v0h − v0) , ∂zλ

(3)⌋

z=0,zM

= 0 . (6.18)

To study briefly these conditions consider a diagonal and constant S and the field v0 = ix of Example 1.We have v0

h = ixM/2, the solution of BVP (6.2a), (6.18), is λ = (xM − x)x/2 which yields v = ixM/2. Thisresult may be inconsistent because if v0 is the real horizontal flow, the true field vT can be U0 = v0 + w0kwith w0 given by (2.4), so that a reasonable approximation of BC (6.1) should allow us to recover U0 fromthe field v = v0 + S−1∇λ. Thus, the BC v · n = v0

h · n on Γ should be used with care.

Remark. The Results 1 to 5 were obtained with a flat topography, but they can be extended to a domain Ωwith a complex terrain h(x, y) by means of a perturbation-theory argument. For instance let S be diagonaland constant, v0 = u0i + v0j, hM = maxΩxy |h(x, y)|, h = h/hM and ε = hM/zM . In terms of the verticalcoordinate σ = zM (z − h)/(zM − h), we get z = (1− εh)σ + zM , and BVP (2.2) takes the form

Lσλ = zσ∇ · v0 in Ωσ

Lσλ =(vT − v0

) · n on Γσ=0

λ = 0 on ΓDσ = Γσ \ Γσ=0

(6.19)

where Ωσ = (x, y) ∈ Ωxy, 0 < σ < zM and Γσ are images of Ω and Γ in the xyσ space, respectively, generalexpressions of Lσ, Lσ, are given in Ref. [35]. A little of algebra yields the decomposition

Lσ = L(0)σ + O(ε) , Lσ = −S−1

3 ∂σ + O(ε) , zσ∇ · v0 = ∂xu0 + ∂yv0 + O(ε)

with L(0)σ = −S−1

1 ∂2x − S−1

2 ∂2y − S−1

3 ∂2σ, zσ = 1 + O(ε), zx = O(ε), zy = O(ε). The analyticity of these

expressions in a vicinity of ε = 0 implies that λ has the form λ = λ(0) +ελ(1) + . . . [38], where λ(0) is solutionof the BVP

L(0)σ λ(0) = ∂xu0 + ∂yv

0 in Ωσ ,∂σλ(0) = 0 on Γσ=0 ,λ(0) = 0 on ΓDσ = Γσ \ Γσ=0 ,

(6.20)

This is BVP1 (2.3) with σ instead of z. On the other hand, the equation ∇ ·U0 = 0 subject to n ·U0 = 0at z = h, has the contravariant form

∂xzσu0 + ∂yzσv0 + ∂σzσU0,3 = 0 with U0,3 |σ=0= 0

where zσU0,3 = w0 − zxu0 − zyv0. Hence we get

w0 = zxu0 + zyv0 −

∫ σ

0(∂xzσu0 + ∂yzσv0)σ=s ds = w0,(0) + O(ε) (6.21)

withw0,(0) = −

∫ σ

0(∂xu0 + ∂yv

0)σ=sds .

Thus Results 1, 2, are valid for terms of order ε0 in v = v0 + S−1∇λ and U0. In meteorology zM can belarge enough to have ε ¿ 1, so that Results 1−5 are essentially valid for a complex topography.

7. Concluding remarks

We have studied the effect of some boundary conditions on the properties of VMCM’s. Proposition 1states that if the BC v · n = vT · n is used on a subset ΓN of Γ, the DBC (1.7) is mathematically correct.

16

However, expressions (3.2), (4.3), show that DBC (1.7) yields a numerical field vn = v0 + S−1∇λn that, ingeneral, does not satisfy ∇ · vn = 0 in a vicinity of ΓD. Some authors [6] have suggested that small or largevalues of matrix elements in S are suitable in some circumstances, but in this work and Ref. [35] we sawthat such values combined with DBC (1.7) generate discontinuities in λn or vn. These results exhibit theknown fact that the solution λ of a BVP like BVP1 or BVP2, looses regularity on the boundary Γ [37], [35,Sect. 4].

To check the accuracy with which a numerical field vn is mass-consistent, we used the flux I (Γi,vn) (4.5)and the fraction of mass ∆M/M (4.6) that flows through the open boundaries in a time tc. The numericalresults of Tables I-V show that the lack of regularity on Γ can reduce significantly the accuracy of numericalapproximations vn to satisfy ∇ ·vn = 0 in the interior of Ω, e.g., in Table IV the flux |I (Γi,vmnl)| increasesfrom 4 × 10−5 to 3 (km3s−1) as Γi → Γ, independently of the matrix elements Si. Table V shows that theuse of the BC λ = 0 in any of the open boundaries can yield a field vn with a poor mass-consistency or nophysical sense for air-quality studies. Of course, boundary effects can be reduced by using a large Ω, but thisoption poses other difficulties such as the extrapolation of v0 in regions with no datum or a larger amountof computations. In summary, the results of sections 3, 4, 5, suggest that the BC λ = 0 should not be used.

The best boundary condition on a VMCM is v · n = vT · n on the whole boundary Γ, which leads tothe NBC (6.2b) and eliminates the use of DBC λ = 0. Since vT · n is, in general, unknown on the openboundaries, it has to be estimated by means of a field that satisfies Eq. (6.4). In meteorology, a simplemass-consistent field is U0 = v0 + w0k. The use of U0 to estimate vT · n guarantees that U0 is a limitingcase of the field v given by BVP3 (Result 5). This reflects a physical consistency because if v0 = u0i + v0jwere the true horizontal field, the true field can be U0. In contrast, Example 3 shows that the use of othermass-consistent fields to estimate vT · n, can exclude the possibility of recovering U0.

The numerical and analytical results of this work show that the condition v · n = U0 · n on Γ improvesthe regularity of λ and the accuracy of numerical fields vn to satisfy ∇ · vn = 0 in the closed domain Ω,e.g., in Table IV the flux I(Γi,v

(3)mnl) goes from −6× 10−7 to −3× 10−14 as Γi → Γ, and the ratio ∆M/M

reported in Table V shows that v(3)mnl is an indeed reliable mass-consistent field for air-quality studies or

other applications. Equation (6.16) shows that λ that looses regularity on Γ, but its average behavior issatisfactory since the flux I(Γ,v(3)

mnl) is almost zero.

Propositions 1 and 2 are a version of the Helmholtz decomposition theorem for vector fields, which isrecovered with S = δij , and state that v = v0 +S−1∇λ represents a family of solenoidal fields parameterizedby S. In meteorology, S has been conceived as a covariance matrix, but numerical results show that thisconception may fail [35, Sect. 2]. Result 5 might suggest that estimating of v(3) is equivalent to thedirect integration of Eq. ∇ · v = 0. However, this equivalence is only apparent and estimating v(3) hasdefinite advantages: (i) solving BVP (6.5) allows us to impose BC’s on different parts of Γ as occurs inoceanography and laboratory experiments, whereas the equation ∇ · v = 0 only permits to impose one BC;(ii) the arbitrariness of S1, S2, S12 in Eq. (2.5) can be exploited to compute a field v(3) that is consistentwith more physical information. For instance, genetic algorithms were used to estimate the elements of adiagonal and constant matrix S. [22,23].

In this work we have worked mainly in a region Ω with flat topography, by the results can be extendedto a complex terrain. In Ref. [36], where a summary of the results of this work was reported, we proposea scheme to compute VMCM’s with the BC v · n = U0 · n on Γ, where the elliptic equation is solvedalmost analytically in meteorological problems with a complex terrain. The scheme is also suitable for data

17

assimilation in oceanography and experimental fluid mechanics. Numerical results of this approach will begiven in a forthcoming work [39].

Acknowledgements

The author wishes to thank professors Jorge E. Sanchez S., Annik Vivier and Ma. T. Nunez, for the revisionof the manuscript.

Appendix.

We begin with a proof of Proposition 1 when the true field vT satisfies vT · n = 0 on ΓN . In this casethe desired field v belongs to the space V = w : ∇·w = 0 in Ω, w · ncΓN

= 0. Consider the Hilbert spaceL2(Ω) =

w : wi ∈ L2(Ω)

with the usual product 〈w | u〉Ω ≡

∫Ω w·u dΩ [40]. If the matrix S = Sij(r)3

i,j=1

is positive-definite and symmetric with continuous elements Sij on Ω, the integral 〈w | u〉ΩS ≡∫Ω w · Su dΩ

is an inner product of L2(Ω) and the norms ‖w‖Ω = 〈w | w〉1/2Ω , ‖w‖ΩS = 〈w | w〉1/2

ΩS , are equivalent. Interms of the norm ‖·‖ΩS the functional J(w) (1.1) has the form

J(w) = ||w − v0||2ΩS .

Lemma: Let S be as in Proposition 1 and let ΓD be a region with a non-zero area. If VS is the closure of Vin L2(Ω) with the norm || · ||ΩS , then for each field v0 in H(div) there is a unique pair (v, λ) ∈ VS ×H1

D(Ω)that satisfies Eq. (1.3) and v is the unique element in VS that minimizes J(w) (1.1).

Proof. (b) Let V⊥S be the orthogonal complement of VS in L2(Ω) with the product 〈· | ·〉ΩS . To characterizeV⊥S we use the fact that the orthogonal complement of V in L2(Ω) with the product 〈· | ·〉Ω is V⊥ =∇q : q ∈ H1

D(Ω)

[41]. Since the norms || · ||Ω, || · ||ΩS , are equivalent, the closure of V in L2(Ω) with thenorm || · ||Ω is VS . For arbitrary elements w, u, in VS and the set S−1∇q : q ∈ H1

D(Ω), respectively, wehave

〈w | u〉ΩS = 〈w | Su〉Ω =⟨w | SS−1∇q

⟩Ω

= 〈w | ∇q〉Ω = 0 .

Therefore V⊥S = S−1∇q : q ∈ H1D(Ω) and the Lemma follows from the decomposition L2(Ω) = VS ⊕ V⊥S

and the Projection Theorem for the Hilbert space L2(Ω) with the product 〈· | ·〉ΩS .

Proof of Proposition 1. Let u be a given element in V∗, i.e., ∇ · u = 0 in Ω, u · n |ΓN= vT · n. We have

u− v0 ∈ H(div) and the Lemma asserts that there exists vh in VS that satisfies

||vh + u− v0||2ΩS = minw∈VS

||w + u− v0||2ΩS . (A1)

Hence the desired field is v = vh + u. To see that v is unique consider a different u ∈ V∗. Then there isa pair vh,λ such that v = vh + u = v0 + S−1∇λ belongs to V∗. The field v − v belongs to VS sinceit satisfies ∇ · (v − v) = 0, (v − v) · n = 0 on ΓN . The field v − v also belongs to V⊥S , since it has theform v − v = S−1∇(λ−λ) with (λ−λ) ∈ H1

D(Ω). Thus v − v ∈ VS ∩ V⊥S = 0 and v = v.

Proof of Proposition 2. (a) We first consider vT · n = 0 on Γ. Let HS be the closure of V = w ∈ C∞0 (Ω) :

∇ · w = 0 in Ω in L2(Ω) with the norm || · ||Ω. As above, the equivalence of the norms || · ||Ω, || · ||ΩS ,and the relation V⊥ =

∇q : q ∈ H1(Ω)

[40, Theorem 2.7], imply that the orthogonal complement of HS

in L2(Ω) with the product 〈· | ·〉ΩS is H⊥S =S−1∇q : q ∈ H1(Ω)

. Hence, the projection theorem implies

that for v0 ∈ L2(Ω), there exists a pair (v, λ) ∈ HS ×H1(Ω) that satisfies (1.3) and v is the unique elementin HS that minimizes J(w) = ||w− v0||2ΩS . (b) The proof of Proposition 2 follows from the last result andan argument similar to the one used in the proof of Proposition 1.

18

Proof of Eq. (2.11). According to Eq. (2.9), the solution of problem (2.8) has the form λij = λ(0)ij +λ

(1)ij S3 +

O(S2

3

)[38]. Replacing into (2.8), we get ∂2

zλ(0)ij = 0 with ∂zλ

(0)ij = 0 at z = 0, zM , hence λ

(0)ij = 0. For λ

(1)ij

we have ∂2zλ

(1)ij = −Fij with ∂zλ

(1)ij = 0 at z = 0, zM , integrating we get

∂zλ(1)ij (z) = −

∫ z

0Fij(s)ds

where

−∫ z

0Fij(s)ds =

∫ z

0

⟨φij ,−∇ · v0

⟩xy

ds =⟨

φij ,−∫ z

0∇ · v0

(x′, y′, s

)ds

⟩

xy

=⟨φij , w

0⟩xy

= w0ij . (A1)

Hence limS3→0 wmn = limS3→0 S−13 ∂zλmn =

∑mnij w0

ijφij (x, y).

Proof of Result 2. There exist s1 ∈ [x, xM ], s0 ∈ [0, x] such that∫ xM

xFjk(s)ϕ1(s)ds = Fjk(s1)

∫ xM

xϕ1(s)ds ,

∫ x

0Fjk(s)ϕ0(s)ds = Fjk(s0)

∫ x

0ϕ0(s)ds .

If we set F1 ≡ Fjk(s1), F0 ≡ Fjk(s0), the equation (3.3b) takes the form

S−13 λjk(x) = −E−1

jk

[ϕ0(x)

1− cosh(rx− rxM )sinh(rxM )

F1 + ϕ1(x)cosh(rx)− 1sinh(rxM )

F0

].

(a) Hence S−13 λjk = 0 for x = 0, xM and any S3 > 0. (b) For 0 < x < xM and S3 → 0 we have r →∞ and

ϕ0(x)1− cosh(rx− rxM )

sinh(rxM )∼ −1

2, ϕ1(x)

cosh(rx)− 1sinh(rxM )

∼ −12.

Thus limα3→0 S−13 λjk(x) = E−1

jk [F0 + F1] /2 =constant6= 0 and limα3→0 w is discontinuous at x = 0, xM .

Proof of Result 4. (i) In Eq. (5.7) the limit S3 → 0 yields r →∞ and the behavior of integrals in Eq. (5.7)for small S3 and 0 < x < xM and is [42]

ϕ0(x)r sinh(rxM )

∼ erx

rerxM,

∫ xM

xFjk(s)ϕ1(s)ds ∼ 2−1erxM e−rx

[Fjk(x)

r+ O

(r−2

)]

ϕ1(x)r sinh(rxM )

∼ e−rx

r,

∫ x

0Fjk(s)ϕ0(s)ds ∼ erx

2

[Fjk(x)

r+ O

(r−2

)],

Therefore

ϕ0(x)r sinh(rxM )

∫ xM

xFjk(s)ϕ1(s)ds ∼ Fjk(x)

2r2+ O

(1r3

),

ϕ1(x)r sinh(rxM )

∫ x

0Fjk(s)ϕ0(s)ds ∼ Fjk(x)

2r2+ O

(r−3

).

Thus λ(2)jk (x) ∼ S1r

−2Fjk(x) + S1O(r−3

)and using r−2 ∼ S3S

−11 E−1

k we get

limS3→0

S−13 λ

(2)jk (x) ∼ E−1

k Fjk(x) for 0 < x < xM .

(ii) For x = 0 and x = xM we have

λ(2)jk (0) =

S1 ϕ0(0)r sinh(rxM )

∫ xM

0Fjk(s)ϕ1(s)ds ∼ S1

[Fjk(0)

r+ O

(r−2

)]

λ(2)jk (xM ) =

S1 ϕ1(xM )r sinh(rxM )

∫ xM

0Fjk(s)ϕ0(s)ds ∼ S1

[Fjk(xM )

r+ O

(r−2

)].

19

Hence limS3→0 S−13 λ

(2)jk (0) ∼ E−1

k Fjk(0) and limS3→0 S−13 λ

(2)jk (xM ) ∼ E−1

k Fjk(xM ), so that limS3→0 S−13 λ

(2)jk (x)

is continuous on [0, xM ] and limS3→0 wnl is continuous at Γx.

Proof of Result 5. (i) For ij = 0 we have E(2)0 = 0 and Eq. (6.10) takes the form ∂2

zψ0 = −S3(F0+w0M,0/zM )

which yields

∂zψ0(z) = −S3

[∫ z

0F0(s)ds +

w0M,0

zMz

].

As in Eq. (A1) we have∫ z

0F0(s)ds =

∫ z

0

⟨∇ · v0 (x, y, s) , φ

(2)0

⟩xy

ds = −⟨w0 (x, y, z) , φ

(2)0

⟩,

hence ∫ z

0F0(s)ds +

w0M,0

zMz = −

⟨w0 (x, y, z) , φ

(2)0

⟩xy

+z

zM

⟨w0 (x, y, zM ) , φ

(2)0

⟩xy

and ∂zψ0 satisfies ∂zψ0 = 0 at z = 0, zM . Thus

ψ0(z) = S3

∫ z

0

⟨w0 (x, y, s) , φ

(2)0

⟩xy

ds− S3z2

2zMw0

M,0 .

(ii) For ij > 0 we have E(2)ij > 0 and the operator Aij = ∂2

z −S3E(2)ij is negative definite, so that BVP (6.10)

has a unique solution which is given by ψij>0 = S3ψij with

ψij(z) =ϕ0(z)

r sinh(rzM )

∫ zM

zfij(s)ϕ1(s)ds +

ϕ1(z)r sinh(rzM )

∫ z

0fij(s)ϕ0(s)ds ,

and fij = Fij +w0

Mij

zMpij , ϕ0 = cosh(rz), ϕ1 = cosh(rz − rzM ), r =

√S3E

(2)ij . To prove (6.11) we need to

compute limS3→0 ψij(z). According to Eq. (6.10) and ψij>0 = S3ψ we have ψij = S3ψ(1)ij +O(S2

3) [38] where

ψ(1)ij is solution of problem ∂2

zψ(1)ij = −Fij − w0

Mij/zM with ∂zψ(1)ij = 0 at z = 0, zM . Hence we get

∂zψ(1)ij = −

∫ z

0Fij(s)ds− w0

Mij

zMz =

⟨w0 (x, y, z) , φ

(2)ij

⟩xy− w0

Mij

zMz,

ψ(1)ij =

∫ z

0

⟨w0 (x, y, s) , φ

(2)ij

⟩xy

ds− qij .

(iii) Therefore

λ(3)ij = ψij + S3qij = S3ψ

(1)ij + O(S2

3) + S3qij = S3

∫ z

0

⟨w0 (x, y, s) , φ

(2)ij

⟩xy

ds + O(S23)

and summing

λ(3)mn =

mn∑

ij

λ(3)ij (z)φ(2)

ij = S3

mn∑

ij=0

[∫ z

0

⟨w0 (x, y, s) , φ

(2)ij

⟩xy

ds + O(S3)]

φ(2)ij

∂zλ(3)mn =

mn∑

ij

λ(3)ij (z)φ(2)

ij = S3

mn∑

ij=0

[⟨w0 (x, y, z) , φ

(2)ij

⟩xy

ds + O(S3)]

φ(2)ij

we get the field v(3)mn = v0 + S−1∇λ

(3)mn and the desired relation (6.11).

20

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[34] M.A. Nunez, G. Ramırez, G.. H. Hernandez, Analisis de metodos de interpolacion de datos de vientobasados en la ecuacion de conservacion de la masa, Reporte de investigacion, Universidad AutonomaMetropolitana, Division de Ciencias Basicas e Ingenierıa, 1–65 (2004).

[35] M.A. Nunez, C. Flores and H. Juarez, A study of hydrodynamic mass consistent models, J. Comput.Met. Sci. Eng. 6, 365-385 (2006).

[36] M. A. Nunez, New scheme to compute mass–consistent models of geophysical flows, in “Proceedingsof the I Workshop on Asymptotics for Parabolic and Hyperbolic Systems”, Laboratoiro Nacional deComputacao Cientıfica, Petropolis-R. J., Brasil, (2008).

[37] K. Rektorys, Variational methods in mathematics, science and engineering (D. Reidel, Dordrecht, 1977).M. Krızek and P. Neittaanmaki, Finite element approximations of variational problems and applications(Logman Scientific Technical, London, 1990).

[38] T. Kato, Perturbation theory for linear operators (Springer, New York, 1965).

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[40] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equation, Theory and Algo-rithms (Springer, Berlin, 1986).

[41] A proof of relation V⊥ =∇q : q ∈ H1

D(Ω)

is given in Ref. [25], pp. 25, 26.

[42] See, e.g., J. D. Murray, Asymptotic analysis (Springer, New York, 1984), p. 35, Eqs. (2.38, 2.39).

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