a new shallow water model with polynomial dependence on depth

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2008; 31:529–549Published online 10 July 2007 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/mma.924MOS subject classification: 76M45; 41A 60; 76B 99; 35Q 35

A new shallow water model with polynomial dependence on depth

Jose M. Rodrıguez1,∗,† and Raquel Taboada-Vazquez2

1Mathematical Methods and Representation Department, University of A Coruna, E.T.S. Arquitectura,Campus da Zapateira, 15071-A Coruna, Spain

2Mathematical Methods and Representation Department, University of A Coruna, E.T.S.I. Caminos,Canais e Portos, Campus de Elvina, 15071-A Coruna, Spain

Communicated by E. Sanchez-Palencia

SUMMARY

In this paper, we study two-dimensional Euler equations in a domain with small depth. With this aim,we introduce a small non-dimensional parameter ε related to the depth and we use asymptotic analysisto study what happens when ε becomes small.

We obtain a model for ε small that, after coming back to the original domain, gives us a shallowwater model that considers the possibility of a non-constant bottom, and the horizontal velocity has adependence on z introduced by the vorticity when it is not zero. This represents an interesting noveltywith respect to shallow water models found in the literature. We stand out that we do not need to makea priori assumptions about velocity or pressure behaviour to obtain the model.

The new model is able to approximate the solutions to Euler equations with dependence on z (reobtainingthe same velocities profile), whereas the classic model just obtains the average velocity. Copyright q2007 John Wiley & Sons, Ltd.

KEY WORDS: shallow waters; asymptotic analysis

1. INTRODUCTION

Our aim is to obtain a shallow water model; so, the domain we consider must have a smalldepth compared with its length, for example, a channel. We represent it by �ε set (see Figure 1)defined by

�ε = {(xε, zε)/xε ∈ [0, L], zε ∈ [H ε(xε), sε(tε, xε)]} (1)

∗Correspondence to: Jose M. Rodrıguez, E.T.S. Arquitectura, Campus da Zapateira s/n, 15071-A Coruna, Spain.†E-mail: [email protected]

Contract/grant sponsor: Ministerio de Educacion y Ciencia of Spain; contract/grant number: MTM2006-14491

Copyright q 2007 John Wiley & Sons, Ltd. Received 30 March 2007

530 J. M. RODRIGUEZ AND R. TABOADA-VAZQUEZ

Figure 1. Domain.

where xε and zε denote the horizontal and vertical coordinate, respectively, zε = H ε(xε) is theequation of the bottom of the channel (supposed known), and zε = sε(tε, xε) is the equation ofthe surface (unknown). We can also define hε(tε, xε) = sε(tε, xε) − H ε(xε) (water depth).

We introduce, now, a non-dimensional parameter ε, which represents the quotient betweencharacteristic depth and length of the domain (as we have previously pointed out ε must be small).Then, we can suppose that H ε(xε) = εH(x), sε(tε, xε) = εs(t, x) and hε(tε, xε) = εh(t, x) (wherex = xε and t = tε are independent of ε). We are just making clear that H ε and hε are of order ε,i.e. they are small compared with the domain length.

The flow is supposed to obey the two-dimensional Euler equations in �ε. The external forcesacting on the fluid are just those due to gravity, that is (both the domain and the functions andvariables involved in this problem depend on ε, and we indicate this dependence with superscript ε):

�uε

�tε+ uε �uε

�xε+ wε �uε

�zε= − 1

�0

�pε

�xε(2)

�wε

�tε+ uε �wε

�xε+ wε �wε

�zε= − 1

�0

�pε

�zε− g (3)

where uε = (uε, wε) = (uε(tε, xε, zε), wε(tε, xε, zε)) is the velocity vector, pε = pε(tε, xε, zε) isthe pressure, �0 denotes the density of the fluid and g is the gravity acceleration (assumed constant).

The fluid is supposed to be incompressible, so it verifies

�uε

�xε+ �wε

�zε= 0 (4)

Let us introduce now the boundary conditions. The pressure must be atmospheric at the surface

pε = pεs at zε = sε(tε, xε) (5)

Copyright q 2007 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2008; 31:529–549DOI: 10.1002/mma

A NEW SHALLOW WATER MODEL WITH POLYNOMIAL DEPENDENCE ON DEPTH 531

(pεs = pε

s (tε, xε) is the atmospheric pressure at the surface; it is supposed to be known, and we

also assume that pεs does not depend on ε, that is, pε

s (tε, xε) = ps(t, x)). The fluid satisfies the

slip condition at the bottom so

uε · nε = 0 at zε = H ε(xε) (6)

where nε denotes the outer unit normal to the domain boundary.In addition, we suppose that the incoming and outcoming flows are known at each instant.

Obviously, other kind of boundary conditions may be taken into consideration.We shall use below the following vorticity equation (see [1]):

��ε

�tε+ uε ��ε

�xε+ wε ��ε

�zε= 0 (7)

where

�ε = �uε

�zε− �wε

�xε(8)

Instead of the classical free surface condition, we have used an equation derived from the massconservation law:

�hε

�tε+ �

�xε

∫ sε

H ε

uε dzε = 0 (9)

Evidently, initial conditions should be imposed too.

2. CONSTRUCTION OF THE REFERENCE DOMAIN

Usually, when using asymptotics to analyse fluids, most of the authors work in the original domain(see, for example, [2, 3]), which in this case depends on parameter ε and time tε, or they supposethe surface to be constant (see, for example, [4]). We shall, however, use the asymptotic techniquein the same way as in [5–8] and related works, that is, we make a change of variable to a referencedomain independent of the parameter ε and the time.

Let �=[0, L] × [0, 1] be the reference domain and let us define the change of variable, from� to �ε:

tε = t

xε = x

zε = ε[H(x) + zh(t, x)](10)

Given any function Fε defined on [0, T ] × �ε, we can introduce another function F(ε) on

[0, T ] × � using the change of variable: F(ε)(t, x, z) = Fε(tε, xε, zε). The relationship between



their partial derivatives is:�Fε

�tε= �F(ε)

�t− z

h

�h�t

�F(ε)

�z= Dt F(ε)

�Fε

�xε= �F(ε)

�x−

H ′ + z�h�x

h

�F(ε)

�z= Dx F(ε)

�Fε

�zε= 1

εh

�F(ε)

�z= 1

εDzF(ε)

where we introduce the notation:

Dt = ��t

− z

h

�h�t

��z

Dx = ��x

−H ′ + z

�h�x

h

��z

Dz = 1

h

��z

(11)

Now, defining,

u(ε)(t, x, z) = uε(tε, xε, zε)

w(ε)(t, x, z) = wε(tε, xε, zε)

p(ε)(t, x, z) = pε(tε, xε, zε)

problem (2)–(9) can be written in the reference domain � as follows, where now dependence onε appears explicitly:

Euler equations:

Dtu(ε) + u(ε)Dxu(ε) + w(ε)1

εDzu(ε) = − 1

�0Dx p(ε) (12)

Dtw(ε) + u(ε)Dxw(ε) + w(ε)1

εDzw(ε) = − 1

�0

1

εDz p(ε) − g (13)

Incompressibility condition:

Dxu(ε) + 1

εDzw(ε)= 0 (14)

Boundary conditions:

p(ε) = ps at z = 1 (15)

w(ε) = εu(ε)H ′ at z = 0 (16)



Vorticity equation:

Dt�(ε) + u(ε)Dx�(ε) + w(ε)1

εDz�(ε)= 0 (17)

Vorticity written in terms of the velocity components:

�(ε)= 1

εDzu(ε) − Dxw(ε) (18)

Equation to determine h:

�h�t

+∫ 1

0

�(u(ε)h)

�xdz = 0 (19)

The change of variable must be applied to the initial conditions too.

3. ASYMPTOTIC ANALYSIS

Let us suppose now that the solution to problem (12)–(19) allows an expansion in powers of ε,that is,

u(ε) = u0 + εu1 + ε2u2 + · · ·w(ε) = w0 + εw1 + ε2w2 + ε3w3 + · · ·p(ε) = p0 + εp1 + ε2 p2 + · · ·�(ε) = ε−1�−1 + �0 + ε�1 + ε2�2 + · · ·

(20)

We replace this expansion in the equations obtained, after the change of variable, in �. Makingthis substitution in (12), we get:

Dtu0 + εDtu

1 + ε2Dtu2 + · · · + (u0 + εu1 + ε2u2 + · · ·)[Dxu

0 + εDxu1 + ε2Dxu

2 + · · ·]

+ (w0 + εw1 + ε2w2 + ε3w3 + · · ·)1ε[Dzu

0 + εDzu1 + ε2Dzu

2 + · · ·]

=− 1

�0(Dx p

0 + εDx p1 + ε2Dx p

2 + · · ·)

We identify the terms multiplied by the same power of ε and we rewrite the previous expressionas follows:

ε−1w0Dzu0 + ε0

(Dtu

0 + u0Dxu0 + w0Dzu

1 + w1Dzu0 + 1

�0Dx p

0)

+ ε

(Dtu

1 + u0Dxu1 + u1Dxu

0 + w0Dzu2 + w1Dzu

1 + w2Dzu0 + 1

�0Dx p

1)



+ ε2(Dtu

2 + u0Dxu2 + u1Dxu

1 + u2Dxu0 + w0Dzu

3 + w1Dzu2

+ w2Dzu1 + w3Dzu

0 + 1

�0Dx p

2)

+ O(ε3) = 0 (21)

A similar expression is obtained from Equation (13):

ε−1(

w0Dzw0 + 1

�0Dz p

0)

+ ε0(Dtw

0 + u0Dxw0 + w0Dzw

1 + w1Dzw0 + 1

�0Dz p

1 + g

)

+ ε

(Dtw

1 + u0Dxw1 + u1Dxw

0 + w0Dzw2 + w1Dzw

1 + w2Dzw0

+ 1

�0Dz p

2)

+ ε2(Dtw

2 + u0Dxw2 + u1Dxw

1 + u2Dxw0 + w0Dzw

3

+w1Dzw2 + w2Dzw

1 + w3Dzw0 + 1

�0Dz p

3)

+ O(ε3) = 0 (22)

Replacing the expansions in powers of ε (20) in (14) and then grouping terms multiplied by thesame power of ε, we have:

ε−1Dzw0 + Dxu

0 + Dzw1 + ε(Dxu

1 + Dzw2) + ε2(Dxu

2 + Dzw3) + O(ε3) = 0 (23)

We repeat the process for the boundary conditions ((15)–(16)) and we arrive at:

p0 + εp1 + ε2 p2 + · · · = ps at z = 1 (24)

w0 + ε(w1 − u0H ′) + ε2(w2 − u1H ′) + ε3(w3 − u2H ′) + · · · = 0 at z = 0 (25)

In the same way, the vorticity equation yields:

ε−2w0Dz�−1 + ε−1(Dt�

−1 + u0Dx�−1 + w0Dx�

0 + w1Dx�−1)

+ Dt�0 + u0Dx�

0 + u1Dx�−1 + w0Dz�

1 + w1Dz�0 + w2Dz�

−1

+ ε(Dt�1 + u0Dx�

1 + u1Dx�0 + u2Dx�

−1 + w0Dz�2 + w1Dz�

1

+ w2Dz�0 + w3Dz�

−1) + ε2(Dt�2 + u0Dx�

2 + u1Dx�1 + u2Dx�

0 + u3Dx�−1

+ w0Dz�3 + w1Dz�

2 + w2Dz�1 + w3Dz�

0 + w4Dz�−1) + O(ε3) = 0 (26)



If we substitute, now, u(ε), w(ε) and �(ε) for their expansions in powers of ε in the relationshipbetween the vorticity and the velocity components (18), we obtain:

ε−1(�−1 − Dzu0) + �0 − Dzu

1 + Dxw0 + ε(�1 − Dzu

2 + Dxw1) + O(ε2) = 0 (27)

Starting off from (19), the equation necessary to calculate the depth, we get:

�h�t

+∫ 1

0

�(hu0)

�xdz + ε

∫ 1

0

�(hu1)

�xdz + ε2

∫ 1

0

�(hu2)

�xdz + O(ε3) = 0 (28)

Since u0, w0, p0, �−1, u1, w1, etc. are independent of ε, once the terms that multiply the samepower of ε are grouped, in the previous equations we have polynomials in ε equated to zero, sotheir coefficients should be zero too. In this way, we arrive at a series of equations that will allowus to determine u0, w0, p0, �−1

i , u1, w1, etc.We begin with the coefficient of ε−2 in Equation (26):

w0Dz�−1 = 0 (29)

Equating the coefficients of ε−1 of Equations (21)–(23) and (26)–(27) to 0 we have:

w0Dzu0 = 0 (30)

w0Dzw0 + 1

�0Dz p

0 = 0 (31)

Dzw0 = 0 (32)

Dt�−1 + u0Dx�

−1 + w0Dx�0 + w1Dx�

−1 = 0 (33)

�−1 − Dzu0 = 0 (34)

Let us see the consequences that can be drawn from the precedent equalities taking into account

p0 = ps at z = 1 (35)

w0 = 0 at z = 0 (36)

(equalities obtained from (24)–(25)). In the first place, (32) and (36) yield:

w0 = 0 (37)

Using (37), equalities (31) and (35) permit us to obtain the zeroth-order term of the pressure:

p0 = ps(t, x, y) (38)

If boundary and initial conditions are appropriate (homogeneous), Equation (33) (keeping inmind w0 = 0) has as only solution �−1 = 0 (see [9, p. 129]). Moreover, it does not seem reasonablethat �−1 �= 0 because in that case, due to (34), u0 would depend on z (contradicting the fact thatwe are working with shallow water) and the vorticity would not be bounded when ε tends to zero,so we assume that

�−1 = 0 (39)



Remark 3.1It should be noted that (39) would be obtained directly supposing in (20) that the expansion for�(ε) begins by the term of order zero in ε.

Now, it follows from (34)

�u0

�z= 0 (40)

Next, we equate the coefficients of ε0 to 0 (Equations (21)–(23) and (26)–(28)) and considering(37)–(40), we have:

�u0

�t+ u0

�u0

�x+ 1

�0

�ps�x

= 0 (41)

1

�0Dz p

1 + g = 0 (42)

�u0

�x+ Dzw

1 = 0 (43)

Dt�0 + u0Dx�

0 + w1Dz�0 = 0 (44)

�0 − Dzu1 = 0 (45)

�h�t

+ �(hu0)

�x= 0 (46)

We continue equating to zero the terms multiplied by ε in Equations (21)–(28). We take intoaccount when we write these terms that w0 = �−1 = �u0/�z = 0 (37), (39) and (40)), and we get:

Dtu1 + u0Dxu

1 + u1�u0

�x+ w1Dzu

1 + 1

�0Dx p

1 = 0 (47)

Dtw1 + u0Dxw

1 + w1Dzw1 + 1

�0Dz p

2 = 0 (48)

Dxu1 + Dzw

2 = 0 (49)

p1 = 0 at z = 1 (50)

w1 − u0H ′ = 0 at z = 0 (51)

Dt�1 + u0Dx�

1 + u1Dx�0 + w1Dz�

1 + w2Dz�0 = 0 (52)

�1 − Dzu2 + Dxw

1 = 0 (53)∫ 1

0

�(hu1)

�xdz = 0 (54)

The process continues with the coefficients of ε2, ε3, . . . .



Now, the integration of (42), setting condition (50), gives us the following expression for p1:

p1 = �0gh(1 − z) (55)

We also integrate (43) with respect to z considering that u0 does not depend on z, and imposing(51) we can find w1:

w1 = u0H ′ − hz�u0

�x(56)

4. FIRST-ORDER APPROXIMATION

We consider the following approximation:

u(ε) = u0 + εu1

w(ε) = w0 + εw1 + ε2w2

p(ε) = p0 + εp1

�(ε) = ε−1�−1 + �0

Terms w0, p0 and �−1 are known ((37)–(39)), while we calculate u0 and h solving (41) and (46):

�u0

�t+ u0

�u0

�x+ 1

�0

�ps�x

= 0

�h�t

+ �(hu0)

�x= 0

and w1 is determined by (56) in terms of u0, H and h.We also have an expression for p1, for which we just need to know the depth of the water (55):

p1 = �0gh(1 − z)

The dependence on z of the term u1 is given by (45):

�u1

�z= h�0 (57)

It is clear from (57) that

u1(t, x, z) = u1(t, x, 0) +∫ z

0h(t, x)�0(t, x, y) dy (58)

but to be able to compute (58) it is necessary to know the dependence on z of �0. Euler equations(2)–(4) transport vorticity (see (7)) and, since our purpose is to find a shallow water model, it seemsreasonable to suppose that the dependence of �0 on z is not very complicated. In addition, from



(44) and (56) we deduce that, if initial and boundary conditions imposed on �0 are polynomialsin z, then �0 is polynomial in z too. Consequently, let us assume that:

�0 =k∑

i=0(zi�0,i ) (59)

Remark 4.1If the function �0(z) is continuous, since it is defined on a closed interval (z ∈ [0, 1]), Weierstrasstheorem guarantees that �0(z) can be approximated by a polynomial in z.

We replace in (44) both �0 and w1 (using (59) and (56), respectively), so that the dependenceon z is explicit; in this way, we reach to:

Dt

[k∑

i=0(zi�0,i )

]+ u0Dx

[k∑

i=0(zi�0,i )

]+ w1Dz

[k∑

i=0(zi�0,i )

]

=k∑

i=0

(zi

��0,i

�t

)− z

h

�h�t

[k∑

i=1(i zi−1�0,i )

]

+ u0

⎡⎢⎢⎣ k∑i=0

(zi

��0,i

�x

)−

H ′ + z�h�x

h

(k∑

i=1(i zi−1�0,i )

)⎤⎥⎥⎦

+ 1

h

(u0H ′ − h

�u0

�xz

)k∑

i=1(i zi−1�0,i ) = 0

Grouping together the terms multiplied by the same power of z (and using to simplify (46)) itturns out:

��0,i

�t+ u0

��0,i

�x= 0 (i = 0, 1, 2, . . . , k) (60)

Remark 4.2The initial and boundary conditions will determine if the terms �0,i (i = 0, 1, 2, . . . , k) vanish ornot.

Once �0,i (i = 0, 1, 2, . . . , k) have been calculated, taking into account (59), we integrate (57)with respect to z:

u1 = u10 + hk∑

i=0

(zi+1

i + 1�0,i

)(61)

where u10(t, x)= u1(t, x, 0) is determined by (47). We substitute in that equation u1, w1 and p1

by expressions (61), (56) and (55), respectively:

Dt

[u10 + h

k∑i=0

(zi+1

i + 1�0,i

)]+ u0Dx

[u10 + h

k∑i=0

(zi+1

i + 1�0,i

)]



+[u10 + h

k∑i=0

(zi+1

i + 1�0,i

)]�u0

�x

+(u0H ′ − hz

�u0

�x

)Dz

[u10 + h

k∑i=0

(zi+1

i + 1�0,i

)]= −g

�s�x

and we rewrite it, grouping together the coefficients of each power of z, as follows:

�u10�t

+ u0�u10�x

+ u10�u0

�x+(

�h�t

+ u0�h�x

+ h�u0

�x

)k∑

i=0

(zi+1

i + 1�0,i

)

+ hk∑

i=0

[zi+1

i + 1

(��0,i

�t+ u0

��0,i

�x

)]

−(

�h�t

+ u0�h�x

+ h�u0

�x

)z

k∑i=0

(zi�0,i

)=−g

�s�x

(62)

Finally, using (46) and (60), Equation (47) yields:

�u10�t

+ u0�u10�x

+ u10�u0

�x= −g

�s�x

(63)

Remark 4.3The term u10 is determined independently of �0 by means of Equation (63).

Subsequently, w2 is calculated from (49) where u1 is replaced using (61):

�w2

�z= −h

{�u10�x

− �h�x

k∑i=0

(i

i + 1zi+1�0,i

)

+ hk∑

i=0

(zi+1

i + 1

��0,i

�x

)− H ′ k∑

i=0(zi�0,i )

}

and we integrate with respect to z imposing the corresponding boundary condition derivedfrom (25):

w2 = u10H′ − hz

�u10�x

+ h�h�x

k∑i=0

(i

(i + 1)(i + 2)zi+2�0,i

)

− h2k∑

i=0

(zi+2

(i + 1)(i + 2)

��0,i

�x

)+ hH ′ k∑

i=0

(zi+1

i + 1�0,i

)(64)

Now, using (38) and (55), we arrive at:

p(ε)= ps + ε�0gh(1 − z) (65)



In the same way, from (37), (56) and (64), we obtain:

w(ε) = εw1 + ε2w2 = ε

(u0H ′ − hz

�u0

�x

)

+ ε2

[u10H

′ − hz�u10�x

+ h�h�x

k∑i=0

(i

(i + 1)(i + 2)zi+2�0,i

)

− h2k∑

i=0

(zi+2

(i + 1)(i + 2)

��0,i

�x

)+ hH ′ k∑

i=0

(zi+1

i + 1�0,i

)]

={u0 + ε

[u10 + h

k∑i=0

(zi+1

i + 1�0,i

)]}εH ′

− εhz

{�u0

�x+ ε

[�u10�x

+ Dx

(h

k∑i=0

(zi+1

i + 1�0,i

))]}+ O(ε2) (66)

hence,

w(ε)= u(ε)εH ′ − εhzDx u(ε) + O(ε2) (67)

Returning to the original domain we obtain the following approximation to the solution:

uε(tε, xε, zε) = u(ε)(t, x, z) = u0(t, x) + εu1(t, x)

wε(tε, xε, zε) = w(ε)(t, x, z) = εw1(t, x, z) + ε2w2(t, x, z)

pε(tε, xε, zε) = p(ε)(t, x, z) = p0(t, x) + εp1(t, x, z)

�ε(tε, xε, zε) = �(ε)(t, x, z) = �0(t, x, y) (68)

The approximation of the pressure in �ε, if the inverse change of variable is used in (65), is

pε = pεs + �0g(s

ε − zε) (69)

Analogously, using the inverse change of variable in (67), we achieve an approximation to thevertical component of the velocity:

wε = uε �H ε

�xε+ (H ε − zε)

�uε

�xε+ O(ε2) (70)

To build the model that we are going to propose, we define

uε(tε, xε) = u(ε)(t, x)= u0(t, x) + εu10(t, x) (71)

From Equations (41) and (63), we get:

�uε

�tε+ uε �uε

�xε= − 1

�0

�ps�x

+ ε

(− �s

�xg

)+ O(ε2) = − 1

�0

�pεs

�xε− g

�sε

�xε+ O(ε2) (72)



and also (from (46))

�hε

�tε+ �(uεhε)

�xε= O(ε2) (73)

If in Equation (60), we make the change of variable and we substitute u0,ε = u0 for uε, theyresult:

��0,i,ε

�tε+ uε ��0,i,ε

�xε= O(ε) (i = 0, 1, 2, . . . , k) (74)

Moreover, uε can be written in terms of uε:

uε = uε +k∑

i=0

((zε − H ε)i+1

(i + 1)(hε)i�0,i,ε

)(75)

(where �0,i,ε (i = 0, 1, 2, . . . , k) represent �0,i (i = 0, 1, 2, . . . , k) after the change of variable (10)).We can summarize the results achieved in the following theorem.

Theorem 4.1If we suppose that there exists asymptotic expansion (20) and that �0 verifies (59), then approxi-mated solution (68), (71) fulfils

�hε

�tε+ �(uεhε)

�xε= O(ε2) (76)

�uε

�tε+ uε �uε

�xε= − 1

�0

�pεs

�xε− g

�sε

�xε+ O(ε2) (77)


ε − zε) (78)

��0,i,ε


�xε= O(ε) (i = 0, 1, 2, . . . , k) (79)

uε = uε +k∑

i=0

((zε − H ε)i+1

(i + 1)(hε)i�0,i,ε

)(80)

wε = uε �H ε

�xε+ (H ε − zε)

�uε

�xε+ O(ε2) (81)

Disregarding in Equations (76), (77) and (81) the terms of order O(ε2), as well as inEquations (79) the terms in ε (although in these equations we neglect the terms of order O(ε),given that �0,i,ε (i = 0, 1, 2, . . . , k) are multiplied by terms of order ε in (80), finally the errorcommitted is of order ε2), the following shallow water model is obtained. Formally, the order ofaccuracy of the model is O(ε2):

�hε

�tε+ �(uεhε)

�xε= 0 (82)

�uε

�tε+ uε �uε

�xε= − 1

�0

�pεs

�xε− g

�sε

�xε(83)




ε − zε) (84)

��0,i,ε


�xε= 0 (i = 0, 1, 2, . . . , k) (85)

uε = uε +k∑

i=0

((zε − H ε)i+1

(i + 1)(hε)i�0,i,ε

)(86)

wε = uε �H ε

�xε− �uε

�xε(zε − H ε) (87)

Remark 4.4The approximation of vorticity �ε verifies

�ε =k∑

i=0

((zε − H ε

hε

)i

�0,i,ε)

(88)

Remark 4.5Approximations (86)–(87) of the velocity components do not verify exactly the incompressibilityequation (4) (it is fulfilled with an error O(ε2)). We can build the following approximation ofvertical velocity wε using Equation (4):

wε = uε �H ε

�xε−∫ zε

0

�uε

�xε(tε, xε, y) dy (89)

that can be calculated exactly in terms of zε.

Equation (89) verifies exactly (4) but it is as (87) an approximation of order ε2 of the exactvertical velocity wε.

5. PROPOSED MODEL IN TERMS OF THE AVERAGE VELOCITY

Model (82)–(87) can be written in terms of the average velocity in the vertical variable. Indeed,taking into account

uε = 1

hε

∫ sε

H ε

uε dzε = 1

hε

∫ sε

H ε

[uε +

k∑i=0

((zε − H ε)i+1

(i + 1)(hε)i�0,i,ε

)]dzε

= uε + hεk∑

i=0

(1

(i + 1)(i + 2)�0,i,ε

)(90)

we can infer uε from the equation above and then substitute in Equations (76)–(81). Beginning bythe first of them:

�hε

�tε+ �(uεhε)

�xε= �hε

�tε+ �(uεhε)

�xε+ �

�xε

[−(hε)2

k∑i=0

(1

(i + 1)(i + 2)�0,i,ε

)]



Clearly,

��xε

[−(hε)2

k∑i=0

(1

(i + 1)(i + 2)�0,i,ε

)]= O(ε2)

Therefore, Equation (76) in terms of the average velocity yields

�hε

�tε+ �(uεhε)

�xε= O(ε2)

and rewriting Equations (77)–(81) in a similar way, we deduce the following:

Theorem 5.1If we assume that there exists asymptotic expansion (20) and that �0 verifies (59), then approximatedsolution (68), (71) (with (90)) fulfils

�hε

�tε+ �(uεhε)

�xε= O(ε2) (91)

�uε

�tε+ uε �uε

�xε= − 1

�0

�pεs

�xε− g

�sε

�xε+ O(ε2) (92)


ε − zε) (93)

��0,i,ε


�xε= O(ε) (i = 0, 1, 2, . . . , k) (94)

uε = uε +k∑

i=0

[((zε − H ε)i+1

(i + 1)(hε)i− hε

(i + 1)(i + 2)

)�0,i,ε

](95)

wε = uε �H ε

�xε− �uε

�xε(zε − H ε) + O(ε2) (96)

If we neglect the terms of order O(ε2) in Equations (91), (92) and (96) and the terms of orderO(ε) in (94), as we have done to obtain (82)–(87), we attain the following shallow water modelexpressed in terms of the depth averaged velocity, whose order of precision, at least formally, isO(ε2) too:

�hε

�tε+ �(uεhε)

�xε= 0 (97)

�uε

�tε+ uε �uε

�xε= − 1

�0

�pεs

�xε− g

�sε

�xε(98)


ε − zε) (99)

��0,i,ε


�xε= 0 (i = 0, 1, 2, . . . , k) (100)



uε = uε +k∑

i=0

[((zε − H ε)i+1

(i + 1)(hε)i− hε

(i + 1)(i + 2)

)�0,i,ε

](101)

wε = uε �H ε

�xε− �uε

�xε(zε − H ε) (102)

Remark 5.1Remark 4.5 is also valid for (101) and (102) approximations of the velocity components.

Remark 5.2Last model is usually written in a conservative form. If we define Qε = hεuε, then Equations (97)and (98) read

�hε

�tε+ �Qε

�xε= 0

�Qε

�tε+ �

�xε

((Qε)2

hε+ 1

2g(hε)2

)= −hε

�0

�pεs

�xε− ghε �H ε

�xε

We have also found, searching a second-order model, an improvement for the pressure (that isobtained without solving new equations) expressed as a function of velocity:

pε = pεs + �0(s

ε − zε)

[g + �uε

�tε�H ε

�xε+ (uε)2

�2H ε

�(xε)2+ uε �H ε

�xε

�uε

�xε

]

+�02

[(hε)2 − (zε − H ε)2][(

�uε

�xε

)2

− �2uε

�tε�xε− uε �2uε

�(xε)2

](103)

6. ANALYTICAL COMPARISON BETWEEN THE SHALLOW WATER MODELPROPOSED AND THE CLASSICAL MODEL

We want to compare model (97)–(102) with the classical shallow water model without viscositythat can be found in the literature (see, for example, [10, p. 581] or [11, p. 38])

�hε

�tε+ �(uεhε)

�xε= 0

�uε

�tε+ uε �uε

�xε+ g

�sε

�xε= 0

(104)

Remark 6.1We can get (104) from (97)–(102) neglecting the vorticity (if it is zero, the horizontal velocitydoes not depend on zε and therefore it is equal to the average horizontal velocity) and supposing



that the atmospheric pressure at the surface does not depend on xε. We deduce then that, if theatmospheric pressure at the surface does not depend on xε, (97)–(102) and (104) are equivalent ifand only if �ε = 0, that is, in case there is no vorticity.

Next, we shall compare Euler equations (see (2)–(4)) and models (97)–(102) and (104) for asimple example that, in our opinion, illustrates how model (97)–(102) improves classical model(104).

If we choose,

u = u0 +k∑

i=0

[(zi+1

(i + 1)Ai− A

(i + 1)(i + 2)

)�i]

, w = 0

H = 0, s = h = A>0

p = ps + �(A − z)g

(105)

with u0, A, �i (i = 0, 1, . . . , k), ps ∈ R, then we have a solution to Euler equations and solutionto (97)–(102), but it is not solution to (104) (because the solution to (104) cannot depend on z)unless �i = 0 (i = 0, 1, . . . , k). That is, models (97)–(102) and (104) capture the exact solution toEuler equations when the velocities are constant, but just model (97)–(102) is able to calculate thesolution to Euler equations (105), since in that case, (104) only obtains the average velocities.

7. NUMERICAL COMPARISON

In this section, we shall compare numerically the classical shallow water model without viscosity(104), which we shall name CM, with the new model proposed in Section 5 (97)–(102), designatedNM from now on. To carry out this comparison, we use the MacCormack scheme (see, for example,[12–14]).

If we try to approximate solution (105) for �i = 0 (i = 0, 1, . . . , k), that is, a solution withconstant velocity and constant depth, both models (CM and NM) compute it exactly.

To show how model (97)–(102) is able to capture exactly solution (105), let us consider a solutionto Euler equations with constant depth and fifth-degree polynomial (in z) horizontal velocity:

u = B + Cz + Dz2 + Ez3 + Fz4 + Gz5, w = 0

H = 0, s = h = A>0

p = ps + �(A − z)g

(106)

In this case, � =C + 2Dz + 3Ez2 + 4Fz3 + 5Gz4. For this computation, we use model CM thatwill provide the average horizontal velocity and model NM approximating � by polynomial ofdegree 0, 1, 2, 3 or 4, with the following values of the constants:

A= 1, B = 5, C = 1, D = 0.5, E = 0.1, F = − 0.2, G = 0.3

The depth is computed exactly by both models. We can see the errors obtained (infinity norm)for the horizontal velocity, considering different values of z, in Table I. We observe that the worst



Table I. Infinity norm errors of example (106) at different depths.

Depth (z) 0 0.25 0.5 0.75 1

CM 7.0× 10−1 4.2× 10−1 6.7× 10−2 3.8× 10−1 1.0× 100

NM (degree-0 �) 8.3× 10−2 2.7× 10−2 6.7× 10−2 1.3× 10−2 2.1× 10−1

NM (degree-1 �) 3.8× 10−2 1.2× 10−2 6.9× 10−3 2.6× 10−3 9.3× 10−2

NM (degree-2 �) 1.8× 10−2 5.0× 10−3 6.9× 10−3 4.4× 10−3 3.7× 10−2

NM (degree-3 �) 9.4× 10−3 2.9× 10−4 0 2.9× 10−4 9.4× 10−3

NM (degree-4 �) 0 0 8.9× 10−16 8.9× 10−16 8.9× 10−16

Table II. Infinity norm global errors of example (106).

CM 1.0× 100

NM (degree-0 �) 2.1× 10−1

NM (degree-1 �) 9.3× 10−2

NM (degree-2 �) 3.7× 10−2

NM (degree-3 �) 9.4× 10−3

NM (degree-4 �) 8.9× 10−16

solution is provided by model CM that just computes the average velocity. If model NM is used,we appreciate that, even in the simpler case in which vorticity is approximated by a constant, theerrors we get are smaller (except for z = h/2= 0.5, where, due to (95), the errors obtained withboth the models are the same). When vorticity is approximated by a polynomial of greater degreethe estimation of the velocity is improved at any depth. We notice that the biggest errors occur atz = 1. In case vorticity is estimated by a polynomial of degree 4, the model NM reaches the exactsolution at any depth (only errors due to numerical precision are present).

We can also compare the errors in the whole domain. Table II shows how the global errordecreases nearly five times when we change from model CM to model NM with � approximatedby a constant; the errors obtained are divided by more than 2 every time the degree of thepolynomial is increased in one unit and it is almost zero if the polynomial degree is 4.

Figure 2 illustrates the accuracy of the polynomial approximations provided by model NM, andFigure 3 is a zoom near z = 1, where the errors are bigger.

As last example, let us consider now the following solution to Euler equations with non-polynomial horizontal velocity:

u = B + CeDz, w = 0

H = 0, s = h = A>0

p = ps + �(A − z)g

(107)

In this case, the vorticity is �=CDeDz . Again, model CM just computes the average velocitywhile we can observe in Table III (for the following values of the constants: A= 1, B = 4, C = 1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5

5.2

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

x

exact uNM approx (degree 0)NM approx (degree 1)NM approx (degree 2)NM approx (degree 3)average u

Figure 2. Solution (106) and its approximations, z ∈ [0, 1].

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 16.25

6.3

6.35

6.4

6.45

6.5

6.55

6.6

6.65

6.7

6.75

x

exact uNM approx (degree 0)NM approx (degree 1)NM approx (degree 2)NM approx (degree 3)

Figure 3. Solution (106) and its approximations, z ∈ [0.9, 1].

and D = 0.5) how model NM is able to obtain an approximated solution as accurate as we wish,choosing the appropriate degree for the polynomial estimation of �. In this example, each time thedegree of � is increased in one unit, the error is reduced more than 10 times.



Table III. Infinity-norm global errors of example (107).

CM 3.5× 10−1

NM (degree-0 �) 3.0× 10−2

NM (degree-1 �) 3.5× 10−3

NM (degree-2 �) 1.8× 10−4

NM (degree-3 �) 1.1× 10−5

NM (degree-4 �) 3.9× 10−7

NM (degree-5 �) 1.6× 10−8

NM (degree-6 �) 4.5× 10−10

8. CONCLUSIONS

We have shown how asymptotic analysis allows us to obtain shallow water model (97)–(102) justassuming the usual ‘ansatz’ (20). We must emphasize that in this model the horizontal velocitydepends on depth if the vorticity is not zero. This fact represents an interesting novelty with respectto the other shallow water models that can be found in the literature.

We have compared the new model analytically and numerically with the classical model withoutviscosity, and we have observed that both models capture the exact solution to Euler equations incase the velocities are constant, but only new model (97)–(102) is able to compute solution (105)to Euler equations if � is not zero, since in this case the classical model only obtains the averagevelocity.

The comparisons with exact solutions to Euler equations (Section 7) prove that our model alwaysachieves better results (see Tables I–III) if the solution depends on z (or equal results in case thesolution is independent of z) than the classical shallow water model, whose errors can be very bigfor values of z away from the medium depth. We have approximated the vorticity by a polynomialin z and we have shown that the bigger the degree of this polynomial is, more accurate solutionswe obtain.

Other examples of application of asymptotic analysis to study shallow waters can be found in[15–18].

ACKNOWLEDGEMENT

This work has been partially supported by MTM2006-14491 project of Ministerio de Educaciony Ciencia of Spain.

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a new shallow water model with polynomial dependence on depth

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