QUARTERLY OF APPLIED MATHEMATICS
VOLUME , NUMBER 0
XXXX XXXX, PAGES 000–000
S 0033-569X(XX)0000-0
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND
MATHEMATICAL PROPERTIES
By
CARINE LUCAS
Laboratoire MAPMO, Université d’Orléans-UFR Sciences, Bât. de Mathématiques-Route de Chartres,
BP. 6759, 45067 Orléans cedex 2, France
Abstract. This paper presents a viscous Shallow Water type model with new Coriolis
terms, and some limits according to the values of the Rossby and Froude numbers. We
prove that the extension to the bidimensional case of the unidimensional results given
by [J.–F. Gerbeau, B. Perthame. Discrete Continuous Dynamical Systems, (2001)]
including the Coriolis force has to add new terms, omitted up to now, depending on the
latitude cosine, when the viscosity is assumed to be of order of the aspect ratio.
We show that the expressions for the waves are modified, particularly at the equator,
as well as the Quasi-Geostrophic and the Lake equations. To conclude, we also study the
mathematical properties of these equations.
Introduction. In many physical domains, we can come upon thin films such as gravity currents of water (rivers, opened or closed pipes), lava flows or blood in vessels. These
thin films are governed by Shallow Water (or Saint-Venant) type equations or Reynolds
type equations according to the flow velocity we consider.
This paper details the results announced in [6], where we only indicated the new models and the main existence result through some key points, but without the complete
derivations and proofs; we have also added here some numerical experiments which confirm what we said. More precisely, this article concerns a new viscous two dimensional
Shallow Water model derived from the three dimensional Navier-Stokes Equations. This
model is used to describe vertically averaged flows in three dimensional shallow domains
in terms of the horizontal mean velocity field and the depth variation, due to the free
surface. The derivation that uses asymptotic analysis is based on the work of GerbeauPerthame [4] in one dimension, which has been extended to two dimensions by Marche
[9]. The small parameter of this study is the aspect ratio, denoted by ε, which is the
ratio between the characteristic depth and the characteristic length of the domain. We
2000 Mathematics Subject Classification. Primary 76M45, 76U05; Secondary 35B40, 35Q35, 46E35.
Key words and phrases. Shallow-Water equations, viscosity, Coriolis force, asymptotics, waves, a priori
estimates, existence of solutions.
E-mail address: Carine.Lucas@univ-orleans.fr
c
XXXX
Brown University
1
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CARINE LUCAS
first consider the case of a viscosity of order of the aspect ratio. The new point in the
corresponding derivation is to remark that, if we want a viscous Shallow Water model
at order ε, we cannot take an approximation of the Coriolis force at the first order: the
second order produces new terms depending on the cosine of the latitude in the two
dimensional Shallow Water model. We also say a word about the case of a viscosity of
order one.
Then we look at some physical quantities and the consequences of this new cosine
term on the dispersion relation for Poincaré waves and on some equatorial waves. More
precisely, we give the equation satisfied by the water height for the linearized rotating
Shallow Water equations: it is a Klein-Gordon type equation, modified by the cosine
terms. We also study equatorial waves to have the largest influence of the new terms,
and we see that they are still trapped in the equatorial waveguide, but their speed has
to be changed.
Next we consider two models derived from the Shallow Water theory in which we add
our new Coriolis terms: the Quasi-Geostrophic model and the Lake equations. The first
one is the result of “filtering out” the fast waves from the rotating barotropic equations.
There is a formal analogy between barotropic Quasi-Geostrophic and incompressible flow:
in the theory of compressible fluid flows, the incompressible limit is obtained by “filtering
out” the “fast” acoustic waves and retaining only the “slow” vorticity modes associated
to convection by the fluid. This analogy originally inspired Charney in 1949 when he
first formulated the Quasi-Geostrophic equation and thus opened the modern area of
numerical weather prediction. To get these equations, we start with the Shallow Water
System, where the Rossby and the Froude numbers are supposed to be small, and ε is
fixed. We obtain two new terms that, in some physical cases, are not negligible (see [7]).
The Lake equations are obtained considering the Froude number as a small parameter
while the Rossby number and ε are fixed. These equations may be seen as Shallow Water
equations that describe the long-time motion of an incompressible fluid contained in a
shallow basin with a slowly spatially varying bottom, a free upper surface and vertical
side walls, under the influence of gravity and in the limit of small characteristic velocities
and very small surface amplitude.
The last part of this article is devoted to the mathematical properties of these models
with new Coriolis terms. The proofs are mainly based on works that have been done
without these new terms: the existence of weak solutions for Shallow Water equations
relies on new a priori estimates obtained thanks to a new entropy inequality. For the
Lake equations, if the water height does not vanish, the proof follows the approach
of Levermore, Oliver and Titi in constructing the solutions as the inviscid limit
of solutions to a system with artificial viscosity. The degenerate case appeals to the
associated Green function. Here, we just have to look at what we can say about our terms
to include them in these demonstrations and we get existence of global weak solutions
for the viscous two dimensional Shallow Water model. We also obtain existence of global
solutions for the Lake equations.
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES
3
1. A new viscous shallow water model. The two dimensional Shallow Water
system is obtained from three dimensional Navier-Stokes equations in a shallow domain.
We look for equations satisfied by the horizontal mean velocity field and the free surface.
We consider 3D Navier-Stokes equations (NSE) for a homogenous fluid:
−
→
∂t U + div(U ⊗ U ) = div σ − 2 Ω × U − f
div U = 0,
for (x, z) in T2 × [b(x), h(t, x)], where U = (u, w) ∈ R2 × R is the fluid velocity, σ =
−
→
−
→
−pId+2µD(U ) is the stress tensor, 2 Ω ×U is the Coriolis force with Ω = Ω(0, cos θ, sin θ),
θ represents the latitude and will be considered as a constant. Finally f = −g t (0, 0, 1)
is the gravity force.
We supplement these equations with the following boundary conditions (see Figure 1):
• at the free surface z = h(t, x):
We usually neglect atmospheric pressure and take
σn = 0,
where n is the unit normal to the surface. Sometimes we can add surface tension
effects, that is,
σn = a κ n,
where a is the capillarity coefficient, and κ is the mean curvature.
But in each case we know that the normal velocity in the referential linked to a
particle moving on the surface is zero:
∂t h + u · ∇x h = w.
• at the bottom z = b(x):
We have Navier type conditions
(σn) · τ1 τ1 + (σn) · τ2 τ2 = k U · τ1 τ1 + k U · τ2 τ2 ,
z
Lchar
n
Hchar
h(t, x)
H(t, x)
b(x)
x = (x1, x2)
Fig. 1. Notations used for the Shallow Water system
4
CARINE LUCAS
where k is the friction coefficient and (τ1 , τ2 ) is a basis of the tangential surface.
We add the non-penetration condition
−u · ∇x b + w = 0.
We can also study turbulent drag terms if we replace k U with kt H U |U | in the
previous expression (see [11]).
In what follows we write these equations in a non-dimensional form, replacing σ with
the expression of the usual Cauchy stress tensor. Next we make an asymptotic development of U and we study its first orders for a viscosity of order ε (geophysical flows).
We get Shallow Water equations by an integration on the water height. We also give the
Shallow Water system for thin films, when the viscosity is of order 1.
1.1. Dimensionless Navier-Stokes equations. We write the Navier-Stokes system and
the boundary conditions in a non-dimensionalized form, using some characteristic scales
specially chosen to get the Shallow Water model.
Let us start with the 3D Navier-Stokes equations for an homogenous fluid, with σ =
−pId + 2µD(U ), where µ represents the viscosity and D(U ) is the symmetric part of the
velocity gradient:
∂t u + u · ∇x u + w∂z u = −∇x p + 2µ divx (Dx (u)) + µ ∂z2 u + µ∇x (∂z w)
−2Ω sin θ u⊥ − 2Ω cos θwe1 ,
∂t w + u · ∇x w + w∂z w = −∂z p + µ ∂z (divx u) + µ ∆x w + 2µ ∂z2 w
divx u + ∂z w = 0,
+2Ω cos θ u1 − g,
where the subscript x denotes the horizontal variable, u1 and u2 are the components of
the vector u, u⊥ is the vector orthogonal to u defined by t (−u2 , u1 ) and e1 is the unit
vector (1, 0).
We introduce the following dimensionless variables and numbers:
x = Lchar x′ ,
z = Hchar z ′
u = uchar u′ ,
µ
,
ν=
Lchar uchar
Lchar ′
t=
t,
uchar
w = wchar w′
p = pchar p′
Ro =
uchar
,
2Lchar Ω
Hchar
≪ 1,
Lchar
with wchar = εuchar ,
with ε =
with pchar = u2char ,
uchar
Fr = √
,
gHchar
where ε is the aspect ratio, Ro the Rossby number, F r the Froude number.
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES
5
We drop the primes to rewrite the 3D NSE:
∂t u + u · ∇x u + w∂z u = −∇x p + 2ν divx (Dx (u)) +
ν 2
∂ u + ν∇x (∂z w)
ε2 z
sin θ ⊥
cos θ
u −ε
we1 ,
Ro
Ro
1
ν
2ν
∂t w + u · ∇x w + w∂z w = − 2 ∂z p + 2 ∂z (divx u) + ν ∆x w + 2 ∂z2 w
ε
ε
ε
1 cos θ
1
+
u1 − 2 2 ,
ε Ro
ε Fr
divx u + ∂z w = 0.
−
(1.1)
(1.2)
(1.3)
We also have to change the boundary conditions. First, we replace σ with its expression
and then change the variables:
• at the free surface z = h(t, x):
−∇x h
1
, and the curvature
The normal outward vector n is n = (1 + (∇x h)2 )− 2
1
is given by κ = div(−n).
The horizontal variable h is rescaled as h = Hchar h′ and consequently we have
−1
′
3
′
κ = ε L−1
char κ = εLchar ∆x′ h up to O(ε ). We also define the non-dimensional
−2
−1
capillarity coefficient A = ε a uchar Lchar to get the dimensionless conditions at
the free surface:
ν
∂z u = 0,
ε2
p + A κ − 2ν ∂z w + ε2 ν ∇x w · ∇x h + ν ∂z u · ∇x h = 0,
p ∇x h + A κ ∇x h − 2νDx (u)∇x h + ν ∇x w +
∂t h + u · ∇x h = w.
(1.4)
(1.5)
(1.6)
• at the bottom z = b(x):
If we write b = Hchar b′ , the non-penetration condition reads
−u · ∇x b + w = 0.
(1.7)
The Navier condition is a not so simple. We choose the tangential vectors
τ1 =
1
|∇x b|
⊥ 1
∇x b
−∇x b
and τ2 = p
2 .
0
|∇x b|2 + |∇x b|4 −|∇x b|
We define K = k u−1
char , Kt = kt Lchar and we get a complex expression we do
not detail here. We perform some approximations in the following and then give
a simplified equality obtained from this expression.
1.2. Hydrostatic approximation. We now use the hydrostatic approximation, that is,
we suppose ε to be small. We only keep the first two orders in Equation (1.2) and we
also drop terms in boundary conditions.
6
CARINE LUCAS
We are led to study the system:
∂t u + u · ∇x u + w∂z u = −∇x p + 2ν divx (Dx (u)) +
ν 2
∂ u + ν∇x (∂z w)
ε2 z
sin θ ⊥
cos θ
u −ε
we1 ,
Ro
Ro
cos θ
1
+ε
u1 ,
∂z p = ν∂z (divx u) + 2ν∂z2 w −
F r2
Ro
divx u + ∂z w = 0.
−
(1.8)
(1.9)
(1.10)
We can simplify the Navier condition
1
2ενDx (u) · ∇x b − εν
∂
u
+
∇
w
z
x
ε2
= −Ku − ε Kt H u|u| + 2ε ν ∂z w ∇x b + ε ν(∇x b · ∂z u)∇x b. (1.11)
At the free surface, Equation (1.6) is not modified but we can rewrite (1.4) as follows:
ν∂z u = −ε2 (p ∇x h + A κ ∇x h − 2νDx (u)∇x h + ν ∇x w)
for z = h(x, t),
and we replace it in Equation (1.5):
p + A κ − 2ν ∂z w
=
=
−ε2 ν ∇x w · ∇x h − ν ∂z u · ∇x h
ε2 (p ∇x h + A κ ∇x h − 2νDx (u)∇x h) for z = h(x, t).
As ε tends to zero, we get
p + A κ − 2ν ∂z w = 0
for z = h(x, t).
(1.12)
We integrate the vertical derivative of the pressure (given by Equation (1.9)) from h to z,
with z between b and h. The value of p at the free surface is given by (1.12) and with
the divergence free condition (1.10) we find the pressure at order ε:
Z
cos θ z
1
(h(t,
x)
−
z)
−
A
κ
−
ν
(div
u)
−
ν
div
u
+
ε
u1 . (1.13)
p(t, x, z) =
x
x
|
z=h(t,x)
F r2
Ro h
As we are looking for equations on the mean velocity and on the evolution of the free
surface, we first integrate the momentum equation (1.8) over the water height (between
z = b(x) and z = h(t, x)).
We apply Leibniz formula and get:
∂t
Z
b
h
u − ∂t h u|z=h + divx
Z
b
h
(u ⊗ u) − ((u · ∇x h) u)|z=h + ((u · ∇x b) u)|z=b
+ (u w)|z=h − (u w)|z=b + ∇x
+ 2ν divx
Z
b
−
h
Z
b
h
p = ∇x h p|z=h − ∇x b p|z=b
ν
∂z u|z=h
ε2
Z
Z
sin θ h ⊥
cos θ h
−
u −ε
we1 .
Ro b
Ro b
Dx (u) − 2ν Dx (u)|z=h ∇x h + 2ν Dx (u)|z=b ∇x b +
ν
∂z u|z=b + ν∇x w|z=h − ν∇x w|z=b
ε2
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES
7
Then, we use the boundary conditions (1.4), (1.6), (1.7), (1.11) to simplify the expressions
at the surface and at the bottom, and we get the integrated momentum equation:
∂t
Z
b
h
u + divx
Z
b
h
(u ⊗ u) + ∇x
Z
h
p
b
Z h
K
u + Kt H u|u|
+ 2ν divx
Dx (u)
= −A κ∇x h − ∇x b p|z=b −
ε
b
|z=b
Z
Z
sin θ h ⊥
cos θ h
+ ν∇x b (2∂z w + ∇x b · ∂z u)|z=b −
u −ε
we1 , (1.14)
Ro b
Ro b
with a new Coriolis term (the last one).
We also want the evolution of the free surface: we integrate the divergence free equation (1.10) from the bottom to the surface, and use Leibniz formula again and the conditions at the surface (1.6) and at the bottom (1.7) to find:
∂t h(t, x) + divx
Z
h(t,x)
u = 0.
(1.15)
b(x)
In the sequel we fix the value of ν, K and Kt with respect to ε and we study the
integrated momentum equation (1.14) and the free surface equation (1.15) when we
approximate u at the first order and at the second order.
1.3. Shallow Water system with a viscosity of order ε. We have already done the main
assumption to get the Shallow Water system, that is, the depth is small compared to the
length of the domain. Now, we make some hypotheses on the physical quantities.
In this part, we consider ν, A, K and Kt of order ε: ν = εν0 , A = εA0 , K = εK0 and
Kt = εK0t .
We develop u, w, H, p, b in powers of ε, that is, u = u0 + εu1 + ε2 u2 + . . . (and so
on) with H(t, x) = h(t, x) − b(x).
1.3.1. At the first order. We want an expression of the first order of the velocity u0 .
We use the horizontal momentum equation (1.8) and the boundary conditions (1.4) and
(1.11). We obtain
∂z2 u = O(ε),
(∂z u)|z=b = O(ε),
(∂z u)|z=h = O(ε),
which means that at the first order u does not depend on z: u0 (t, x, z) = u0 (t, x), still
unknown. As we are looking for the dynamics on u0 , we study the previous equations at
the first order. We can already rewrite the evolution equation (1.15) as
∂t H 0 + divx (H 0 u0 ) = 0.
Then we have the expression of p at the first order with (1.13):
p0 (t, x, z) = F r−2 (h − z).
(1.16)
8
CARINE LUCAS
We replace this value in the integrated momentum equation (1.14) and we find the
equality:
1
∇x (H 0 )2
2 F r2
1
sin θ 0 0 ⊥
= − 2 H 0 ∇x b0 − K0 u0 −
H u . (1.17)
Fr
Ro
Equations (1.16)–(1.17) form the Shallow Water system at the first order in non-dimensional variables.
If we go back to dimensional variables we have the Shallow Water system at the first
order for a viscosity of order ε:
∂t (H 0 u0 ) + divx (H 0 u0 ⊗ u0 ) +
∂t H + divx (Hu) = 0,
(1.18)
g
2
⊥
∂t (Hu) + divx (Hu ⊗ u) + ∇x H = −gH∇x b − ku − 2Ω sin θHu . (1.19)
2
These equations are the classical generalization to the 2D case of the ones given by
A. J.-C. de Saint-Venant in [12]. But there are no viscosity effects in this system:
following [4], we are led to study the equations we obtain at the second order to see the
viscous terms in our Shallow Water system.
1.3.2. At the second order. We study here the equations we can obtain at the second
order because we want a viscous system.
We will denote by a bold letter the approximation of the variables at order ε (for
example, u1 = u0 + εu1 ) and by a bar the mean value on the water height:
Z h
1
u dz.
ū :=
H(t, x) b
We rewrite the divergence condition at the second order
H 1 ū1 ) = O(ε2 ).
∂tH 1 + divx (H
(1.20)
As before, we rewrite the momentum equation (1.8) but a order ε, and, with the Shallow Water system at the first order (1.16)–(1.17), we find an expression of the second
derivative of u:
ν 2
K0
∂ u = − 0 u0 + O(ε).
ε2 z
H
We integrate this equality from b to z (for z between b(x) and h(t, x)) with the boundary
condition (1.11). We integrate it again from b to z to find an approximation of u at the
second order:
Z s − b0
K0 0 z
ds + O(ε2 )
u
1−
u = u1 |z=b + ε
ν0
H0
b
K0
z − b0
= u1 |z=b 1 + ε
(z − b0 ) 1 −
+ O(ε2 ).
ν0
2H 0
With this expression, we can compute the mean value of u:
K0 H 0
+ O(ε2 ).
ū = u1 |z=b 1 + ε
ν0 3
One can notice that the mean value of u2 is equal to the square of the mean value of u
up to O(ε2 ). We have the same property for u ⊗ u. This is used in the sequel.
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES
9
We also have the value of p at the second order with Equation (1.13) and we can
replace it in the integrated momentum equation (1.14). Using again the divergence free
condition (1.10) to express w0 as a function of h0 and u0 , we get
1
H 1 )2 − εA0 ∇x (H 0 ∆x h0 )
∇x (H
2F r2
cos θ
u1 |)|z=b
−2εν0 ∇x (H 0 divx u0 ) − ε
∇x u01 (H 0 )2 = −(K0u1 + εK0t H 0u1 |u
2Ro
1
cos θ 0 0
H
0
0
− εA0 ∆x h − 2εν0 divx u − ε
u H − εA0 ∆x h0 ∇x h0
−∇x b
F r2
Ro 1
+ 2εν0 divx (H 0 Dx u0 )
+εν0 ∇x b0 2∂z w0 + ∇x b0 · ∂z u0 |
H 1 ū ⊗ ū) +
H 1 ū) + divx (H
∂t (H
z=b
sin θ 1 ⊥
cos θ 0 2
cos θ 0
−
H ū + ε
(H ) e1 divx u0 − ε
H e1 ∇x b0 · u0 + O(ε2 ).
Ro
2Ro
Ro
The velocity at the bottom u|z=b is given by
−1
εK0 H 0
u1 |z=b = ū 1 +
+ O(ε2 ),
ν0 3
which leads to:
−1
εK0 H 1
1
1 2
H ) = −K0 ū 1 +
H ū ⊗ ū) +
H ū) + divx (H
∇x (H
∂t (H
2F r2
ν0 3
1 −2
εK0 H
−εK0tH 1 1 +
ū|ū| + εA0H 1 ∇x ∆xH 1
ν0 3
1
1
H 1 divx ū) + 2εν0 divx (H
H 1 Dx ū)
+εA0H 1 ∇x ∆x b + 2εν0 ∇x (H
(1.21)
cos θ 1 2
cos θ 1
cos θ
H 1 )2 + ε
H ) e1 divx ū − ε
∇x ū1 (H
(H
H e1 ∇x b · ū
+ε
2Ro
2Ro
Ro
1
sin θ 1 ⊥
cos θ
H
−
−ε
H ū − ∇x b
ū1H 1 + O(ε2 ).
2
Ro
Fr
Ro
Equations (1.20)–(1.21) form the Shallow Water system at the second order in nondimensional variables, with new cosine terms.
Finally, let us go back to the dimensional form to get the viscous Shallow Water system
at the second order:
∂t H + divx (Hu) = 0,
(1.22)
g
∂t (Hu) + divx (Hu ⊗ u) + ∇x H 2 = −α0 (H) u − α1 (H)H u |u|
2
+aH∇x ∆x H + 2µ∇x (Hdivx u) + 2µ divx (HDx u)
+Ω cos θ ∇x u1 H 2 + Ω cos θ H 2 e1 divx u − 2Ω sin θHu⊥
−2Ω cos θ He1 ∇x b · u + 2Ω cos θ u1 H ∇x b + aH∇x ∆x b − gH ∇x b,
where
α0 (H) =
k
,
1 + kH
3µ
α1 (H) = kt
1+
kH
3µ
2 .
(1.23)
10
CARINE LUCAS
We now have the viscous terms we could not find at the first order. We remark that
the cosine part of the Coriolis force must then be taken into account. We briefly consider
another physical case, where the viscosity is of order one.
1.4. If the viscosity is of order one. We have looked at a viscosity of order ε in the
previous part and found that going to the second order was necessary to see viscous
terms. Here, we keep all the parameters unchanged but we modify the viscosity.
If the viscosity is of order one, we find again:
∂z2 u(t, x, z) = O(ε),
∂z u(t, x, 0) = O(ε),
∂z u(t, x, h(t, x)) = O(ε)
that is, the main order of the velocity does not depend on z: u0 (t, x, z) = u0 (t, x, 0)
which is unknown.
As before, we get Equation (1.16) and p(0) .
The integration of the horizontal momentum equation at the first order leads to
g
∂t (Hu) + divx (Hu ⊗ u) + ∇x H 2 = −2Ω sin θHu⊥ + aH∇x ∆x b
2
−gH ∇x b − k u + aH∇x ∆x H + 2µ∇x (Hdivx u) + 2µ divx (HDx u).
Note that we have viscous terms in this equality, which is obtained at the first order;
there is no need for a correcting term on u at the second order to see viscosity effects.
We obtained Shallow Water models with viscous terms. When the viscosity is of order
one, the first order of the system is sufficient, and the Coriolis terms reduce to the usual
ones. But when the viscosity is of order ε, it is necessary to go at the second order, and
then we must take into account the cosine effect of Coriolis terms: this creates a new
model.
As our new terms are linked to the cosine of the latitude, the first idea is to study
this new model where the cosine plays a great role, that is, near the equator. Therefore,
in the following, we describe various waves and we emphasize equatorial waves obtained
when the tangential projection of the Coriolis force vanishes.
2. Waves for rotating shallow water equations. This study is based on [8],
where the results are obtained without the cosine terms. We consider the non-dimensional
Shallow Water equations at the second order (Equations (1.20)–(1.21)), without viscosity,
surface tension or turbulent drag.
This section is divided into three parts: first, we write the potential-vorticity conservation for the new rotating Shallow Water equations. Then we study the linearized
equations and give the Poincaré waves and their dispersion relation. The last results are
about equatorial waves: at the equator, the Coriolis force is not complete, the tangential
part vanishes. As a physical consequence of this simplification, we see that the tropics
behave as a waveguide, and we prove that the wave speed depends on the cosine term.
2.1. Potential vorticity conservation. The vorticity, defined by ω = curlu, satisfies the
following equation:
∂t ω + u · ∇ω + ωdivu
=−
cos θ
sin θ
divu + ε
curl (H∇u1 + 2u1 ∇h + He1 divu − 2e1 ∇b · u) .
Ro
2Ro
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES 11
The water height equation allows us to write:
curl (H∇u1 + 2u1 ∇h + He1 divu − 2e1 ∇b · u)
= ∂x2 H divu + u · ∇∂x2 H + 2∂x2 b divu + 2u · ∇∂x2 b + ∂t ∂x2 H,
and, replacing it in the whole equation, we find
cos θ
sin θ
−ε
∂x2 (H + 2b)
(∂t + u · ∇) ω +
Ro
2Ro
cos θ
sin θ
−ε
∂x2 (H + 2b) divu = 0.
+ ω+
Ro
2Ro
We multiply this expression by H and add the water height conservation multiplied by
(−2Ro ω − 2sin θ + ε cos θ∂x2 (H + 2b)) /(2Ro). We divide the result by H 2 and get
2Ro ω + 2 sin θ − ε cos θ∂x2 (H + 2b)
= 0.
(∂t + u · ∇)
2Ro H
This relation expresses the conservation of a new potential vorticity defined by
2b)) /(2Ro H).
(2Ro ω + 2 sin θ − ε cos θ∂x2 (H + 2b)
Compared with the usual definition (see [8] for example), we find a new term, the bold
one, depending on the cosine of the latitude.
2.2. Linear theory and Poincaré waves. In order to derive the linearized rotating Shallow Water equations, we consider the non-dimensional Shallow Water system at the second order, with a flat bottom. We look at small perturbations around the constant
background state (H = H c , u = 0) of the form:
H = H c + δpH p,
u = δ p up ,
with δ p ≪ 1,
and we keep the first order in δ p :
∂t H p + H c divup = 0,
cos θ c p
cos θ c
sin θ p⊥
1
∇H p = ε
H ∇u + ε
H e1 divup −
u .
∂t up +
2
Fr
2Ro
2Ro
Ro
(2.1)
(2.2)
From these equations, we can derive Poincaré waves, also called inertio-gravity waves.
We differentiate in time Equation (2.1), replace the term div∂t u using Equation (2.2),
and derive again in time. We get
∂t3 H p =
1
H c ∆(∂t H p )
F r2
− εH c 2
cos θ
sin θ
cos θ
∂t ∆up1 − εH c 2
∂x1 ∂t divup − H c
∂t curlup .
2Ro
2Ro
Ro
If we define ω = δ p ω p = δ p curlup , the potential vorticity conservation can be linearized
as
sin θ
cos θ
∂x ∂t H p = 0.
∂t ω p − c ∂t H p − ε
H Ro
2Ro 2
12
CARINE LUCAS
We also have from Equations (2.1) and (2.2) the equalities
1
∂x ∂t (∂t H p ),
Hc 1
∂t ∆up1 = ∂t (∂x1 divup − ∂x2 ω)
cos θ 2
sin θ
1
∂x2 (∂t H p ) − c ∂x2 (∂t H p ),
= − c ∂x1 ∂t (∂t H p ) − ε
H
2Ro
H Ro
∂x1 ∂t divup = −
which give
2
c cos θ
∂t − εH
∂x
(∂t H p )
2Ro 1
=
2 !
2
sin θ
cos
θ
1
p
c
c
∆(∂t H ) −
H + εH
(∂t H p ).
F r2
2Ro
Ro
We can also define a new function depending on new variables:
c cos θ
p
p
p
H (x1 , x2 , t) = G (y1 , y2 , s) = G x1 + εH
t, x2 , t .
2Ro
Then, the function Gp (y1 , y2 , s) = (εH c cos θ/(2Ro)∂y1 + ∂s ) Gp (y1 , y2 , s) satisfies the
Klein-Gordon Equation:
2 !
2
p
sin θ
1
2
2
c
c cos θ
2 p
Gp .
+
∂
∂
−
G
H
+
εH
∂s G =
y2
y1
F r2
2Ro
Ro
To get the dispersion relation, we write H p (x, t) = H0 exp i k1 x1 + k2 x2 − W̃ t
or
Gp (y, s) = G0 exp (i(K1 y1 + K2 y2 − W s)), with H0 = G0 , K1 = k1 , K2 = k2 and
W = W̃ + εH c cos θk1 /(2Ro).
We obtain:
v
u
2 !
2
u
cos
θ
sin θ
cos θ
1
c
t
c
c
W̃ (k) = −εH
|k|2 +
k1 ±
H + εH
2Ro
F r2
2Ro
Ro
1.0
1.0
1.0
1.0
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.0
0.0
0.0
0.0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−0.8
−1.0
0
−1.0
0
−1.0
0
5
10
15
5
10
15
5
10
15
−1.0
0
5
10
15
Fig. 2. Wave with (solid line) and without (dotted line) cosine effect
for a latitude equal to 30o , at times 10s, 40s, 70s and 100s.
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES 13
1.0
1.0
1.0
1.0
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.0
0.0
0.0
0.0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1.0
0
5
10
15
−1.0
0
5
10
15
−1.0
0
−0.8
5
10
15
−1.0
0
5
10
15
Fig. 3. Wave with (solid line) and without (dotted line) cosine effect
for a latitude equal to 45o , at times 10s, 40s, 70s and 100s.
or
v
u
u
W (K) = ±t
2 !
2
sin θ
cos θ
1
2
c
c
|k| +
H + εH
.
F r2
2Ro
Ro
If we want the dimensional results, we just have to replace εH c cos θ/(2Ro) by H c Ω cos θ,
sin θ/Ro by 2Ω sin θ and 1/F r2 by g. Then we can look at the difference due to the
cosine effect: we plot H p in the dimensional case, with k1 = 1, k2 = 0, H c = 1000 m,
Ω = 7 ∗ 10−5 s−1 , g = 9.81 ms−2 and at the latitude θ = π/6 or π/4, with and without
the new terms. We see, Figures 2 and 3, that the discrepancy between the two curves in
not negligible.
2.3. Equatorial waves. To see the effects of the cosine terms, we can study waves
where the latitude vanishes, near the equator. We start with the linearized equations to
obtain Klevin waves and mixed Rossby-gravity waves.
Kelvin waves are an important part of the observational record of the equatorial ocean.
We prove that they are trapped in the equatorial wave guide and that their speed depends
on the cosine term. We study flows which are parallel to the equator, that, is with u2 = 0,
and we use the β-plane approximation in the linearized equations (2.1)–(2.2) to get:
1
1
∂x2 H p + ε
H c ∂x2 up1 ,
2
Fr
2Ro
1
1
∂x1 H p = ε H c ∂x1 up1 ,
∂t up1 +
2
Fr
Ro
∂t H p + H c ∂x1 up1 = 0
βx2 up1 = −
We introduce the characteristic variables:
c2
q = c H p + up1 ,
H
r=
c1 p
H − up1 ,
Hc
which satisfy
∂q
∂q
=0
+ c1
∂t
∂x1
(eastward),
and
∂r
∂r
= 0 (westward),
− c2
∂t
∂x1
14
CARINE LUCAS
where the speeds are defined by
s
H c2
1
Hc
2
+
ε
−ε
H c,
c1 =
F r2
(2Ro)2
2Ro
c2 =
s
H c2
1
Hc
2
+
ε
+ε
H c.
F r2
(2Ro)2
2Ro
These two waves should be combined to find a general solution to our problem. But we
must seek their structure in the second space variable.
For the eastward group (r ≡ 0), if we write q = G(x2 )q(x1 −c1 t) we find, with previous
equations, an expression for G:
−β
G(x2 ) = G0 exp
x2 2 ,
c1 + c2
which means that these waves are trapped in the equatorial waveguide.
For the westward group (q ≡ 0), we find an unphysical solution.
Then we conclude that Kelvin waves only propagate in the eastward direction, with
the speed c1 , are non-dispersive and are trapped in the equatorial waveguide.
We can also look at mixed Rossby-gravity waves, or Yanai waves. We write up1 and
H p in terms of the characteristic variables q and r, and we modify the linearized Shallow
Water equations:
∂q
c1 + c2 ∂up2
∂q
+
− βx2 up2 = 0,
+ c1
∂t
∂x1
2
∂x2
∂r
∂r
c1 + c2 ∂up2
+
+ βx2 up2 = 0,
− c2
∂t
∂x1
2
∂x2
c1 − c2 ∂(c1 q − c2 r)
c1 c2 ∂(q + r) βx2 (c1 q − c2 r)
∂up2
+
+
= 0.
+
∂t
c1 + c2 ∂x2
c1 + c2
2(c1 + c2 )
∂x2
We have r ≡ 0 so the Rossby-gravity waves must satisfy
c1 + c2 ∂up2
+ βx2 up2 = 0,
2
∂x2
β
x2 and q of the form q =
which gives an expression for up2 : up2 = A(x1 , t) exp − c1 +c
2 2
β
2
B(x1 , t)x2 exp − c1 +c
, where A and B are linked as follows:
x
2 2
∂t B + c1 ∂x1 B = 2βA,
∂t A = −
c1
B.
2
We finally obtain an expression for the velocity field:


−β
2ωx2
2
exp
x
sin(kx
−
ωt)
−
1
2
c
c
+c
1
1
2
,
up = 
2
sin(kx
−
ωt)
x
exp c1−β
1
+c2 2
with the dispersion relation: ω(ω − kc1 ) = βc1 . These waves are dispersive, but, as
Kelvin waves, they are trapped in the equatorial waveguide.
We thus see that the new cosine terms give new velocities for equatorial waves that
depend on the aspect ratio.
Let us now seek some equations (Quasi-Geostrophic, Lake) that are usually derived
from the Shallow Water model. If we use the first order of the Shallow Water system,
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES 15
those equations remain unchanged. But we see in the following how the new terms modify
those equations when we consider the second order system.
3. The quasi-geostrophic limit. The Quasi-Geostrophic model is used for modeling and forecasting mid-latitude oceanic and atmospheric circulation. It is obtained from
the Shallow Water system if the Rossby and Froude numbers are small1.
We consider the dimensionless Shallow Water system (1.20)–(1.21). We write an
asymptotic development u = u0 + η u1 + . . . , H = 1 + η F H 1 + . . . where we suppose
Ro = η, F r2 = F η 2 , b = η b̃, with η ≪ 1, F = (2ΩLchar )2 /(gHchar ) and ε fixed.
We have to study the following equations:
∂t H + divx (Hu) = 0,
(3.1)
1
H∇x H =
∂t (Hu) + divx (Hu ⊗ u) +
F η2
sin θ
cos θ
cos θ
−
H u⊥ + ε
e1 H 2 divx u + ε
∇x (H 2 u1 )
η
2η
2η
(3.2)
−α̃0 (H) u + 2νdivx (HD(u)) + 2ν∇x (H divu) + A H∇x ∆x H
+A H∇x ∆x b −
cos θ
cos θ
1
h ∇x b − ε
He1 ∇x b · u + ε
u1 H ∇x b.
F η2
η
η
At the first and second orders, Equation (3.1) gives
divx u0 = 0
and
F ∂t H 1 + divx u1 + F ∇x H 1 · u0 = 0.
We also look at the first and second orders of the momentum equation (3.2). The first
order gives
cos θ
∇x b̃
⊥
1
∇x H + sin θ − ε
∂x2 u0 +
= 0.
2
F
Then we take the curl (i.e., −∂x2 of the first component + ∂x1 of the second one) of the
second order and get
(∂t + u0 · ∇x )(curl u0 ) = −α̃0 (1)curl u0 + ν∆(curl u0 )
+ sin θF (∂t H 1 + u0 · ∇x H 1 ) − sin θ F u0 · ∇x H 1
+εF
cos θ
1
∂x2 (∂t H 1 + u0 · ∇x H 1 ) − ∇⊥
x H · ∇x b̃
2
0
+ε cos θ ∂x2 (u0 · ∇x b̃) + ε cos θ ∇⊥
x u1 · ∇x b̃.
Let us remark that
sin θ u0 · ∇x H 1 − ε
cos θ
1 ∇x b̃
(∂x2 u0 ) · ∇x H 1 + ∇⊥
=0
xH ·
2
F
and
0
(∂x2 u0 ) · ∇x b̃ + ∇⊥
x u1 · ∇x b̃ = 0.
1Note that the Quasi-Geostrophic equations could also be obtained directly from the Navier-Stokes
equations, considering the Rossby number comparable to the aspect ratio. This work is still under
consideration.
16
CARINE LUCAS
1
We define ψ by u0 = ∇⊥
x ψ, which implies H = (sin θ − ε cos θ∂x2 /2) ψ − b̃/F , and we
find
2
2
2 cos θ
0
2
∂x2 ψ − sin2 θ F ψ
(∂t + u · ∇x )
∂x1 + 1 + ε F
4
cos θ
+ sin θ − ε
∂x2 b̃ = −α̃0 (1) ∆ψ + ν∆2 ψ. (3.3)
2
Equation (3.3) is the non-dimensional Quasi-Geostrophic equation obtained from the
viscous Shallow Water system at the second order with a variable topography. The new
terms due to the Coriolis force are ε2 F cos2 θ∂x22 ψ/4 and ε cos θ∂x2 b̃/2. The “unusual”
sin θ coefficient is linked to an “unusual” Rossby number expression; in the “usual” case,
the term sin θ is replaced by 1, and cos θ by 1/ tan θ. We could overcome this problem
writing the dimensional form of the Quasi-Geostrophic equation as in [7]. Note that
the numerical experiments presented in this paper show that the cosine effect has a
non-negligible influence for long times.
4. The Lake equations. We now consider other equations directly linked to the
Shallow Water system: the Lake equations, where the domain is a shallow basin with a
slowly spatially varying bottom. They are derived for a small Froude number.
To get the Lake equations we start with the second order Shallow Water system in
non-dimensional form and we suppose there are no friction and viscosity:
∂t H + div(Hu) = 0,
1
H∇H =
F r2
1
sin θ
cos θ
− 2 H ∇b −
H u⊥ + ε
∇ u1 H 2
Fr
Ro
2 Ro
cos θ 2
cos θ
cos θ
+ε
H e1 divu − ε
He1 ∇b · u + ε
u1 H ∇b.
2 Ro
Ro
Ro
∂t (Hu) + div(Hu ⊗ u) +
Let Ro and ε be fixed and F r2 = δ ≪ 1: we are led to study the following equations:
∂t H + div(Hu) = 0,
(4.1)
1
∂t (Hu) + div(Hu ⊗ u) + H∇H =
δ
1
sin θ
cos θ
− H ∇b −
(4.2)
H u⊥ + ε
∇ u1 H 2
δ
Ro
2 Ro
cos θ 2
cos θ
cos θ
+ε
H e1 divu − ε
He1 ∇b · u + ε
u1 H ∇b.
2 Ro
Ro
Ro
As we have done before, we develop the variables in powers of the small parameter δ:
u = u0 + δu1 . . . , H = H 0 + δH 1 . . . .
At the first order, the momentum equation (4.2) gives H 0 ∇(H 0 + b) = 0 so H 0 + b
does not depend on spatial variables: H 0 + b = f (t).
Then we look at the first order of Equation (4.1): f ′ (t) + div((f + b)u0 ) = 0 and we
integrate it in space considering periodic conditions. We get f ′ (t) = 0 and H 0 + b is a
constant (in space and in time); we can suppose its value is equal to 1.
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES 17
With this result, we can reformulate Equation (4.1):
div(H 0 u0 ) = 0.
(4.3)
At the second order, Equation (4.2) gives
H 0 ∂t u0 + H 0 (u0 · ∇)u0 + H 0 ∇H 1 = −
sin θ 0 0 ⊥
H u
Ro
cos θ 0
H H 0 ∇u01 + 2 u01 ∇H 0 + H 0 e1 divu0 − 2 e1 ∇b · u0 + 2 u01 ∇b .
2 Ro
Using the divergence relation (4.3), we remark that
+ε
1
1 0 0
H ∇u1 + u01 ∇H 0 + H 0 e1 divu0 − e1 ∇b · u0 + u01 ∇b
2
2
= (∂x2 H 0 + ∂x2 b +
H0
H0
⊥
⊥
∂x2 )u0 =
∂x2 u0 ,
2
2
and we get
∂t u0 + (u0 · ∇)u0 + ∇P = −
cos θ 0
sin θ
−ε
H ∂x2
Ro
2 Ro
⊥
u0 ,
P = H 1,
(4.4)
which forms, with Equation (4.3), the Lake equations.
Now we want to express these equations in a vorticity formulation. We first recall how
we get it in the “usual” case and then, we study our new term in cos θ.
The “usual” Lake equations are (4.3) and
⊥
∂t u0 + (u0 · ∇)u0 + ∇P + u0 = 0,
with div(H 0 u0 ) = 0,
P = H 1,
and we know how to write it in vorticity formulation. We take the curl of the first
equation and divide it by H 0 . We use the equality H 0 div u = −u · ∇H 0 and we get:
We define Π =
1
H 0 curlu
∂t
curlu curl(u · ∇u) u · ∇H 0
+
−
= 0.
H0
H0
H 02
+
1
H0
and we rewrite the system:
∂t Π + u · ∇Π = 0,
1
u = 0 ∇⊥ ψ.
H
In our case, to get the vorticity formulation of the Lake equations (4.3)–(4.4), we have
⊥
to look at the new term H 0 ∂x2 u0 .
With the divergence free condition, we find
1
⊥
curl(H 0 ∂x2 u0 )
H0
=
=
∇H 0 · ∂x2 u
+ div(∂x2 u2 )
H0
∇H 0 · ∂x2 u
u · ∇H 0
.
−
∂
x2
H0
H0
Then, we write the relation
1
⊥
curl(H 0 ∂x2 u0 ) = −u · ∂x2
H0
∇H 0
H0
= −u · ∇
∂x2 H 0
H0
,
18
CARINE LUCAS
and we find a new system:
∂t Π + u · ∇Π = 0,
1
u = 0 ∇⊥ ψ,
H
1
sin θ 1
cos θ ∂x2 H 0
where Π = 0 curl u +
+ε
,
0
H
Ro H
2 Ro H 0
Π|t=0 = Π0 ,
which is the vorticity formulation of the Lake equations obtained from our new Shallow
Water system.
5. If the latitude is not constant. At the beginning, we supposed the latitude to
be constant. But we can do exactly the same development if the latitude is not constant.
The only change is that we must keep the gradient of cos θ u1 H 2 . We find the viscous
Shallow Water system
∂t H + divx (Hu) = 0,
g
∂t (Hu) + divx (Hu ⊗ u) + ∇x H 2 = −α0 (H) u − α1 (H) H u |u|
2
+aH∇x ∆x H + 2µ∇x (Hdivx u) + 2µ divx (HDx u)
+Ω ∇x cos θ u1 H 2 + Ω cos θ H 2 e1 divx u − 2Ω sin θHu⊥
(5.1)
(5.2)
−2Ω cos θ He1 ∇x b · u + 2Ω cos θ u1 H ∇x b + aH∇x ∆x b − gH ∇x b.
We can also deduce from this system the Quasi-Geostrophic equation for a varying latitude. At the beginning, in dimensional variables, we replace 2Ω sin θ by 2Ω sin θ0 + βx2
and 2Ω cos θ by 2Ω cos θ0 − β tan θ0 x2 (β-plane approximation, see [11]). We define a
new non-dimensional parameter β ′ = βL2char /Uchar , and we find the non-dimensional
Quasi-Geostrophic equation for a variable latitude:
2
2
2 cos θ0
0
2
∂x2 ψ − sin2 θ0 F ψ
(∂t + u · ∇x )
∂x1 + 1 + ε F
4
cos θ0
+ sin θ0 − ε
∂x2 b̃ + βx2 = −α̃0 (1) ∆ψ + ν∆2 ψ. (5.3)
2
6. Mathematical properties of these models. In this part, we study mathematical properties of the previous models. Our goal is to establish the existence of global
weak solutions for the Shallow Water system and for the Lake equations.
6.1. Existence of global solutions for shallow water equations. The study of the Shallow Water model has been done without the new Coriolis terms by F. Marche and
P. Fabrie in [10] using Gronwall’s lemma. We remark that the cosine effect involves
derivatives (of order one in space) of the height and the velocity, and no control is a
priori available on them. Then we are led to use an appropriate interpolation lemma on
these terms in order to be able to apply Gronwall’s lemma in the same way as [10].
The idea of the demonstration is first to get estimates thanks to the classical energy
inequality and to a BD entropy type inequality. If we build sequences of approximate
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES 19
regular enough solutions as in [2], then we pass to the limit in the approximate problem
and get the existence result.
Remark 6.1. An important property to be noted is that our new Coriolis terms do not
contribute in the energy estimates. Consequently, results based on this type of estimates
remain unchanged.
6.1.1. Theorem. We consider the following Shallow Water system in a bounded domain D with periodic boundary conditions:
∂t H + div(Hu) = 0,
1
∇H 2 = −α̃0 (H)u − α̃1 (H)Hu|u|
2F r2
cos θ
+AH∇∆H + AH∇∆b + ν div(HD(u)) + ε
∇ u1 H 2
2Ro
cos θ 2
cos θ
sin θ
+ε
H e1 divu − ε
He1 ∇b · u −
Hu⊥
2Ro
Ro
Ro
cos θ
H
−ε
u1 H .
−∇b
F r2
Ro
∂t (Hu) + div(Hu ⊗ u) +
(6.1)
We suppose the bottom to be regular enough (b ∈ H 3 (D)) and the initial condition
√
1
on the water height to satisfy: H(t = 0) = H0 ≥ 0, H0 ∈ Hper
(D), H0 ∈ L2per (D),
−1
2
1
1
H0 u0 ∈ Lper (D), ln K0 H0 α̃0 (H0 ) ∈ Lper (D). We prove the theorem:
Theorem 6.2. If A > 0, α̃0 (H) > 0 and α̃1 (H) ≥ 0, with H0 satisfying the above
conditions, then there exists a global weak solution of (6.1).
6.1.2. A priori estimates.
Energy Inequality. The energy inequality associated to System (6.1) is classically given
by
1 d
2 dt
Z
Z
H2
2
2
2
α̃1 (H) H|u|3
α̃0 (H)|u| +
+ H|u| + A|∇H| +
F r2
D
D
D
Z
Z
ν
∇b
t
2
+
· u.
H A∇∆b −
H|∇u + ∇u| ≤
4 D
F r2
D
Z In order to get better estimates, we write the BD entropy of our system.
Entropy Inequality.
(6.2)
20
CARINE LUCAS
Proposition 6.3. The entropy inequality for System (6.1) reads:
1 d
2 dt
Z
ν
H2
2
+
+
A|∇H|
H|u + ν∇ ln H|2 +
H|∇u − t ∇u|2
F r2
4 D
D
Z
Z
d
−νK0
ln K0−1 H α̃0 (H) + ν
α̃′0 (H)∇H · u
dt D
D
Z
Z
Z
|∇H|2
|∇2 H|2 + ν
α̃1 (H)|u|u · ∇H + νA
+ν
2
D Fr
D
D
Z
Z
sin θ
cos θ
+ν
u⊥ · ∇H − νε
e1 · ∇H Hdiv u
Ro D
2Ro D
Z
Z
cos θ
cos θ
−νε
∇(H 2 u · e1 ) · ∇ ln H + νε
∇b · u e1 · ∇H
2Ro D
Ro D
Z
Z
Z
cos θ
u · e1 ∇b · ∇H +
α̃0 (H)|u|2 +
α̃1 (H) H|u|3
−νε
Ro D
D
D
Z ∇b
A∇∆b −
≤
· (Hu + ν∇H).
F r2
D
Z (6.3)
The proof relies on the following lemma:
Lemma 6.4. We have the two following equalities:
1 d
2 dt
Z
2
H|∇ln H| +
D
Z
D
H∇div u · ∇ ln H +
Z
D
H∇u : ∇ ln H ⊗ ∇ ln H = 0
(6.4)
and
1 2d
ν
2 dt
Z
Z
α̃1 (H)|u|u · ∇H
α̃0 (H)u · ∇ ln H + ν
H|∇ ln H|2 + ν
D
D
D
Z
Z
Z
|∇H|2
sin θ
|∇2 H|2 + ν
+νA
+
ν
u⊥ · ∇H
2
F
r
Ro
D
D
Z
Z D
cos θ
cos θ
−νε
e1 · ∇H Hdiv u − νε
∇(H 2 u · e1 ) · ∇ ln H
2Ro D
2Ro D
Z
Z
Z
d
cos θ
= −ν
u · ∇H + ν
H∇u : t ∇u − νε
∇b · u e1 · ∇H
dt D
Ro D
D
Z
Z
∇b
cos θ
A∇∆b −
· ∇H.
u · e1 ∇b · ∇H + ν
+νε
Ro D
F r2
D
Z
Remark 6.5. Our new terms only give us four terms in the second relation.
Proof. For the first equality, we refer to [1].
(6.5)
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES 21
For the second one, we multiply the momentum equation by ν∇ ln H
∇H ⊗ ∇H
(∂t u + (u · ∇)u) · ∇H + ν
ν
D(u) : ∇∇H −
H
D
D
Z
Z
Z
α̃1 (H)|u|u · ∇H
α̃0 (H)u · ∇ ln H + ν
|∇2 H|2 + ν
+νA
D
D
D
Z
Z
Z
sin θ
cos θ
|∇H|2
⊥
+
ν
u
·
∇H
−
νε
e1 · ∇H Hdiv u
+ν
2
Ro D
2Ro D
D Fr
Z
Z
cos θ
cos θ
−νε
∇(H 2 u · e1 ) · ∇ ln H + νε
∇b · u e1 · ∇H
2Ro D
Ro D
Z Z
∇b
cos θ
A∇∆b −
−νε
· ∇H.
u · e1 ∇b · ∇H = ν
Ro D
F r2
D
Z
2
Z
We simplify this expression using
Z
D
H∇u : ∇ ln H ⊗ ∇ ln H =
Z
D
D(u) : ∇∇H +
Z
D
Z
D(u) :
D
∇H ⊗ ∇H
,
H
∇div u · ∇H = 0,
and we add Equation (6.4) multiplied by ν 2 . We obtain:
Z
|∇H|2
|∇2 H|2
+ νA
H|∇ ln H| + ν
2
D
D Fr
D
Z
Z
Z
sin θ
α̃1 (H)|u|u · ∇H + ν
α̃0 (H)u · ∇ ln H + ν
+ν
u⊥ · ∇H
Ro D
D
D
Z
Z
cos θ
cos θ
−νε
e1 · ∇H Hdiv u − νε
∇(H 2 u · e1 ) · ∇ ln H
2Ro D
2Ro D
Z
Z
cos θ
cos θ
+νε
∇b · u e1 · ∇H + νε
u · e1 ∇b · ∇H = I,
Ro D
Ro D
1 2d
ν
2 dt
Z
2
Z
where
Z ∇b
· ∇H
(∂t u + (u · ∇)u) · ∇H + ν
A∇∆b −
F r2
D
D
Z Z
Z
∇b
d
t
A∇∆b −
· ∇H.
H∇u : ∇u + ν
u · ∇H + ν
I = −ν
dt D
F r2
D
D
I = −ν
Z
22
CARINE LUCAS
Proof of Proposition 6.3. We write Equation (6.5) as:
1 d
2 dt
Z
α̃0 (H)u · ∇ ln H
H|u + ν∇ ln H|2 + ν
D
D
Z
Z
Z
|∇H|2
+ν
α̃1 (H)|u|u · ∇H + νA
|∇2 H|2 + ν
2
D
D
D Fr
Z
Z
sin θ
cos θ
+ν
u⊥ · ∇H − νε
e1 · ∇H Hdiv u
Ro D
2Ro D
Z
Z
cos θ
cos θ
∇(H 2 u · e1 ) · ∇ ln H + νε
∇b · u e1 · ∇H
−νε
2Ro D
Ro D
Z
cos θ
−νε
u · e1 ∇b · ∇H
Ro D
Z Z
Z
∇b
1 d
A∇∆b −
=
· ∇H,
H∇u : t ∇u + ν
H|u|2 + ν
2 dt D
F r2
D
D
Z
and we add Equality (6.2) to get the result .
A priori Estimates. We consider (Hk )k and (uk )k two sequences of weak solutions of (6.1)
and satisfying (6.2) and (6.3). The energy inequality (6.2) classically gives the following
estimates:
1
• (Hk )k in L∞ (0, T ; Hper
(D)),
√
∞
Hk uk k in L (0, T ; (L2per (D))2 ),
•
√
•
Hk (∇uk + t ∇uk ) k in L2 (0, T ; (L2per (D))4 ),
• α̃0 (Hk )1/2 uk k in L2 (0, T ; (L2per (D))2 ),
• α̃1 (Hk )1/3 Hk 1/3 uk in L3 (0, T ; (L3per (D))2 ).
k
We do not detail the proof here. We have more information thanks to the entropy
inequality (6.3):
−1,q
• (∂t Hk )k in L∞ (0, T ; Wper
(D)), ∀q < 2,
2
2
• (uk )k in L (0, T ; (Lper (D))2 ),
1/3
• hk uk in L3 (0, T ; (Lsper (D))2 ), ∀s < 3,
k
1
(α̃0 (Hk ))k in L∞ (0, T ; Hper
(D)),
∞
1
(α̃1 (Hk ))k in L (0, T ; Hper (D))),
√ ∇ Hk k in L∞ (0, T ; (L2per (D))2 ),
2
(Hk )k in L2 (0, T ; Hper
(D)),
√
t
Hk (∇uk − ∇uk ) k in L2 (0, T ; (L2per (D))4 ) and consequently, with the results
√
Hk ∇uk k in L2 (0, T ; (L2per (D))4 ),
from the energy,
1,1
• (Hk uk )k in L2 (0, T ; (Wper
(D))2 ).
•
•
•
•
•
Proof. For the first five estimates, we refer to [10]. The last ones are based on the
entropy inequality (6.3), but are more intricate than without the cosine terms: we cannot
√
directly write that ( Hk ∇uk )k is bounded in L2 (0, T ; (L2per (D))4 ). We are led to combine
terms on the right-hand side with terms on the left-hand side.
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES 23
From the results on Hk and uk , we know that (Hk uk )k is in L2 (0, T ; (Lpper (D))2 ) for
all p < 2. We can also write:
Z tZ
ν
α̃
(H
)u|u|
·
∇H
1
k
k
0
D
p
p
≤ Ck Hk uk k2L2 (0,T ;(L2 (D))2 ) + δ1 k Hk ∇uk k2L2 (0,T ;(L2 (D))4 ) .
So for small enough δ1 and for ν1 > 0, we can prove as in [10] that the following inequality
holds for all p < 2:
Z tZ
Z
Z tZ
2
1
|∇2 Hk |2 + ν1
Hk |uk + ν∇ ln Hk |2 + νA
Hk ∇u − t ∇u
2 D
0
D
0
D
Z Z
Z tZ
cos
θ
sin θ t
uk ⊥ · ∇Hk − νε
e1 · ∇Hk Hk div uk
+ν
Ro 0 D
2Ro 0 D
Z Z
Z Z
cos θ t
cos θ t
−νε
∇(Hk 2 uk · e1 ) · ∇ ln Hk + νε
∇b · uk e1 · ∇Hk
2Ro 0 D
Ro 0 D
Z Z
cos θ t
∇b uk · e1 ∇b · ∇Hk ≤ C1 + C2 A∇∆b −
−νε
Ro 0 D
F r2 (L2 (D))2
kHk uk kL2 (0,T ;(Lp (D))2 ) + k∇Hk kL2 (0,T ;(L2 (D))2 ) .
Then weZhave
Coriolis terms:
Z to study the
t
• u⊥
k · ∇Hk ≤ kuk kL2 (0,T ;(L2 (D))2 ) k∇Hk kL∞ (0,T ;(L2 (D))2 ) ,
0
D
Z t Z
• e1 · ∇Hk Hk div uk 0
D
Z t
p
p
≤
k∇Hk k(L2 (D))2 k Hk kL∞ (D) k Hk div uk kL2 (D)
0 Z
Z t p
t
p
1
≤
k Hk div uk k2L2 (D) .
k∇Hk k2(L2 (D))2 k Hk k2L∞ (D) + δ2
4δ2 0
0
For small enough δ2 , the last term can be absorbed into the left-hand part. Let
us look Rat the other term: with the estimates from the energy, we just have to
t √
control 0 k Hk k2L∞ (D) . For all δ > 0, we can write:
p
k Hk k2L∞ (D) = kHk kL∞ (D) ≤ kHk kH 1+δ (D) .
p
Thanks to Sobolev embedding, we get: k Hk k2L∞ (D) ≤ k∇Hk kL2/(1−δ) (D) . Then
we have the following inequalities:
p
δ
k Hk k2L∞ (D) ≤ k∇Hk kL2/1−δ (D) ≤ Cδ k∇Hk k1−δ
L2 (D) k∇Hk kH 1 (D)
(2−2δ)/(2−δ)
≤ Cδ′ k∇Hk kL2 (D)
+ δ3 k∇Hk k2L2 (D) + δ3 k∆Hk k2L2 (D) .
So when we integrate in time, for small enough δ3 , the last term can be taken
up
Z tby
Z the left-hand expression.
• ∇(Hk 2 uk · e1 ) · ∇ ln Hk 0
D
Z t Z
2
= 2uk · e1 (∇Hk ) + Hk ∇Hk · ∇(uk · e1 ).
0
D
The second part can be controlled exactly as the previous term. For the first
24
CARINE LUCAS
one, we write:
Z
uk · e1 (∇Hk )2 ≤ kuk kL2 (D) k∇Hk k2L4 (D) .
D
As before, we know that k∇Hk k2L4 (D) ≤ Ck∇Hk kL2 (D) k∇Hk kH 1 (D) .
Then we have:
Z t Z
2 2uk · e1 (∇Hk ) ≤ C ′ k∇Hk k2L∞ (0,T ;L2 (D)) kuk k2L2 (0,T ;L2 (D))
0
D
+ δ4 k∇Hk k2L2 (0,T ;L2 (D)) + δ4 k∆Hk k2L2 (0,T ;L2 (D)) ,
and
term can be moved to the left.
Z Zfor a small δ4 , the last
t
• ∇b · uk e1 · ∇Hk 0
D
≤ k∇bk(L∞ (D))2 kuk kL2 (0,T ;(L2 (D))2 ) kHk kL∞ (0,T ;H 1 (D)) ,
Z t Z
• uk · e1 ∇b · ∇Hk 0
D
≤ k∇bk(L∞ (D))2 kuk kL2 (0,T ;(L2 (D))2 ) kHk kL∞ (0,T ;H 1 (D)) .
√
Gathering all these relations, we get three new estimates on (∇ Hk )k , (Hk )k and on
√
( Hk ∇uk )k .
√
√
Hk ∇uk + uk · ∇Hk .
The last one is finally obtained writing ∇(Hk uk ) as Hk
A Better Property for Hk uk . In order to pass to the limit, we need better estimates on
the product Hk uk . We have:
Proposition 6.6. There exists a positive constant C such that
1/3
kτη (Hk uk ) − Hk uk kL∞ (0,T −η;(Wper
1,∞
(D))′ ≤ Cη
where τη , for η > 0 is the translation operator τη v(t, x) = v(t + η, x).
If for 1 ≤ q ≤ +∞ and 0 < ζ < 1, for a Banach space E we denote by Nqζ (0, T ; E) the
Nikolskii space defined as:
o
n
Nqζ (0, T ; E) = f ∈ Lq (0, T ; E) s. t. ∃C > 0 : kτη f − f kL∞ (0,T −η;E) ≤ Cη ζ ,
1/3
1,∞
it gives that (Hk uk )k is bounded in N∞ (0, T ; Wper
).
Proof. We multiply System (6.1) by a test function ψ and we integrate over D and
from t to t + η. We find (see [10]):
Z
Z t+η
(τη (Hk uk )(t) − Hk uk (t)) ψ ≤
g(s)ds
kψkW 1,∞ (D)
D
t
Z
Z
Z
Z
ε cos θ t+η
sin θ t+η
⊥ 2
H
u
ψ
H
divu
e
ψ
+
+
k k k 1 k
Ro t
2Ro
D
t
D
Z
Z
Z
Z
ε cos θ t+η
ε cos θ t+η
+
Hk ∇b · uk e1 ψ +
Hk (uk )1 ∇bψ Ro t
Ro
D
t
D
Z
Z
ε cos θ t+η
Hk2 (uk )1 ∇ψ ,
+
Ro t
D
with g bounded in L3/2 (0, T ). Let us consider the new terms:
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES 25
•
•
•
•
•
Z
Hk u ⊥
k ψ ≤ kHk uk kL1 (D) kψkL∞ ,
ZD
p
p
Hk2 divuk e1 ψ ≤ kHk kL6 (D) k Hk kL6 (D) k Hk divuk kL2 (D) kψkL6 (D) ,
ZD
Hk ∇b · uk e1 ψ ≤ kHk uk kL1 (D) k∇bkL∞ (D) kψkL∞ (D) ,
ZD
Hk (uk )1 ∇bψ ≤ kHk uk kL1 (D) k∇bkL∞ (D) kψkL∞ (D) ,
ZD
p
p
Hk2 (uk )1 ∇ψ ≤ k Hk uk kL2 (D) k Hk kL8 (D) kHk kL8 (D) k∇ψkL4 (D) .
D
Then we can conclude that there exists g̃ bounded in L3/2 (D) such that:
Z
Z t+η
(τη (Hk uk )(t) − Hk uk (t)) ψ ≤
g̃(s)ds
kψkW 1,∞ (D) ,
D
t
which gives
1/3
kg̃kL3/2 (0,T ) .
kτη (Hk uk )(t) − Hk uk (t)k(Wper
1,∞
(D)′ ) ≤ η
6.1.3. Convergence and compactness. We are now able to pass to the limit in the
different terms. In [10], they proved that all the terms, except the ones linked to Coriolis,
converge to the expected limit. Here, we study the other ones.
Properties on Hk and uk . First, we have uk ⇀ u in L2 (0, T ; (L2per (D))2 ).
About the sequence (Hk )k , as we proved the inequalities kHk kL∞ (0,T ;H 1 (D)) ≤ C and
k∂t Hk kL∞ (0,T ;H −2 (D)) ≤ C, thanks to the compactness results [13] we find Hk → H in
C 0 (0, T ; H s (D)), for all s < 1.
Finally on the product Hk uk ,using both Proposition 6.6 and Sobolev properties, we
get Hk uk → Hu in L2 (0, T ; (L2per (D))2 ).
Study of the New Terms. For the usual Coriolis term, we can write, thanks to the above
2
⊥
2
2
results, Hk u⊥
k → Hu in L (0, T ; (Lper (D)) ).
The terms with the bottom are also quickly treated:
Hk ∇b · uk e1 → H∇b · ue1 in L2 (0, T ; (L2per (D))2 ),
and Hk (uk )1 ∇b → Hu1 ∇b in L2 (0, T ; (L2per (D))2 ).
The last two terms read:
(uk )1 Hk2 → u1 H 2 in L1 (0, T ; (L1 (D))2 ),
√
√
and, as Hk divuk ⇀ Hdivu in L2 (0, T ; L2(D)) and Hk → H in C 0 (0, T ; H s (D)), for
all s < 1, we get
Hk2 divuk e1 → H 2 divue1 in L1 (0, T ; L1(D)).
6.1.4. End of the proof. We have proved that from a sequence verifying Equations
(6.2) and (6.3) we can extract a subsequence that strongly converges to a weak solution
of (6.1). As in [2], we can construct sequences of approximate smooth enough solutions
of (6.1) using a regularization process. Considering such sequences, we classically get the
existence result.
26
CARINE LUCAS
6.2. Existence of a solution for the Lake equations. Just as in the Shallow Water
system case, the existence of a solution for the Lake equations has been proved in [5] in
the non-degenerate case, and in [3] for the degenerate case, without the term in cos θ.
We see here how to adapt these proofs to our case.
6.2.1. If H 0 does not vanish. We rewrite the vorticity formulation as
∂t Π + u · ∇Π = 0,
u = K̃Π,
Π|t=0 = Π0 .
The operator K̃ is a linear operator such that u satisfies div(h0 u) = 0. It can be seen as
a perturbation of the inverse of the operator curl.
In [5], C. D. Levermore, M. Oliver and E. S. Titi consider the following system:
∂t Π + u · ∇Π = 0,
u = KΠ,
(6.6)
Π|t=0 = Π0 ,
in a domain D, where K is a linear operator such that we have the condition div(h0 u) = 0.
In our case, the operator K̃ can be deduced from the operator K because K̃Π is the
sum of KΠ and terms that do not depend on the unknowns. Then the estimate on K is
modified in
kK̃ΠkW 1,p ≤ cpkΠkLp + F,
where F only depends on θ, ε and H 0 .
We can do the same reasoning: we first consider the problem with artificial viscosity
∂t Πν + u · ∇Πν − νH −1 ∇ · (H∇Πν ) = 0.
If the initial value of Πν is in H01 (D), we can prove there exists a unique solution Πν in
C 0 ([0, +∞[ , H01 (D)) ∩ L2loc ([0, +∞[ , H 2 (D) ∩ H01 (D)) for each ν. This result is obtained
by construction of a family of approximate solutions (solutions in the subset spanned by
the first n eigenvectors) and we pass to the limit thanks to compactness.
Then we study the inviscid limit: we have compactness of the sequence Πν which
enables us to pass to the limit and claim that if the initial value is in L∞ (D), there exists
Π in C 0 ([0, +∞[ , w⋆ − L∞ ([0, +∞[ × D) ∩ L∞ ([0, +∞[ × D)).
In the study of the uniqueness, the new Coriolis terms do not appear because we compute
the difference between two solutions.
We can thus conclude that, if the initial vorticity Π0 is in L∞ (D), we have existence
and uniqueness of global solutions to the Lake equations.
6.2.2. If H 0 vanishes. This part is directly linked to the estimates proved by Bresch
and Métivier in [3] for the system
div(H 0 u) = 0,
curl u = f,
p
0
2
where f is in L (D) and H u is in L (D).
So with the same arguments we can deduce estimates on the solution.
COSINE EFFECT ON SHALLOW WATER EQUATIONS AND MATHEMATICAL PROPERTIES 27
About the existence of global solutions, the use of an artificial viscosity can also be
adapted: we define Hν0 = H 0 + ν and we change the Lake equations for ∂t (Hν0 Πν ) +
Hν0 uν · ∇Πν − νdiv(Hν0 ∇Πν ) = 0. But the new terms of our model (compared to [3])
only add the x2 derivative of Hν0 ; so we pass to the limit in the same way.
The proof of uniqueness is based on Yudovitch’s work: let u1 and u2 be two solutions
of the Lake equations
cos θ
sin θ
−ε
H∂x2 u⊥
∂t u1/2 + u1/2 · ∇u1/2 + ∇P1/2 = −
1/2 .
Ro
2 Ro
The function u defined by u = u1 − u2 is solution of
cos θ
sin θ
−ε
H∂x2 u⊥ .
∂t u + u · ∇u1 + u2 · ∇u + ∇P = −
Ro
2 Ro
We multiply by Hu and we get:
Z
Z
1 d
H|u|2 + u · ∇u1 Hu = 0
2 dt
As we have
Z
2
√
√
u · ∇u1 Hu ≤ k Huk p ∞ (k Huk2 2 )1− p1 k∇u1 kLp
L
L
2
√
√
1
≤ k HukLp ∞ (k Huk2L2 )1− p (pC + F ),
we get the following inequality
√
1
d √
k Huk2L2 ≤ (k Huk2L2 )1− p (pC + F )M
dt
which gives u = 0.
So we conclude with the theorem:
Theorem 6.7. For all initial data Π0 ∈ L∞ (D) there exists a unique weak solution (u, Π)
for the Lake Equations, with Π belonging to C0 ([0, T ]; L∞(D) weak ⋆), verifying Π/H 0
in L∞ ([0, T ] × D) and H 0 u in C 0 [0, T ]; L2(D) .
Furthermore, this
solution is regular, that is, Π/h0 is in C 0 ([0, T ]; Lr (D)) and u in
C 0 [0, T ]; W 1,r (D) for all finite r.
Acknowledgments. The author would like to thank D. Bresch for fruitful discussions.
References
[1] D. Bresch and B. Desjardins. Existence of global weak solutions for a 2D viscous shallow water
equations and convergence to the quasi-geostrophique model, Comm. Math. Phys., 238 1-2 (2003),
211–223.
[2] D. Bresch and B. Desjardins. On the construction of approximate solutions for the 2D viscous
Shallow Water model and for compressible Navier-Stokes models, J. Math. Pure Appl., 86 (2006),
362–368.
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