Diploma report Implementation and verification of a simple ... - LPAS

Diploma report
Implementation and verification of a simple
non-hydrostatic atmospheric finite volume model
Catherine Pomezny
Supervisors:
Oliver Fuhrer, Alain Clappier
Air and Soil Pollution Laboratory (lpas)
October 2005 to February 2006

Contents
1 Introduction 4
2 Formulation 4
2.1 Continuous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Hyperbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Derivation of Euler equations . . . . . . . . . . . . . . . . . . . . . . . 5
3 Implementation 8
3.1 Finite volume discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Stability and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Numerical fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.1 Flux Averaging Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.2 Central Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.3 Lax-Friedrichs Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.4 Richtmeyer Two-Step Lax-Wendroff Method . . . . . . . . . . . . . . 14
3.4 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4.1 Forward Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4.2 Leapfrog Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4.3 Third-order Runge-Kutta Scheme . . . . . . . . . . . . . . . . . . . . . 15
3.5 Dimensional splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.6 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.6.1 Gas dynamics (c = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.6.2 Unstratified atmosphere (g = 0) . . . . . . . . . . . . . . . . . . . . . 17
3.6.3 Stratified atmosphere (N = const) . . . . . . . . . . . . . . . . . . . . 17
3.7 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.7.1 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 19
3.7.2 Solid wall boundary conditions . . . . . . . . . . . . . . . . . . . . . . 19
3.7.3 Solid wall with hydrostatic pressure . . . . . . . . . . . . . . . . . . . 20
3.7.4 Solid wall with background . . . . . . . . . . . . . . . . . . . . . . . . 20
3.7.5 Prolongation of hydrostatic balance . . . . . . . . . . . . . . . . . . . 20
3.8 Damping terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.8.1 Time filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.8.2 Spatial filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Results 23
4.1 Comparison of different methods . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Hydrostatic balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.4 Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.5 Buoyant bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 Conclusion 38
2

6 Acknowledgements 38
A Derivation of the differential form from the integral form 39
3

1 Introduction
Numerical models of the atmosphere often solve the Euler equations on grids which conform
to the geometry of the boundary, leading to irregular-shaped grid cells. Even though facilitating
the formulation of the lower boundary conditions, this can lead to a myriad of numerical
truncation errors related to the grid deformation, especially over steep and complex topography.
Another approach is to keep the grid Cartesian, but implement a cut-cell approach for
the lower boundary conditions. To this end, a straightforward two-dimensional finite volume
discretization of the fully non-hydrostatic Euler equations on a Cartesian grid was developped.
In the first part of this report, discretization and implementation are described. In
the second part, sensitivity to numerical filtering, source term formulation and time-stepping
method is discussed. Model verification is presented using hydrostatic equilibrium, sound
wave and buoyant bubble simulations.
2 Formulation
2.1 Continuous equations
The Euler equations describe the flow phenomena of compressible inviscid gas dynamics.
They model the conservation of mass, momentum and energy. It is a system of nonlinear
hyperbolic equations. The nonlinearity of these equations leads to some computational
challenges. And the use of a method which is able to treat strong gradients is required.
2.1.1 Hyperbolic systems
Hyperbolic systems of partial differential equations can be used to model a large variety of
phenomena in different disciplines. An important class of homogeneous hyperbolic equations
are the conservation laws, since many of the partial differential equations arising in fluid
dynamics can be expressed as a system of conservation laws. In two dimensions, a first-order
system of conservation laws has the form
q t + f(q) x + g(q) z = 0, (1)
where q = q(x, z, t) : R × R × R → R m is a m-components vector representing the unknown
functions of conserved quantities we want to determine and f(q) and g(q) are flux functions
in the x- and z-directions. This equation states that the rate of change of q at each point
is determined by the fluxes at that point. Equation (1) can be rewritten in the quasilinear
form
q t + f ′ (q)q x + g ′ (q)q z = 0, (2)
where the Jacobian matrices f ′ (q) and g ′ (q) are constant m × m real matrices. This suggests
that the equation is hyperbolic if the matrix ⃗n · ⃗f ′ (q) = n x f ′ (q) + n z g ′ (q) is diagonalizable
with real eigenvalues and m corresponding linearly independant eigenvectors, for any
arbitrary direction ⃗n = (n x , n z ). This means that any vector in R m can be uniquely decomposed
as a linear combination of these eigenvectors and this provides the decomposition into
distinct waves. The corresponding eigenvalues give the speeds at which each wave propagates.
4

In practice, however, many physical problems give rise to nonlinear conservation laws
where f(q) and g(q) are not linear functions of q. In this case, the solution deforms as it
evolves and, in particular, shock waves can form, across which the solution is discontinuous.
If a function contains a discontinuity, it cannot be the solution to a partial differential equation
in the classical sense, because derivatives are not defined at the discontinuity. Instead,
the solution is required to satisfy the corresponding, more fundamental, integral equation.
For any arbitrary spatial domain Ω, the integral formulation of a conservation law has
the form
∫∫
∫
d
q(x, z, t)dxdz = − ⃗n ·
dt
⃗f(q)ds, (3)
Ω
where ∂Ω denotes the boundary of the domain, ⃗ f(q) = (f(q), g(q)) is the flux vector and
⃗n(s) = (n x (s), n z (s)) the outer unit normal vector to ∂Ω at point (x(s), z(s)) on ∂Ω. If each
component of q represents the density of a conserved quantity, then equation (3) states that
the total mass of this quantity within the domain Ω can change only due to the flux across
the boundary ∂Ω of the domain.
In the case where the domain is rectangular, Ω = [x 1 , x 2 ] × [z 1 , z 2 ], the normal vector
always points in either the x- or z-direction. The formula (3) simplifies and the normal flux
is given by f(q) along the left and right edges and by g(q) along the top and bottom :
d
dt
∫∫
Ω
q(x, z, t)dxdz =
∫ z2
z
∫ 1
x2
x 1
∂Ω
f(q(x 1 , z, t))dz −
g(q(x, z 1 , t))dx −
∫ z2
z 1
f(q(x 2 , z, t))dz +
∫ x2
x 1
g(q(x, z 2 , t))dx. (4)
In general, the derivation of conservation laws from physical principles gives first an integral
form and the differential equation is derived from this by imposing smoothness assumptions
on the solution (see appendix A). An important point is that, unlike the differential form,
the integral form can be satisfied by piecewise-continuous functions.
2.1.2 Derivation of Euler equations
Euler equations consist in a system of conservation laws that model the conservation of mass,
momentum and energy and where the laws of physics determine the flux functions. In this
section, these equations will be derived in two dimensions.
Consider a fluid in two dimensions. The description of mass flow requires two quantities:
the density ρ(x, z, t) and the velocity vector ⃗u = (u(x, z, t), w(x, z, t)) at any point (x, z)
and time t. Then the total mass of fluid in an area [x 1 , x 2 ] × [z 1 , z 2 ] of the region occupied
by the fluid at time t is given by the integral of the density
∫ z2
∫ x2
z 1
x 1
ρ(x, z, t)dxdz. (5)
If there is no creation or destruction of fluid quantities within this area, then the total mass
can change only due to the fluxes, or flow of particles, through the boundary of the area. We
5

have
∫
d z2
∫ x2
ρ(x, z, t)dxdz = F 1 (z, t) − F 2 (z, t) + G 1 (x, t) − G 2 (x, t), (6)
dt z 1 x 1
where F i (z, t) are the flux functions at the lateral boundaries x i , i = 1, 2, and G i (x, t) the
fluxes at the top and bottom boundaries z i , i = 1, 2. Note that F 1 (z, t) corresponds to the
flux to the right and −F 2 (z, t) represents the flux to the left.
Equation (6) represents the integral from of a conservation law for the mass and the
flux functions are to be related to ρ(x, z, t) in order to obtain an equation that might be
solvable for ρ. The mass flux through boundary x i is associated with the mass transported
with velocity u through x i so that F is given by ρu. Similarly, G is given by ρw. The integral
form of the conservation law becomes then
d
dt
∫ z2
∫ x2
z 1
x 1
ρ(x, z, t)dxdz = −ρu∣ x 2
− ρw∣ z 2
. (7)
x 1 z 1
Assuming that ρ, u and w are smooth functions, the integral form of the conservation law can
be transformed into a partial differential equation (see appendix A). The differential form of
the conservation law for ρ is then
ρ t + (ρu) x + (ρw) z = 0. (8)
This equation called the continuity equation models the conservation of mass.
In addition to ρ(x, z, t), the velocity components u(x, z, t) and w(x, z, t) are also to be determined
and two equations are needed for this. The velocities themselves are not conserved
quantities, but the x-momentum ρu and the z-momentum ρw are. Each of these momenta
are transported with the flow and, by analogy with the density flux in (8), this will give
contributions of ((ρu)u) x + ((ρu)w) z and ((ρw)u) x + ((ρw)w) z to the x- and z-momentum
respectively. Beside this, there are additional microscopic terms for the momentum fluxes
that are due to the pressure in the fluid. These microscopic fluxes are caused by the fact that
all nearby molecules are not moving at the same speed in a gas. Pressure variation in the
x-direction, measured by p x appears in the equation for ρu, while p z appears in the equation
for ρw.
In the vertical direction, there are external gravity forces acting on the gas that accelerate
downward individual molecules. These forces must be integrated over the domain
and this contribution added to the integral conservation law for z-momentum (if z is aligned
with the gravitational acceleration). Thus the differential form of the z-momentum equation
will contain a source term of s = −ρ∇Φ, where Φ = gz is the gravitational potential.
The differential forms of the conservation laws for momentum are then
(ρu) t + (ρu 2 + p) x + (ρuw) z = 0, (9)
(ρw) t + (ρwu) x + (ρw 2 + p) z = −ρ∇Φ. (10)
6

In developing the conservation laws for momentum a new variable, the pressure, has been
introduced. A fourth differential equation is therefore needed and a fourth conserved quantity
is the energy. The total density of energy, denoted ρe, consists of two terms
ρe = ρe int + 1 2 ρ(u2 + w 2 ), (11)
where ρe int is the internal energy of the molecular motion and 1 2 ρ(u2 + w 2 ) is the kinetic
energy of the macroscopic motion of the fluid. Moreover, the total energy ρe will contain a
contribution due to the potential so that
ρe → ρe + ρΦ. (12)
The total energy is advected with the flow, leading to macroscopic contributions to the
energy flux of (ρe)u in x-direction and (ρe)w in z-direction. In addition, the microscopic momentum
fluxes lead to fluxes of energy due to work done by pressure of pu and pw respectively.
The differential form of the conservation law for total energy is then
(ρe + ρΦ) t + [(ρe + ρΦ + p)u] x + [(ρe + ρΦ + p)w] z = 0. (13)
Taking out the terms due to the contribution of the potential energy yields
(ρe) t + [(ρe + p)u] x + [(ρe + p)w] z + (ρΦu) x + (ρΦw) z = −(ρΦ) t (14)
Finally, putting equations (8), (9), (10) and (14) together gives the system of Euler
equations in two dimensions
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
ρ ρu
ρw 0 0 0 0
⎢ ρu
⎥
⎣ρw⎦+
⎢ ρu 2 + p
⎥
⎣ ρuw ⎦ + ⎢ ρuw
⎥
⎣ ρw 2 + p ⎦ + ⎢ 0
⎥
⎣ 0 ⎦ + ⎢ 0
⎥
⎣ 0 ⎦ = ⎢ 0
⎥
⎣−ρ∇Φ⎦ + ⎢ 0
⎥
⎣ 0 ⎦ . (15)
ρe (ρe + p)u (ρe + p)w ρΦu ρΦw 0 −∂ t (ρΦ)
t
This can be written in the form
x
z
x
q t + f(q) x + g(q) z + f corr (q) x + g corr (q) z = s(q) + r(q), (16)
with q the vector of conserved quantities ,f(q) and g(q) the flux functions and s(q) the source
term due to gravity. f corr (q) and g corr (q) are the correction fluxes and r(q) is the correction
term that are due to the contribution of the potential energy in the energy term.
⎡ ⎤
ρ
q = ⎢ ρu
⎥
⎣ρw⎦ ,
ρe
⎡
f(q)= ⎢
⎣
f corr (q)= ⎢
⎣
ρu
ρu 2 + p
ρuw
(ρe + p)u
⎡
0
0
0
ρΦu
⎤
⎥
⎦ ,
⎤
⎥
⎦ ,
⎡
g(q)= ⎢
⎣
gcorr (q)=
7
z
ρw
ρuw
ρw 2 + p
(ρe + p)w
⎡
⎢
⎣
0
0
0
ρΦw
⎤ ⎡ ⎤
0
⎥
⎦ , s(q)= ⎢ 0
⎥
⎣−ρ∇Φ⎦
0
⎤ ⎡ ⎤
0
⎥
⎦ , r(q)= ⎢ 0
⎣ 0
−∂ t (ρΦ)
⎥
⎦ . (17)

To close the system of equations, the pressure p must be specified as a function of ρ,
ρu, ρw and ρe in order that the fluxes are functions of the conserved quantities alone. This
additional equation is called the equation of state and depends on physical properties of the
gas. For a polytropic ideal gas :
ρe int =
p
γ − 1 , (18)
where γ is the ratio of specific heats C p /C v . The equation for the pressure is then
p = (ρe − 1 2 ρ(u2 + w 2 ))(γ − 1). (19)
The form (15) is the differential form of the Euler equations, which holds in the usual sense
only where the solutions are smooth. But the Euler equations are clearly nonlinear, since
products of the unknowns appear. In this case, discontinuities can develop spontaneously in
the flow quantities.
As seen in the previous section, at a discontinuity in q, the partial differential equations
(1) do not hold in the classical sense, while the more fundamental integral conservation
laws (3) do still make sense. For this reason, classical finite difference discretization methods,
in which derivatives are approximated by finite differences, can be expected to break down
near discontinuities in the solution. To compute solutions containing discontinuities, finite
volume discretization methods, which are based on the integral formulation instead of the
differential equation, are much more effective.
3 Implementation
3.1 Finite volume discretization
In finite volume methods, the spatial domain is subdivided into grid cells (finite volumes)
and cell averages of the solution q are approximated over each of these volumes. Considering
a Cartesian grid, the regular grid cell C ij is given by C ij = [x i−1/2 , x i+1/2 ] × [z j−1/2 , z j+1/2 ].
The numerical approximation to q over the (i, j) grid cell at time t is
Q ij (t) ≈ 1
∆x∆z
∫∫
C ij
q(x, z, t)dxdz, (20)
where ∆x = x i+1/2 − x i−1/2 and ∆z = z j+1/2 − z j−1/2 are the length of the cell in the x- and
z-directions.
Approximating the integral form of the conservation law (4) over the volume of each
grid cell yields
∫∫
∫
d
zj+1/2
∫ zj+1/2
q(x, z, t)dxdz = f(q(x
dt
i−1/2 , z, t))dz − f(q(x i+1/2 , z, t))dz +
C ij z j−1/2
z j−1/2
∫ xi+1/2
x i−1/2
g(q(x, z j−1/2 , t))dx −
8
∫ xi+1/2
x i−1/2
g(q(x, z j+1/2 , t))dx, (21)

and this expression can be used to obtain a numerical method.
A possible approach would be to develop a fully discrete method, discretized in both
space and time. An other possibility, which will be done here, is to perform the discretization
process in two stages, first discretizing only in space, leaving the problem continuous in time.
This leads to a system of ordinary differential equations (ODEs) in time. The advantage of
this semidiscrete method is that the system of ODEs can then be solved by any ODE method.
Rearranging expression (21) and dividing by the cell area ∆x∆z gives
∫∫
1 d
q(x, z, t)dxdz =
∆x∆z dt C ij
[ ∫
− 1 zj+1/2
∫ zj+1/2
f(q(x
∆x∆z
i+1/2 , z, t))dz − f(q(x i−1/2 , z, t))dz
z j−1/2
z j−1/2
− 1
∆x∆z
[ ∫ xi+1/2
x i−1/2
∫ xi+1/2
g(q(x, z j+1/2 , t))dx − g(q(x, z j−1/2 , t))dx
x i−1/2
]
]
. (22)
Equation (22) is then used to derive a system of ordinary differential equations for the evolution
of cell averages Q ij (t)
d
dt Q ij(t) = − 1
∆x (F i+1/2,j − F i−1/2,j ) − 1
∆z (G i,j+1/2 − G i,j−1/2 ), (23)
where F i−1/2,j = F i−1/2,j (Q(t)) is some approximation to the flux along the interface x =
x i−1/2 and G i,j−1/2 = G i,j−1/2 (Q(t)) an approximation along the interface z = z j−1/2 :
F i−1/2,j (Q(t)) ≈ 1
∆z
∫ zj+1/2
z j−1/2
f(q(x i−1/2 , z, t))dz,
G i,j−1/2 (Q(t)) ≈ 1 ∫ xi+1/2
g(q(x, z
∆x
j−1/2 , t))dx.
x i−1/2
(24)
Equation (23) represents a standard finite volume method capable of treating discontinuities.
The values Q ij are advanced in time using approximations to the fluxes through the edges of
the grid cells. (This is illustrated in Figure 1.)
Finite volume methods are appropriate for problems where some quantity should be
conserved because they guarantee exact conservation numerically. Therefore they are said
to be in conservative form. Summing ∆x∆z d dt Q ij(t) from (23) over all of the cells in the
domain, i.e. from cell I 1 to cell I 2 in x-direction and from cell J 1 to cell J 2 in z-direction,
yields
∆x∆z
∑
I 2 ∑
i=I 1
J 2
j=J 1
d
dt Q ij(t) = −∆z
−∆x
9
∑J 2
j=J 1
(F I2 +1/2,j − F I1 −1/2,j)
I 2 ∑
i=I 1
(G i,J2 +1/2 − G i,J1 −1/2), (25)

Figure 1: Illustration of a finite volume method where the average value Q ij is updated by
the fluxes at the cell edges.
which is a discrete approximation to the integral equation (22). The sum of the flux differences
cancels out except for the fluxes at the extreme edges x I1 −1/2, x I2 +1/2, z J1 −1/2 and
z J2 +1/2. Therefore there will be exact conservation of the total quantity within the computational
domain and this quantity will vary only due to the fluxes at the boundary.
In order to obtain a fully discrete numerical method the second step is to discretize (23)
in time. Some techniques of time integration will be discussed in section 3.4. One possibility
of discretization is given here as an example and a fully discrete method is derived.
Let ∆t = t n+1 − t n be the length of a time step, and let Q n ij represent the fully discrete
approximation to Q ij (t n ). Then the time discretization of (23) is done by simply taking the
difference between the values Q n+1
ij
and Q n ij and dividing by ∆t. The numerical method thus
obtained has the form
Q n+1
ij
= Q n ij − ∆t
∆x (F i+1/2,j n − F i−1/2,j n ) − ∆t
∆z (Gn i,j+1/2 − Gn i,j−1/2
), (26)
where the numerical fluxes Fi−1/2,j n and Gn i,j−1/2
are some approximations to the time averages
of the fluxes along the cell interfaces over the time interval [t n , t n+1 ]:
Fi−1/2,j n ≈ 1 ∫ tn+1
∫ zj+1/2
f(q(x
∆t∆z
i−1/2 , z, t))dzdt,
t n z j−1/2
G n i,j−1/2 ≈ 1 ∫ tn+1
∫ xi+1/2
g(q(x, z
∆t∆x
j−1/2 , t))dxdt.
t n x i−1/2
(27)
If these average fluxes can be computed based on the values Q n , then we will have a fully
discrete method.
For hyperbolic problems, information propagates with finite speed, so F n
i−1/2,j and Gn i,j−1/2
can be assumed to be based only on the cell averages on both sides of the interfaces x i−1/2
10

and z j−1/2 respectively. They can be expressed as
F n
i−1/2,j = F(Qn i−1j , Qn ij ),
G n i,j−1/2 = G(Qn ij−1 , Qn ij ). (28)
where F and G are some numerical flux functions. It follows that, in (26), Q n+1
ij
will depend
only on the five values Q n i−1j , Qn ij−1 , Qn ij , Qn i+1,j and Qn i,j+1 at the previous time level.
The methods thus obtained are explicit methods with a three-point stencil in both x- and
z-directions.
3.2 Stability and convergence
In finite volume methods, one difficulty is to determine good numerical flux functions F
that approximate the correct fluxes reasonably well. One essential requirement is that the
numerical solution should converge to the true solution of the differential equation as the grid
is refined, i.e as ∆x, ∆z and ∆t go to zero. This demands two conditions for the numerical
method: consistency and stability.
3.2.1 Consistency
Consistency for a numerical method means that the method should approximate the differential
equation well locally. For a finite volume method, the numerical flux functions F and G
are expected to reduce to the true fluxes f and g for the case of constant flow. If q(x, z, t) ≡ q,
then q does not change with time or height and the first integral (27) simply reduces to f(q).
On the other hand, as Q n i−1j = Qn ij = q, the numerical flux function F of (28) is expected to
reduce to
F(q, q) = f(q) (29)
for any value of q. Moreover, continuity in this function is also generally required as Q i−1j
and Q ij vary, so that F(Q i−1j , Q ij ) → f(q) as Q i−1j , Q ij → q.
3.2.2 Stability
Stability for a numerical method means that the small errors made in each time step do
not grow too fast in later time steps. A necessary condition for stability and hence convergence
that must be satisfied by any finite volume method is that the numerical domain of
dependence of the method contains the true domain of dependence of the partial differential
equation, at least in the limit as ∆x, ∆z and ∆t go to zero. This condition is known as the
CFL condition after Courant, Friedrichs and Lewy who first recognized it. In one dimension,
the CFL condition can be interpreted intuitively as requiring that the distance v∆t traveled
by a wave in one time step not exceed one spatial step ∆x.
To illustrate this, consider the case of advection with the constant velocity u > 0 in
x-direction. In this way, the exact solution simply translates at speed u and travels a distance
u∆t in one time step. In one dimension, the last term in the method (26) drops. Thus
the value Q n+1
i
depends only on the three values Q n i−1 , Qn i and Q n i+1 at the previous time
step and the computation of the numerical flux at x i−1/2 is assumed to be based on Q n i−1
11

and Q n i alone. This is effectively true if the time step is small enough so that information
propagates less than one grid cell in a single time step, and this is the situation shown in
figure 2(a). In figure 2(b), a larger time step is used so that u∆t is bigger than ∆x. In this
case, the true flux clearly depends on Q n i−2 . For this reason, the one-dimensional version of
method (26) would most probably be unstable when applied with such a large time step, if
the numerical flux was computed with Q n i−1 and Qn i only.
Figure 2: Characteristics for the advection equation with u > 0, showing the information
that flows into cell C i during a single time step. (a) For a small enough time step, the flux
at x i−1/2 depends only on Q n i−1 . (b) For a larger time step, the flux should also depend on
Q n i−2 .
Still in one dimension, the quantity
cn = v max∆t
∆x , (30)
where v max is the maximum characteric speed occuring in the solution, is called the Courant
number for a particular calculation. Thus, a necessary condition for stability of three-point
stencil methods is that the Courant number be no greater than 1. For methods with fivepoint
stencils in which Q n+1
i
depends also on Q n i−2 and Qn i+2 , the CFL condition would give
cn 2.
In two dimensions, the shortest spatial step is in the diagonal of the cell and it is given
by 1 2√
∆x 2 + ∆z 2 . The courant number can then be expressed as
cn =
2v max∆t
√
∆x 2 + ∆z 2 . (31)
For the method (26), the condition cn 1 should be satisfied. Again this is only a necessary
condition for stability and it is not always sufficient to guarantee stability.
Rigorous stability analysis is often only possible for linear equations. In the case of
nonlinear equation, the CFL condition may however provide a good indication for the size of
the temporal and spatial steps to use.
12

3.3 Numerical fluxes
There are different ways in which the numerical flux functions of (28) can be defined, leading
to different finite volume methods. Some possibilities of fluxes that fulfil the consistency
requirement are given in this chapter.
3.3.1 Flux Averaging Scheme
The average flux Fi−1/2,j n at x i−1/2 was assumed to be based only on the values Q n i−1j and
Q n ij . The simplest way to compute the numerical flux functions is to take the arithmetic
average of the flux evaluations at Q n i−1j and Qn ij
F n
i−1/2,j = F(Qn i−1j, Q n ij) = 1 2 [f(Qn i−1j) + f(Q n ij)]. (32)
Taking the corresponding definition for the flux functions G and replacing in (26) yields the
following method
Q n+1
ij
= Q n ij − ∆t
2∆x [f(Qn i+1j) − f(Q n i−1j)] − ∆t
2∆z [g(Qn ij+1) − g(Q n ij−1)]. (33)
However, for hyperbolic problems this method is generally unstable, even when the CFL
condition is satisfied.
3.3.2 Central Difference Scheme
In the Central Difference scheme, the average of the cell values Q n i−1j and Qn ij at the left- and
right-hand side of the interface x i−1/2 is first computed. Then the flux at x i−1/2 is evaluated
for this value
Fi−1/2,j n = F(Qn i−1j, Q n ij) = f( 1 2 (Qn i−1j + Q n ij)). (34)
This scheme has the advantage of being second-order accurate and, when used with an
appropriate time integration scheme, it generally leads to methods that are stable (if the
CFL condition is respected).
3.3.3 Lax-Friedrichs Method
The numerical flux in the classical Lax-Friedrichs method is defined as
F n
i−1/2,j = F(Qn i−1j, Q n ij) = 1 2 [f(Qn i−1j) + f(Q n ij)] − ∆x
2∆t (Qn ij − Q n i−1j). (35)
Replacing in (26), with an analogous definition for G, gives the general form of the Lax-
Friedrichs method
Q n+1
ij
= Q n ij +1 2 (Qn i−1j − 2Q n ij + Q n i+1j) − ∆t
2∆x [f(Qn i+1j) − f(Q n i−1j)]
+ 1 2 (Qn ij−1 − 2Q n ij + Q n ij+1) − ∆t
2∆z [g(Qn ij+1) − g(Q n ij−1)]. (36)
The Lax-Friedrichs flux (35) is similar to the Averaging flux (32) but with an additionnal
diffusion term which has the effect of damping the instabilities arising in (33). Therefore, the
Lax-Friedrichs method is generally stable provided the CFL condition is satisfied.
13

3.3.4 Richtmeyer Two-Step Lax-Wendroff Method
The Richtmeyer Two-Step Lax-Wendroff method is of second order. This is achieved by using
a better approximation to the average flux (27). In this method a half step in time is first
made using the Lax-Friedrichs method at the cell interfaces (∆x, ∆z and ∆t are replaced
by 1/2∆x, 1/2∆z and 1/2∆t). The numerical approximation of q at the midpoint in time
t n+1/2 = t n + 1 2∆t is given by
and the flux is evaluated at this point
Q n+1/2
i−1/2j = 1 2 (Qn i−1j + Q n ij) − ∆t
2∆x [f(Qn ij) − f(Q n i−1j)], (37)
F n
i−1/2,j = f(Qn+1/2 i−1/2j ) (38)
Although more accurate than the Lax-Friedrichs method, this method often leads to oscillations
in solutions.
3.4 Time integration
In section 3.1, a system of ordinary differential equations (23) for the evolution of cell averages
Q ij (t) in time was derived. Our model solves the partial differential equation (16) that
contains source and correction terms in addition to the flux terms. This is done using a
splitting technique, first solving
and then making the post update
q t + ˜f(q) x + ˜g(q) z = s(q), (39)
q t = r(q). (40)
The flux terms in (39) are tilded as, in addition to the regular fluxes, they contain the flux
correction terms due to the potential energy.
Performing a finite volume discretization for (39) yields the following system of ODEs
dQ
dt = RHS x(Q) + RHS z (Q) + S(Q), (41)
where RHS x (Q) and RHS z (Q) consist in the x- and z-flux terms and S(Q) is the source term.
For simplicity, these three terms can be replaced by F(Q) giving
dQ
dt
= F(Q). (42)
The discretization in time can then be done by using any standard method for systems of
ODEs. Some time-discretization methods will be presented in this section.
14

3.4.1 Forward Scheme
The forward scheme is the most natural time-discretization method. It is a single-stage twotime-level
scheme as it advances the variables from time level t n to time level t n+1 and the
right-hand sides of the equations are generally evaluated at t n . It has the form
Q n+1 − Q n
∆t
= F(Q n ). (43)
The forward scheme is of first order and it is said to be explicit, as the calculation of Q n+1
does not depend on Q n+1 . Explicit methods generally require much less computation per
time step than implicit methods, in which the calculation of Q n+1 does depend on Q n+1 .
A quite natural way to derive a numerical method is to use a forward-in-time approximation
in combination with a spatially centered approximation, such as the central difference
scheme defined in section 3.3.2. Unfortunately this method suffers from severe stability problems.
But there are other explicit schemes, such as Lax-Friedrichs and Lax-Wendroff that
may be used with a forward approximation and are stable with reasonable size time steps.
3.4.2 Leapfrog Scheme
The Leapfrog scheme is an explicit three-time-level method. With this method, the variables
are stepped forward from time level t n−1 to time level t n+1 where the right-hand sides of the
corresponding equations are evaluated at the mid-time level t n . The Leapfrog method has
the general form
Q n+1 − Q n−1
= F(Q n ). (44)
2∆t
Its advantages are that it is stable when used togheter with spatially centered approximations,
it is of second order, and it requires only one function evaluation per time step. The weakness
of the Leapfrog scheme is its undamped computational mode. Numerical noise is generated
by numerical dispersion leading to the initiation and amplification of nonlinear instabilities.
A way to avoid the growth of the computational mode will be given in section 3.8.
3.4.3 Third-order Runge-Kutta Scheme
Higher-order time-integration methods are in general not used because of limitations on
computational ressources. Nevertheless, the third-order Runge-Kutta scheme has interesting
characteristics: it is stable with strongly damped computational modes. Furthermore, it can
be computed in a low-storage variant:
k 1 = ∆tF(Q n ),
k 2 = ∆tF(Q 1 ) − 5k 1 /9,
k 3 = ∆tF(Q 2 ) − 153k 2 /128,
Q 1 = Q n + k 1 /3,
Q 2 = Q 1 + 15k 2 /16,
Q n+1 = Q 2 + 8k 3 /15.
(45)
If m is the size of Q, this variant, called the Williamson-Runge-Kutta scheme, uses only 2m
storage locations, m for each of the arrays k and Q, which are overwritten three times during
each step. The storage requirement of the Williamson-Runge-Kutta scheme is the same as
that of the forward and the standard leapfrog schemes. This method offers the possibility
15

to achieve better than second-order accuracy without overstepping the storage limitations.
However, the third-order Runge-Kutta scheme has not yet been implemented in our model.
It will be used in a future version.
3.5 Dimensional splitting
In our model, equation q t + f(q) x + g(q) z = s(q) must be solved. The straightforward
approach is to evaluate the three terms f, g, s at the same time and thus updating q with the
three contributions at once. However an other possibility is to split the problem into three
sub-problems and solve them in turn
x-sweeps: q t + f(q) x = 0, (46)
z-sweeps: q t + g(q) z = 0, (47)
source-addition: q t = s(q). (48)
The application of a splitting technique generally gives more accurate results as the values
q are updated between the computation of each term. The successive steps will be detailed
using a forward-in-time integration.
In the x-sweeps we start with cell averages Q n ij at time t n and solve the one-dimensional
problem (46) along each row of cells C ij with j fixed, computing an intermediate solution Q ∗ ij :
Q ∗ ij = Q n ij + ∆t RHS x (Q n ). (49)
In the z-sweeps the values Q ∗ ij are then used as data and (47) is solved along each column of
cells with i fixed. This results in Q ∗∗
ij :
Finally the values Q ∗∗
ij
Q ∗∗
ij = Q ∗ ij + ∆t RHS z (Q ∗ ). (50)
are used to solve (48) and the new value Qn+1
ij
at t n+1 is obtained
Q n+1
ij
= Q ∗∗
ij + ∆t S(Q ∗∗ ). (51)
Without flux splitting, the information that is contained in the adjacent cells C i−1j and
C ij−1 is transmitted to cell C ij at the same time. However this does not take into account the
information that is contained in the diagonally adjacent cell C i−1j−1 , which is transported by
diagonal winds. If a flux splitting technique is used and x-sweeps are preceeding z-sweeps,
then information contained in the diagonally adjacent cell C i−1j−1 is first transmitted to cell
C ij−1 during the x-sweep and it is then further conveyed to cell C ij during the z-sweep.
3.6 Initial conditions
To determine the quantities q(x, z, t) for all times t > t 0 , initial conditions at time t 0 must be
specified. An initial ”background” state is computed for the whole domain and cell averages
are determined for every grid cell. This background state is stored in a variable q 0 , as it
might be further needed for boundary conditions computation (section 3.7) and filtering
(section 3.8). Then, a perturbation is often initiated in one of the variables q depending on
the problem to be solved. Different possibilities of initial states are given is this section.
16

3.6.1 Gas dynamics (c = 1)
A simple way to initialise the grid cells is to set ρ 0 = ρe 0 = 1 and ρu 0 = ρw 0 = 0. This
initial conditions were used in particular during the different stages of model implementation
in order to facilitate the verification of its functioning.
3.6.2 Unstratified atmosphere (g = 0)
A more ”physical” but still unstratified atmosphere is initialised by setting ρ 0 equal to
1.25 kgm −2 in all grid cells. Lateral and vertical wind speed intensities u 0 and w 0 are then
chosen and the initial momenta are given by the products ρ 0 u 0 and ρ 0 w 0 . To initialise the
energy variable, a value for the temperature T 0 must also be specified an it is assumed to be
constant in the whole domain. The energy is computed from
ρe 0 = ρ 0 C v T 0 + 1 2 ρ 0(u 2 0 + w 2 0). (52)
with ρe int = C v T where C v = 717 Jkg −1 K −1 is the specific heat at constant volume.
3.6.3 Stratified atmosphere (N = const)
In the absence of atmospheric motion, there is a balance between vertical pressure gradient
and buoyancy forces. In the atmosphere, the pressure of air decreases with increasing altitude.
This creates an upward force, called the pressure gradient force. The force of gravity, on the
other hand, almost exactly balances this out. Explicitly, hydrostatic balance is expressed as
∂p
∂z
= −ρg. (53)
The two variables, pressure and density, are related through the ideal gas law
p = ρR d T, (54)
where R d = 287 Jkg −1 K −1 is the gas constant for dry air. The temperature can be expressed
as
T = πθ, (55)
where θ and π are the temperature and pressure that a parcel of dry air at temperature T and
pressure p would have if it were brought at a standard pressure p s . θ is called the potential
temperature and π the Exner function which is defined explicitely as
π =
( p
p s
) Rd /C p. (56)
p s is the pressure at ground level and C p = 1004 Jkg −1 K −1 the specific heat at constant
pressure. A last equation that is relevant for hydrostatic balance is
N 2 = g θ
where N is the buoyancy frequency usually taken to be 0.01 s −1 .
∂θ
∂z , (57)
17

In order to compute hydrostatic balance we need to express θ and π as functions of z
only. θ(z) can be obtained by integrating equation (57) in z :
( N
2 )
θ(z) = θ s exp
g z . (58)
where θ s = θ(0). Substituting (54) in (53) and then writing this in terms of π and θ yields
Integrating in z gives the equation for π
π(z) = 1 −
∂π
∂z = − g
C p θ . (59)
g2
(1 − exp(−N2 z)), (60)
C p θ s N 2 g
and this equation is used to compute hydrostatic stratification. Then performing a backsubstitution
T = πθ,
p = p s π Cp/R d,
ρ =
R p
d T ,
(61)
ρe =
γ − p
1 ,
yields the values for the components of q.
3.7 Boundary conditions
In an atmospheric limited area model, only the lower boundary is physical. The other sides
of the spatial domain are unbounded, meaning that the fluid flows in an exterior domain,
and artificial boundaries must be introduced. In order to deal with the boundaries of the
computational domain boundary conditions must be specified, for both artificial and physical
boundaries.
F n
In the methods studied here, the value Q n ij
is updated by computing the fluxes F
n
i−1/2,j ,
i+1/2,j , Gn i,j−1/2 and Gn i,j+1/2 . For doing that the neighboring cell values Qn i−1j , Qn i+1j , Qn ij−1
and Q n ij+1 are needed. But for the cells at the edges of the computational domain one or
more of these values are missing. The general approach to this problem is to extend the
computational domain to include a few additional cells, called ghost cells, on each side.
Figure 3 shows a portion of a grid on the computational domain that consists of interior
cells labelled by i = 1, 2, ..., N x and j = 1, 2, ..., N z and that is extended by introducing
two ghost cells on each side, for i = −1, 0 and i = N x + 1, N x + 2 and for j = −1, 0 and
j = N z + 1, N z + 2. The values in the ghost cells are set at the beginning of each time step
based on data in the interior cells in a manner that depends on the boundary conditions.
This provides the neighboring cell values needed in updating the cells near the boundary of
the domain.
The number of ghost cells depends on the stencil of the method. For a three-point method,
only one ghost cell on each side is needed to compute the fluxes, while two are required with
a five-point stencil.
18

Figure 3: The lower left corner of a typical computational domain. Interior grid cells (in
black) are labeled with i = 1,2,... and j = 1,2,... . The domain is extended with two rows of
ghost cells (in grey) on each side.
3.7.1 Periodic boundary conditions
For a two-dimensional domain [a, b] × [c, d], the periodic boundary conditions in x-direction
can be expressed as q(a, z, t) = q(b, z, t). Therefore the values Q n 0j in the column of ghost
cells before the left-hand boundary should be equal to the values Q n N xj
in the last column of
internal cells. Therefore we set Q n 0j = Qn N xj
before computing the fluxes. For a method in
which two ghost cells are needed, these are filled by setting
for j = 1, 2, ..., N z .
Q n −1,j = Qn N x−1,j , Qn 0,j = Qn N x,j , Qn N x+1,j = Qn 1,j , Qn N x+2,j = Qn 2,j (62)
In z-direction, the periodic boundary conditions are given by q(x, c, t) = q(x, d, t) and
the ghost cells are filled with a similar procedure
Q n i,−1, = Qn i,N z−1 , Qn i,0 = Qn i,N z
, Q n i,N z+1 = Qn i,1 , Qn i,N z+2 = Qn i,2 (63)
for i = −1, 0, 1, 2, ..., N x + 1, N x + 2.
Note that in the second direction the procedure is applied for the interior cells as well as
the ghost cells (i = −1, 0 and i = N x + 1, N x + 2) in order to have all the ghost cells filled,
including the sixteen corner cells.
3.7.2 Solid wall boundary conditions
A possible condition for the boundaries is to consider a solid wall that the fluid cannot cross.
This means that the velocity normal to the boundary must be zero. Therefore the solid wall
boundary conditions can be written as ⃗u · ⃗n = 0.
Consider first the lateral boundaries of the domain, where the x-component, u, of the
19

velocity should be zero. The ghost cells are filled by extrapolating all components of Q in
a symmetric manner, except for the x-momentum ρu that must also be negated so that it
vanishes at the boundary
for j = 1, 2, ..., N z .
Q n −1,j = Qn 2,j , Qn 0,j = Qn 1,j , Qn N x+1,j = Qn N x,j , Qn N x+2,j = Qn N x−1,j (64)
A similar extrapolation can be performed for the ghost cells at the top and bottom
edges and the z-component of momentum ρw should be negated.
3.7.3 Solid wall with hydrostatic pressure
In many fluid dynamics models, the pressure is made to decrease with altitude in order to
mimic the true atmospheric conditions. In this case, the usual solid wall boundary conditions
are not very appropriate for the top and bottom boundaries as, by symmetry, the pressure
would be decreasing instead of increasing under the bottom boundary. For this reason, the
solid wall conditions must be adapted to take into account the increase of pressure with lowering
height.
Integrating equation (53) that describes hydrostatic balance over two cells yields a way to
extrapolate the components of Q that correspond to the energy (and therefore to the pressure)
in the ghost cells under and above the computational domain. For the energy components in
the two bottom cells this gives
ρe i,−1 = ρe i,1 + 2∆z ρ i,0g
(γ − 1) , ρe i,0 = ρe i,2 + 2∆z ρ i,1g
(γ − 1)
for i = −1, 0, 1, 2, ..., N x + 1, N x + 2. A similar extension for ρe i,Nz+1 and ρe i,Nz+2 in the top
ghost cells is performed. The other components of Q in the ghost cells obey the usual solid
wall extension of section 3.7.2.
3.7.4 Solid wall with background
Instead of straightly performing the extrapolation of the values Q n , the solid wall extension
can be done on the difference between the current and the background states Q n − Q 0 . Then
(65)
Q n i,−1 = Q0 i,−1 + Qn i,2 − Q0 i,2 , Qn i,0 = Q0 i,0 + Qn i,1 − Q0 i,1 (66)
for i = −1, 0, 1, 2, ..., N x + 1, N x + 2. This technique permits to preserve the hydrostatic
tendency of pressure that is to increase with lowering altitude as only the differences from the
hydrostatic background state are copied from the interior cells and added to the background
values in the ghost cells.
3.7.5 Prolongation of hydrostatic balance
An other way to deal with hydrostatic situations is to perform an extrapolation of the pressure
under and above the boundaries of the domain. Hydrostatic balance is computed with the
help of accessory variables among which the potential temperature θ and the Exner function
20

π. In order to fill the ghost cells, a first order extrapolation of θ, based on fitting a linear
function through the interior solution is done. For the bottom cells this gives
θ i,0 = θ i,1 − ∆θ i,3/2 = θ i,1 − (θ i,2 − θ i,1 ) = 2θ i,1 − θ i,2 ,
θ i,−1 = θ i,0 − ∆θ i,1/2 = θ i,0 − (θ i,1 − θ i,0 ) = 2θ i,0 − θ i,1 , (67)
for i = −1, 0, 1, 2, ..., N x + 1, N x + 2. Using these values, the Exner function can be computed
for the ghost cells by integrating equation (59).
2g
π i,0 = π i,1 + ∆z
C p (θ i,0 + θ i,1 ) ,
π 2g
i,−1 = π i,0 + ∆z
C p (θ i,−1 + θ i,0 )
(68)
for i = −1, 0, 1, 2, ..., N x + 1, N x + 2. Finally the components of Q are found using the set of
equations (61).
3.8 Damping terms
Non-linear instabilities and small-scale computational noise can be controlled by introducing
artificial numerical damping terms acting on the computational grid. Numerical noise of such
type is continuousely generated by numerical dispersion, especially with the Leapfrog time
integration using centerd difference operators for advection.
3.8.1 Time filtering
A way to control the growth of the computational mode when using a Leapfrog integration
(section 3.4.2) is to apply a centered second-order time filter to the mid-time level t n after
each Leapfrog step. The Robert-Asselin time filter has the form
˜Q n ij = Q n ij + µ(Q n+1
ij
− 2Q n ij +
˜Q
n−1
ij
), (69)
where Q n ij is the solution at time t n, ˜Qn ij the solution after filtering and µ is a positive real
coefficient that determines the strength of the filter. Typical values of µ are between 0.05
and 0.2. Note that the last term in (69) is the usual finite-difference approximation to the
second derivative and preferentially damps the high-frequency components. Therefore, the
filter permits to control rapid oscillations and instabilities of the computational mode, which
are generated by the Leapfrog integration. The temporal decoupling of the physical and the
computational modes of the numerical solution is thus avoided.
3.8.2 Spatial filtering
In a simulation, numerical dispersion continuously generates numerical noise of wavelength
near the two grid interval (at the 2∆x-scale), especially when using centered difference
schemes. These schemes preserve the amplitude of each wave component, but the various
components propagate at different speeds and thus the superposition of these components
does not properly represent the true solution anymore. It is therefore often useful to add
artificial diffusion in order to damp this small scale noise. This is generally realized by
introducing a scale-selective spatial filter of the form
dQ
dt = (−1)m/2+1 K m ∇ m Q, (70)
21

where m is the order of the diffusion term, that must be even, and K m is the corresponding
diffusion coefficient.
Note that these filters not only damp the two grid interval wavelength but may also
act on longer wave components. However, the fourth and sixth order filters are more scaleselective
than the second order filter as they have very little damping for these longer waves.
Nevertheless, they have the disadvantage to introduce spatial oscillations and thus creating
possible new maxima or minima. This does not happen with a second order filter as it has
a Lapacien form and thus it is always monotonic: it simply transfers quantities from regions
with higher values to regions with smaller values.
In the case of our model, a second order spatial filter is applied to the variables ρ, ρu
and ρw after each time step. It is not applied to ρe because pressure, which is related to
energy, should not be smoothed. If it was so, the pressure values would be moved from their
equilibrium values and this would lead to the creation of winds. This filter has the form
˜Q n+1
ij
= Q n+1
ij
+ ν 8 (Qn−1 i+1j − 2Qn−1 ij
+ Q n−1
i−1j + Qn−1 ij+1 − 2Qn−1 ij
˜Q
n+1
+ Q n−1
ij−1
), (71)
where Q n+1
ij
is the solution at time t n+1 ,
ij
the solution after filtering and ν is a coefficient
that determines the strength of the filtering. If ∆x = ∆z, ν can be expressed as
ν = 8K ∆t
∆x 2 . (72)
It follows from equation (72) that, for a given ν, the physical diffusion K depends on the
lenght of the time step and the cells size.
22

4 Results
4.1 Comparison of different methods
Several types of numerical fluxes and various time integration methods were implemented
and can be combined to yield different models.
A model using Leapfrog time integration and central differences in space with initial
hydrostatic equilibrium conditions is found stable for integration times up to ten hours if a
numerical smoothing filter is applied. However stability cannot be achieved with the Lax-
Friedrichs and Lax-Wendroff methods. In the simulations, the pressure is continuousely
increasing with time. In the Lax-Wendroff method a half step in time is first made approximating
q at the cell interfaces, to be consistent it seems that the source term s(q) and possibly
also the correction term r(q) should be introduced when doing this forward integration. This
will be implemented in a future version.
Concerning the boundary conditions, three different possibilities for filling the ghost cells
can be used in models using hydrostatic equilibrium. These are described in section 3.7.3,
3.7.4 and 3.7.5. However the solid wall with hydrostatic pressure boundary conditions and
the prolongation of hydrostatic balance boundary conditions appear to give values for the
ghost cells that are not in exact prolongations with the values in the interior cells. In this
case wind are created at the boundaries of the domain that can lead to instabilities. The
solid wall with background boundary conditions give good results as they better preserve the
hydrostatic tendency of the pressure.
The model with dimensional splitting (section 3.5) is suffering from stability problems.
In simulations with hydrostatic equilibrium, the winds that are generated are of two orders
of magnitude bigger than without splitting, after an integration of 20 s. These wind velocities
are even bigger in the regions near the boundaries. The boundary conditions were
first suspected as they must be applied between the computation of each sub-problem and,
therefore, they are applied several times inside a single time step. An attempt to solve this
problem was made by computing the x-fluxes (in the x-sweep step) for the whole domain
including the ghost cells so that they are updated and the boundary conditions must not be
reapplied before the z-sweep is performed. However the results obtained are not better with
this technique and another cause for the instabilities must be found. Another possibility to
control these instabilities is that numerical diffusion should be possibly introduced as a fourth
sub-equation in the splitting. In the current version, it is only applied after the time step is
done. This will be implemented and tested in a future version of our model.
23

4.2 Hydrostatic balance
Hydrostatic equilibrium in vertical direction z can be computed using equation (60). However,
even if the initial pressure is analytical, there will be small numerical imprecisions due
to discretization into finite volumes. Indeed, the pressure is generally not a linear function
of height and small errors arise when approximating average values over each cells. These
imprecisions then lead to the creation of winds at the beginning of the simulation. The maximum
velocity of these winds then increases to stabilise at a constant mean value around
which it oscillates with time.
This behaviour is due to the integration schemes themselves and their associated local
truncation error. In order to obtain good results, numerical filters are used that smooth the
solutions and therefore reduce the wind velocities. Sensitivity tests to filtering using hydrostatic
equilibrium in one dimension were done. The results are presented on semilogarithmic
plots of |w| max versus time, where |w| max is the maximum of z-velocity values over the whole
domain. The plots are shown in Figure 4 and Figure 5. The model is integrated using a
Leapfrog scheme together with centered-in-space differences and the boundary conditions are
a solid wall with background.
The models using a Leapfrog scheme afford an undamped computational mode that leads
to the creation of oscillations and instabilities in the solution. These unwanted effects can
be controlled by applying an Asselin time filter after each step (see section 3.4.2). The value
of the filter coefficient µ must be chosen so that the filter strongly damps the computational
mode while having only little impact on the physical mode. A series of simulations is done for
various values of µ and the results are shown in Figure 4. It is observed that the amplitude
of the oscillations are reduced as µ is increased. However, at the same time, the average
value of |w| max increases a little bit. This can be seen by comparing the plots with time
filter constants of 0.05 and 0.4 in Figure 4. Although the curve for µ = 0.4 is smoother, it
is also higher, meaning that the velocities are bigger. With very large values of µ (e.g. 0.6),
the velocities even continuously increase with time to reach values of 10 −3 ms −1 after 9000 s.
Filter coefficients µ in the range 0.05 - 0.2 will be used in our model.
A second order spatial filter as given by (71) can be used to damp the 2∆x-scale noise
generated by numerical dispersion. Results for different values of the filter coefficient ν are
presented in Figure 5 showing that the velocities can be reduced by increasing the strength of
the filter. However, beside reducing the 2∆x noise, the filtering may also act on longer wave
components, which does contain physical meaningful information and therefore should not
be affected by the filter. In the last plot of Figure 5, the decrease of |w| max with time seems
to indicate that these longer wave components are smoothed by the filter. A very strong filter
may therefore create too much diffusion and lead to the loss of physical information. For
this reason the strength of the filter should be adapted according to the problem being solved.
Sensitivity to source term formulation is also tested and is presented in Figure 6. Three
different techniques for the interpolation of ρ are compared: Midpoint, Simpson rule over
one cell and Simpson rule over two cells. In Midpoint, the average cell value of ρ is used to
compute the source term, leading to a first order approximation. The value of the source term
24

time filter cst = 0
time filter cst = 0.15
10 −5
10 −5
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
10 −7
10 −7
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
time filter cst = 0.05
time filter cst = 0.2
10 −5
10 −5
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
10 −7
10 −7
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
time filter cst = 0.075
time filter cst = 0.4
10 −5
10 −5
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
10 −7
10 −7
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
time filter cst = 0.1
time filter cst = 0.6
10 −4 t [s]
10 −2 t [s]
10 −3
10 −5
log 10
|w| max
[m/s]
10 −6
log 10
|w| max
[m/s]
10 −4
10 −5
10 −6
10 −7
10 −7
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Figure 4: Comparison of the maximum z-velocities w obtained for differents values of the
Asselin time filter constant µ in a one-dimensional hydrostatic model with ν = 0.002.
25

spatial filter cst = 0
spatial filter cst = 0.01
10 −5
10 −5
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
10 −7
10 −7
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
spatial filter cst = 0.0008
spatial filter cst =0.04
10 −5
10 −5
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
10 −7
10 −7
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
spatial filter cst = 0.002
spatial filter cst = 0.1
10 −5
10 −5
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
10 −7
10 −7
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
spatial filter cst = 0.006
spatial filter cst = 0.8
10 −5
10 −5
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
log 10
|w| max
[m/s]
10 −4 t [s]
10 −6
10 −7
10 −7
10 −8
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Figure 5: Comparison of the maximum z-velocities w obtained for differents values of the
spatial filter constant ν in a one-dimensional hydrostatic model with µ = 0.1.
26

in the (ij) grid cell is then s ij = −ρ ij g. In Simpson rule, ρ is computed from its interpolating
polynomial of degree two: ∫ b
[ (
b−a
a
ρ(z)dz ≈
6 ρ(a) + 4ρ a+b
) ]
2 + ρ(b) . In Simpson over one cell,
a and b are located at the edges of the cell, while in Simpson over two cells, a and b are
situated in the neighboring cells.
The results indicate a huge sensitivity to the method used for source term interpolation,
as the maximum velocities drop from 10 −2 ms −1 with Midpoint and Simpson over one
cell to 10 −6 ms −1 with Simpson over two cells. A possible explanation for this is that applying
Simpson rule over two cells rather than one single cell might produce a ”smoothing” effect
and this may be the reason of the dramatical decrease of the maximum velocities.
midpoint
10 0 t [s]
10 −1
log 10
|w| max
[m/s]
10 −2
10 −3
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Simpson with 1 grid cell
10 0 t [s]
10 −1
log 10
|w| max
[m/s]
10 −2
10 −3
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Simpson with 2 grid cells
10 −4 t [s]
10 −5
log 10
|w| max
[m/s]
10 −6
10 −7
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Figure 6: Comparison of the maximum z-velocities obtained for differents source term formulations
in a one-dimensional hydrostatic model with µ = 0.1 and ν = 0.01.
27

4.3 Sound waves
An acoustic wave is a very small pressure disturbance that propagates through a compressible
gas. This causes infinitesimal changes in the density and pressure of the gas because of little
motions of the gas with small values of the horizontal and vertical velocities u and w. Most
sound waves are linear phenomena. The disturbances from the background state are very
small so that products of powers of the perturbation amplitude can be ignored. Therefore
the quantities q can be written as q(x, z, t) = q 0 + q ′ (x, z, t), where q 0 is the background state
and q ′ is the perturbation.
As test case, a Gaussian pressure perturbation (with σ= 400 m) is superimposed onto
a one-dimensional hydrostatic background for which surface pressure and temperature values
of 10 5 Pa and 273, 15 K are chosen. The initial perturbation is located in the middle of the
computational domain and, in order for it to be linear (that is p ′ /p

400
t=0s
250
t=9s
350
200
300
150
250
100
p’[Pa]
200
150
p’[Pa]
50
100
0
50
−50
0
0 2500 5000 7500 10000 12500 15000
z[m]
−100
0 2500 5000 7500 10000 12500 15000
z[m]
250
t=18s
250
t=33s
200
200
150
150
100
100
p’[Pa]
50
p’[Pa]
50
0
0
−50
−50
−100
0 2500 5000 7500 10000 12500 15000
z[m]
−100
0 2500 5000 7500 10000 12500 15000
z[m]
Figure 7: Evolution of an initial pressure perturbation concentrated in the middle of the
computational domain into distinct waves propagating in opposite directions. Gray solid
line: ∆z = 200 m, dashed line: ∆z = 100 m, dotted line: ∆z = 50 m, dashed-dotted line:
∆z = 25 m, black solid line: ∆z = 12.5 m.
permits to explain why the upper wave is slower.
Caracteristics for the simulation with 12.5 m-cells resolution at t = 9 s are given in Table 1.
The velocity of little motions in the gas, |w ′ |, is found to be bigger in the upper perturbation.
Although the same quantity of energy should be propagating in both directions, this difference
may be explained by the fact that the density is smaller in the upper region yielding a
bigger velocity w ′ .
caracteristics at t = 9s lower peak upper peak
z(t) 4775.0000 10075.0000
T 0 (z) 238.9642 199.0194
ρ 0 (z) 0.7704 0.4038
w ′ (z, t) - 0.8243 1.2403
p ′ (z, t) 194.1356 146.0013
Table 1: Caracteristics for the simulation with 12.5 m-cells resolution at t = 9 s
From linear acoustics, a relation between the pressure perturbation p ′ and the velocity
perturbation w ′ at position z is given by
p ′ = ±w ′ Z 0 . (74)
29

where Z 0 = Z 0 (z) = ρ 0 (z)c 0 (z) is the impedance of the medium. This equation might provide
an explanation for the difference in size between the two peaks. Calculations of p ′low and p ′up
are made using the values of w ′ , ρ 0 and c 0 in Table 1 and give 196.7960 Pa and 141.6410 Pa
for the lower and upper wave respectively. These values are close to the values of p ′ obtained
in the simulation that figures in Table 1.
As second test case, a top hat perturbation with half width of 800 m is initialised in
the middle of the domain. This shape of perturbation is generally more difficult to simulate
than a Gaussian shape as it features steep slopes. In this case, oscillations generally appear
near the bottom and the top of these steep slopes. The background state and the integration
method are the same than in the previous example.
400
t=0s
300
t=9s
350
250
300
200
250
150
p’[Pa]
200
150
p’[Pa]
100
100
50
50
0
0
0 2500 5000 7500 10000 12500 15000
z[m]
−50
0 2500 5000 7500 10000 12500 15000
z[m]
300
t=18s
250
t=33s
250
200
200
150
150
p’[Pa]
100
p’[Pa]
100
50
50
0
0
−50
0 2500 5000 7500 10000 12500 15000
z[m]
−50
0 2500 5000 7500 10000 12500 15000
z[m]
Figure 8: Evolution of a top hat shaped pressure perturbation with 12.5 m resolution.
Only a single simulation with 12.5 m resolution is presented here as an illustration. The
evolution of the perturbation is shown in Figure 8. As the perturbation is superimposed
onto a stratified background, the top does not appear flat in the first plot in 8. In the plots
for t = 9, 18 and 33 s, the above mentioned oscillations can be observed and are relatively
pronounced. The different amplitudes and propagation speeds of the two waves are analogous
with the previous case. Finally, in the plot for t = 33 s the slope at the top of the perturbation
is inverted due to the reflection at the boundary.
Further tests with various values of filter coefficient should be performed in order to
attempt to reduce the oscillations. Besides this, simulations with higher grid resolutions
should be done as they might provide better results.
30

4.4 Advection
Advection of a density perturbation is done in a two-dimensional fluid of otherwise constant
density (ρ = 1.25 kgm −2 ) that flows with constant velocity. The inital background state is
the unstratified atmosphere described in section 3.6.2. A lateral wind speed of u = 20 ms −1
is chosen (w = 0) and the temperature is set to 273.15 K in the whole domain. The density
perturbation is initialised as a two-dimensional Gaussian distribution with 1/e-width of 400 m
and maximum height of 10 −4 kgm −2 and centered in the middle of the domain. In order to
avoid vertical acceleration of the density perturbation due to buoyancy forces, the gravitation
acceleration is set to zero for these experiments. The model is integrated using Leapfrog
scheme and space centered differences, and periodic boundary conditions are applied in both
directions. An Asselin time filter with coefficient µ = 0.1 and a spatial filter with ν = 0.01
are also used.
A reference solution is computed with a grid resolution of 25 m and a time step of 2.5
10 −2 s. It is obtained by progressively halving the grid-cells size, starting from ∆x = ∆z =
100 m with ∆t = 0.1 s, and helding the ratio of ∆t/∆z constant.
The evolution of the density perturbation can be seen in the ρ ′ solution from the reference
simulation at t = 0, 20, 60, 100, 200 and 300 s in Figure 9. Contour interval is
10 −5 kgm −2 and the contours represent values starting from 0 kgm −2 for the external line
in the first plot of Figure 9 to 9 10 −5 kgm −2 for the smallest circle at the center of the perturbation.
In addition, the u ′ , w ′ and p ′ fields from the reference simulation at 100 s are
shown together in Figure 10. As the density perturbation is advected to the right with the
fluid, small perturbations develop in the background density forming curved stripes. This
can be attributed to pressure perturbations caused by the movement of the circular density
perturbation. Then, as the bubble reaches the right-hand side boundary, it reenters on the
other side of the domain due to the periodic boundary conditions. At 100 s, it is back in
the center of the domain. A second and a third revolution are completed at 200 and 300 s.
The propagation speed of the front of the perturbation is found to be of 20 ms −1 and the
maximum u and w velocities are of 2.5 10 −9 ms −1 and 2 10 −9 ms −1 respectively.
By comparing reference solutions at 0, 100, 200 and 300 s in Figure 9, a deformation
of the perturbation with time is observed: the front becomes sharper. This deformation is
more significant in ρ ′ solutions from simulations with lower resolutions, which are shown at
100 s in Figure 11. Therefore the resolution should be high enough to preserve correct information.
An other noticeable change is that the maximum value of density, in the central region
of the perturbation, becomes smaller with time. (The number of contour lines in the
plots of Figure 9 are of nine for 0 s and 100 s, eight for 200 s and seven for 300 s.) Again, this
is more pronounced in the ρ ′ solutions from simulations with lower resolutions that are shown
in Figure 11. This decrease of intensity in the center of the perturbation may be due to the
diffusion introduced by the numerical smoothing filter. Even though the filter coefficient ν is
the same for all simulations, it is important to remember that the physical diffusion K depends
on the length of the spatial and temporal steps. Equation (72) is used to calculate the
31

physical diffusion factor for the different resolutions and gives values of K 100m = 125 m 2 s −1 ,
K 50m = 62.5 m 2 s −1 and K 25m = 31.25 m 2 s −1 . As the size of grid cells is halved, K is also
divided by two, and therefore less diffusion is generated with higher grid resolution.
1000
∆x=∆z=25m , t=0s
1000
∆x=∆z=25m , t=100s
800
800
600
600
z[m]
z[m]
400
400
200
200
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
∆x=∆z=25m , t=20s
1000
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
∆x=∆z=25m , t=200s
1000
800
800
600
600
z[m]
z[m]
400
400
200
200
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
∆x=∆z=25m , t=60s
1000
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
∆x=∆z=25m , t=300s
1000
800
800
600
600
z[m]
z[m]
400
400
200
200
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
Figure 9: Plots of ρ’ at 0, 20, 60, 100, 200 and 300 s for the 25 m resolution solution. Contour
interval is 10 −5 kgm −2 and the contours are centered around 0 kgm −2 .
32

1000
u’ ∆x=∆z=25m t=100s
800
600
z[m]
400
200
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
w’ ∆x=∆z=25m t=100s
1000
800
600
z[m]
400
200
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
p’ ∆x=∆z=25m t=100s
1000
800
600
z[m]
400
200
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
Figure 10: Plots of u ′ (CI = 0.5 10 −9 ms −1 ), w ′ (CI = 0.5 10 −9 ms −1 ) and p ′ (CI = 0.5
10 −7 Pa) at 100 s for 25 m resolution (CI: contour interval).
33

1000
∆x=∆z=100m , t=100s
800
600
z[m]
400
200
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
∆x=∆z=50m , t=100s
1000
800
600
z[m]
400
200
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
∆x=∆z=25m , t=100s
1000
800
600
z[m]
400
200
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
x[m]
Figure 11: Plots of ρ’ at 100 s for 100, 50 and 25 m resolution.
34

4.5 Buoyant bubble
In order to study the behaviour of numerical methods, a standard non-linear test problem
was designed for the ’Workshop on Numerical Methods for Solving Nonlinear Flow Problems’
that was held at the NCSA in 1990 [8]. This test-problem consists in a density current in an
otherwise homogeneous and isentropic two-dimentional fluid. It is initiated as a cold bubble
of air that subsequently descends to the ground. Then, as the density current spreads out
laterally at the bottom boundary, Kelvin-Helmholtz shear instability rotors form along the
top of the cold air boundary. A grid-converged solution for this problem could be found in
the literature and is shown in Figure 12.
Figure 12: Plots of θ’ at 0, 300, 600 and 900 s for the 25 m resolution compressible reference
solution [8]. Contour interval is 1 ◦ C and the contours are centered around 0 ◦ C.
A background state that is in hydrostatic equilibrium is chosen but the potential temperature
θ is set constant over the whole domain (N = 0). Then the Exner function π and
therefore also the pressure are computed from (59) with θ constant. A value of 10 5 Pa is
chosen for the surface pressure and θ is set equal to the surface temperature of T s = θ =
300 K. The lateral boundary conditions for the problem are the solid wall, and the vertical
boundary conditions the solid wall with background, both are described in section 3.7. The
size of the domain is of 6.4 km in the vertical direction and of 51.2 km in the lateral direction
centered at x = 0 km. The temperature perturbation is specified by the following function
⎧
⎨ 0 if L > 1,
∆T =
⎩−15 cos(πL) + 1
(75)
if L 1,
2
where
L =
√
( x − x c
)
x 2 + ( z − z c
)
r z 2 , (76)
r
35

x c = 0 m, x r = 4000 m, z c = 3000 m and z r =2000 m. The thermal perturbation has a
minimum temperature of -15 ◦ C and is centered at x = 0 m and z = 3000 m. These initial
conditions provide the negative buoyancy necessary to initiate a density current.
The simulations are initiated with a thermal perturbation as described above and our
model is integrated using Leapfrog and centered-in-space differences. Asselin and spatial
filters are used to prevent oscillations in solutions. The value µ = 0.1 is chosen for all simulations,
while ν is adjusted according to cells size and time step in order to always obtain a
diffusion coefficient K of 75 m 2 s −1 , as this particular value is chosen for the reference problem
in [8].
Simulations are made using 80, 130 and 200 m resolution. According to [8], the flow features
of the test problem should be adequately resolved for the first simulation and marginally
resolved for the two others. The θ ′ solutions at 0, 300, 600 and 900 s from the three simulations
are shown in Figure 13 and Figure 14.
z[m]
4000
3000
2000
1000
0
80m
t=0s
z[m]
4000
3000
2000
1000
0
80m
t=300s
z[m]
4000
3000
2000
1000
0
80m
t=600s
z[m]
4000 80m
3000
t=900s
2000
1000
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
x[m]
Figure 13: Plots of θ’ at 0, 300, 600 and 900 s for the 80 m resolution solution.
interval is 1 ◦ C and the contours are centered around 0 ◦ C.
Contour
Our model seems to be capable of correctly reproducing the basic characteristics of the
flow in the test problem, even though the propagation speed of the density current is found
to be slightly bigger in our model and the front of the perturbation ends a little further than
in the reference model.
In the solution from the 80 m resolution simulation, the three rotors are correctely reproduced.
In the solution from the 130 m resolution simulation, the formation of the first
rotor at 600 s is well reproduced. However, the third rotor is missing in the solution at 900 s.
With 200 m resolution, the rotors are even less developed and there are significant quantitatives
and qualitative differences between our solutions and the 25 m reference solution.
36

4000
4000
z[m]
3000
2000
1000
0
130m
t=0s
z[m]
3000
2000
1000
0
200m
t=0s
4000
4000
z[m]
3000
2000
1000
0
130m
t=300s
z[m]
3000
2000
1000
0
200m
t=300s
4000
4000
z[m]
3000
2000
1000
0
130m
t=600s
z[m]
3000
2000
1000
0
200m
t=600s
z[m]
4000
3000
2000
1000
130m
t=900s
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
x[m]
z[m]
4000 200m
3000
t=900s
2000
1000
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
x[m]
Figure 14: Plots of θ’ at 0, 300, 600 and 900 s for the 130 and 200 m resolution solutions.
Contour interval is 1 ◦ C and the contours are centered around 0 ◦ C.
As mentioned above, our simulations were made using the same diffusion constant (K
= 75 m 2 s −1 ) as in the test problem in [8]. In consequence, our results are relatively noisy.
In order to reduce this noise the value of the spatial filter coefficient ν should be increased.
Results of a 130 m resolution simulation where ν is increased by a factor ten (ν=0.07) and
the solution previousely obtained for the 130 m resolution simulation (ν=0.007, Figure 14)
are shown togheter in Figure 15.
z[m]
4000
3000
2000
1000
130m
t=900s
ν=0.007
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
x[m]
z[m]
4000
3000
2000
1000
130m
t=900s
ν=0.07
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
x[m]
Figure 15: Plots of θ’ at 900 s for the 130 m resolution solutions with ν=0.007 (left) and
ν=0.07 (right).
The solution with the stronger filter coefficient appears to be very smoothed. In this case
it seems that part of the physical information is lost: the rotors form much slowlier. Moreover
the propagation speed is reduced.
These observations indicate that the results may be improved by adding more diffusion.
The noise could be attenuated and the propagation speed reduced. So that our results would
be even closer to the 25 m reference solution in Figure 12.
37

5 Conclusion
A two-dimensional finite volume method modeling the Euler equations on a Cartesian grid
was developped. Several types of numerical fluxes and various time integration methods were
implemented.
The model using Leapfrog time integration and central differences in space with initial
hydrostatic equilibrium conditions was found stable for integration times up to ten hours if
a numerical smoothing filter was applied. Sensitivity to numerical filtering was tested using
a one-dimensional model and indicated that the oscillations in solutions that occur with
Leapfrog scheme could be controlled using an Asselin time filter. Small-scale noise generated
by numerical dispersion could be reduced by applying of a smoothing filter. Results also
indicated that the source term formulation has a important effect on the velocities of the
winds generated by numerical imprecisions.
Grid-converged solutions could be obtained for various problems. In one space dimension
the simulation of sound waves gave satisfying results for a Gaussian perturbation and the
behaviour of the numerical solutions is in agreement with physical intuition. In two dimensions,
advection of a density perturbation demonstrated the ability of the model to reproduce
correct propagation speed. Finally, numerical simulations of a density-current test-problem
showed that the model is capable of correctly reproducing the essence of all the important
physics of the flow.
Further improvements can be made by solving the instability problems arising with the
splitting technique. To this end, changes could be done in the splitting method to include the
numerical diffusion term. Besides this the order of the time integration can also be improved
using a third-order Runge-Kutta scheme for example. These two modifications should permit
to gain in accuracy.
The numerical model that was developed here was found to work well and gave satisfying
results. As next step, the implementation of a cut-cell boundary condition can be
performed as this would permit to treat complex topography on the lower boundary.
6 Acknowledgements
A would like to thank Oliver Fuhrer for his help and guidance during these four months. I
would also like to thank Alain Clappier for offering me the possibility to do my diploma work
in the LPAS group and the people in the group for their support.
38

A
Derivation of the differential form from the integral form
The transformation of the integral form of a conservation law
∫∫
∫
d
q(x, z, t)dxdz = − ⃗n ·
dt
⃗f(q)ds,
Ω
into a partial differential equation is possible provided that the functions q(x, z, t) and f(q)
are sufficiently smooth. First, since Ω is independant of t, the time derivatives may be taken
inside the integral in the left-hand side. If q and f are smooth, we can use the divergence
theorem to convert the integral over ∂Ω into an integral over Ω
∫
∫∫
⃗n · ⃗f(q)ds = ⃗∇ · ⃗f(q)dxdz,
where the divergence of ⃗ f is
∂Ω
Ω
∂Ω
⃗∇ · ⃗f(q) = ∂
∂x f(q) + ∂ ∂z g(q).
Thus the integral form can be rewritten as
∫∫
∂
Ω ∂t
∫∫Ω
q(x, z, t)dxdz = − ⃗∇ · ⃗f(q)dxdz.
This leads to
∫∫
Ω
[ ∂ ∂t q + ⃗ ∇ · ⃗f(q)]dxdz = 0.
Since this integral must be zero over an arbitrary region Ω, the integrand must be identically
zero, and this finally gives the differential equation
∂
∂t q + ∂
∂x f(q) + ∂ g(q) = 0,
∂z
which is the differential form of the conservation law.
39

References
[1] D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics,
Springer, New York, 1999.
[2] H. Forrer, Second Order Accurate Boundary Treatment for Cartesian Grid Methods,
Seminar fuer Angewandte Mathematik Eidgenoessische Technische Hochschule, Zuerich,
Research Report No 96-13, 1996.
[3] E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation
Laws, Springer, New York, 1996.
[4] J. R. Holton, An Introduction to Dynamic Meteorology, Elsevier Academic Press, New
York, fourth edition, 2004
[5] R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhaeuser Verlag, Basel,
1990.
[6] R. J. LeVeque, D. Calhoun, Cartesian Grid Methods for Fluid Flow in Complex Geometries,
IMA Workshop on Computational Modeling in Biological Fluid Dynamics,
1999.
[7] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University
Press, Cambridge, 2002.
[8] J. M. Straka, R. B. Wilhelmson, L. J. Wicker, K. K. Droegemeier, Numerical Solutions
of a Non-linear Density Current: a Benchmark Solution and Comparisons, Int. Jour. for
Numerical Methods in Fluids, vol.17, 1-22(1993).
[9] G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974.
40