A perfectly matched layer approach to the linearized

shallow water equations models

I.M. Navon

Department of Mathematics and

Computational Science and Information Technology

Florida State University

Tallahassee, FL 32306-4130

B. Neta

Naval Postgraduate School

Department of Mathematics

Monterey, CA 93943

M.Y. Hussaini

Computational Science and Information Technology

Florida State University

Tallahassee, FL 32306-4120

November 26, 2002

1

Abstract

A limited-area model of linearized shallow water equations (SWE) on an f-planefor a rectangular domain is considered. The rectangular domain is extended toinclude the so-called perfectly matched layer (PML) as an absorbing boundary con-dition. Following the proponent of the original method, the equations are obtainedin this layer by splitting the shallow water equations in the coordinate directions andintroducing the absorption coeÆcients. The performance of the PML as an absorb-ing boundary treatment is demonstrated using a commonly employed bell-shapedGaussian initially introduced at the center of the rectangular physical domain.

Three typical cases are studied:

� A stationary Gaussian where adjustment waves radiate out of the area.

� A geostrophically balanced disturbance being advected through the boundaryparallel to the PML. This advective case has an analytical solution allowingus to compare forecasts.

� The same bell being advected at an angle of 45 degrees so that it leaves thedomain through a corner.

For the purpose of comparison, a reference solution is obtained on a �ne grid onthe extended domain with the characteristic boundary conditions.

We also compute the r.m.s. di�erence between the 48-hour forecast and theanalytical solution as well as the 48-hour evolution of the mean absolute divergencewhich is related to geostrophic balance.

We found that the PML equations for the linearized shallow water equations onan f-plane support unstable solutions when the mean ow is not unidirectional. Useof a damping term consisting of a 9-point smoother added to the discretized PMLequations stabilizes the PML equations. The re ection/transmission is analyzedalong with the case of instability for glancing propagation of the bell disturbance.A numerical illustration is provided showing that the stabilized PML for glancingbell propagation performs well with the addition of the damping term.

2

1 Introduction

In a limited-area numerical weather prediction model, the lateral boundaries are not

physical boundaries, and they require arti�cial boundary conditions so that the problem

is well-posed and the solution in the limited area remains uncontaminated and consistent

with the global solution. As such the treatment of lateral boundaries with the non-

re ecting or absorbing boundary conditions has been the subject of continuing interest

since the early days of has been the subject of continuing interest since the early days of

numerical weather prediction.

Several good reviews are available on the topic of both physical and arti�cial bound-

ary conditions (Givoli and Harari, 1998; Turkel, 1983; Givoli, 1991; Mcdonald, 1997;

and Tsynkov, 1998). Givoli and Harari (1998) have edited a special issue of Computer

Methods in Applied Mechanics and Engineering on the subject of boundary conditions

for exterior wave propagation problems. Turkel (1983) provided an early review on the

out ow boundary conditions in the context of computational aerodynamics. Givoli (1991)

reviews nonre ecting boundary conditions for the wave problems, discusses local and non-

local boundary conditions for physical and arti�cial boundaries in the context of problems

from di�erent disciplines. McDonald's (1997) review is con�ned to lateral boundary con-

ditions for operational regional forecast models. Kalnay (2001) presents the state of art

of limited area boundary conditions as used in meteorology. The most comprehensive

survey to date of arti�cial boundary conditions is due to Tsynkov (1998). He provides a

comparative assessment of the current methods for constructing the arti�cial boundary

conditions and divides them into two categories { local and global arti�cial boundary

conditions. Local arti�cial boundary conditions are algorithmically simple but relatively

less accurate than the global ones that are highly accurate and robust but computation-

ally expensive if not impractical. His di�erence potential approach although nonlocal is

said to be computationally inexpensive and easy to implement. Any practical algorithm

should be a compromise between these two categories. The sponge-layer approach may

be viewed as a compromise between local and nonlocal approaches. Although it involves

no global integral relations along the boundary, a certain amount of nonlocality persists

since the computational domain is arti�cially enlarged to include the sponge layer wherein

the model equations are solved using a numerical method close or identical to the one

3

employed in the interior domain.

The approaches of Givoli and Keller (l989) typify the global arti�cial boundary con-

ditions. They seem to work only in speci�c geometries and they are obviously not very

popular. Typical examples of local approaches to arti�cial boundary conditions are those

of Gustafson and Sundstrom (1978) and Engquist and Majda (1977, 1979). These meth-

ods can be viewed as a generalization of the Sommerfeld radiation condition and the

traditional characteristic boundary conditions. The so-called transparent boundary con-

ditions of McDonald (2001a, 2001b, 2002) and of Holstad and Lie (2001) and Lie (2001)

applied to the shallow water equations belong to this category.

In the bu�er/sponge layer approach, the computational domain is abutted on by

the arti�cial layers in which the waves are either damped or accelerated to supersonic

out ow (Perkey and Kreitzberg, 1976), Davies (1976), Israeli and Orszag (1981). The

boundary relaxation scheme of Davies (1976, 1985) is such an approach, and it is most

frequently used for limited area forecasting using mesoscale model. Typically the forecast

equations at the boundary are modi�ed by the addition of a Newtonian relaxation term

that damps the di�erences between the mesoscale and host model at in ow boundaries

while mitigating the e�ects of overspeci�cation at the out ow boundaries.

The perfectly matched layer (PML) method recently introduced by Berenger (1994)

as an absorbing boundary condition in the context of electromagnetic wave propagation

is an improvement on the sponge layer method allowing all outgoing disturbances trans-

mitted into the PML absorbing computational domain to be absorbed by the layer with

no re ection. The parameters of the PML are chosen such that the wave either never

reaches the external boundary, or, even if it reaches the boundary and re ects back, its

amplitude is negligibly small by the time it reaches the interface between the absorbing

layer and interior domain. This means that all outgoing disturbances transmitted into

the absorbing layer are damped out. Hu (1996) was the �rst to apply the PML approach

to aeroacoustic problems using the linearized Euler equations then (1996b) extending his

work to nonuniform mean ow for the nonlinear Euler equations. (Clement, 1996; Karni,

1996; R. Koslo� and D. Koslo�, 1986; Collino, 1996; Hayder et al., 1999; Hayder and

Atkins, 1997). The work of Hayder et al. (1999) is the �rst to demonstrate the viability

of the PML method in the applications to nonlinear Euler equations. A preliminary work

4

of Darblade et al. (1997) implements the PML method to the linearized shallow water

equations model in oceanography.

Hu (2001) presents a new stable PML formulation for the linearized Euler equations

in unsplit physical variables and provides a modi�cation to render the proposed scheme

strongly well-posed by the addition of arbitrarily small terms. Abarbanel and Gottlieb

(1997) provide the general mathematical analysis of the PML method while Abarbanel

et al (1999) provide a well posed version of PML for advective acoustics. Abarbanel and

Gottlieb (1998) provide the mathematical framework for use of PML in computational

acoustics. The well posedness of PML for linearized Euler equation and for the Cauchy

problem is discussed in Rahmouni (2000) and Metral and Vacus (1999), respectively. The

PML approach has been shown to provide signi�cantly better accuracy than most other

arti�cial boundary conditions in many applications.

The paper is organized as follows. In section 2, we introduce the PML approach to

the linearized two-dimensional shallow-water equations on an f-plane for the purpose of

analysis. The implications of the split shallow-water model and its weak hyperbolicity are

discussed shedding some light on the stability issues of this approach. Using a MAPLE

symbolic manipulator we obtain a dispersion relation for the linearized PML split shallow

water equations system. The solution of plane waves propagating in the perfectly matched

layer is also discussed. The weak well-posedness of the split-PML linearized shallow water

equations on an f-plane is provided in section 2.5.

In Chapter 3 we provide a description of the numerical testing using a widely employed

bell shape Gaussian (McDonald, 2000) at the center of the domain. The �rst test consists

of an adjustment case which is not in geostrophic balance and we compare the PML

results with its known asymptotic solution.

We then proceed to test an advective case of the bell shape Gaussian propagating

in parallel to the PML that has an analytical solution with which we can compare our

forecasts in terms of rms error as well as the vanishing of the mean absolute divergence.

Note that since the system is in geostrophic balance, the analytical divergence is always

zero.

This is followed by a test of propagation of the bell shaped Gaussian at an angle with

the PML yielding unstable solutions of the PML equations. (See also Hu, 1996; Tam et

5

al., 1998) An analysis is carried out to understand and explain the underlying reasons for

the instability. By adding arti�cial selective damping the mild instabilities for a PML of

20 grid spacings are suppressed when the distribution of � in the PML layer is adequately

chosen.

2 The perfectly matched layer methodology

The perfectly matched layer (PML) technique was �rst introduced by Berenger (1994)

for electromagnetic wave propagation governed by the Maxwell equations. Hu (1996) and

Hayder et al. (1999) extended it to the Euler equations.

The perfectly matched layer is an absorbing boundary condition allowing all outgoing

disturbances transmitted into the PML to be damped out and no wave is transmitted

back into the computational domain. The parameters of the PML are chosen such that

the wave either never reaches the external boundary, or, even if it reaches the boundary

and re ects back, its amplitude is negligibly small by the time it reaches the interface

between the absorbing layer and interior domain. In what follows we will brie y present

the framework of the PML method for the linearized shallow-water (S-W) equations on

an f-plane.

2.1 Linearized S-W equations on an f plane

The full nonlinear shallow water equations including the Coriolis factor are

@

@tu+ u

@

@xu+ v

@

@yu+

@

@x�� fv = 0 (1)

@

@tv + u

@

@xv + v

@

@yv +

@

@y�+ fu = 0 (2)

@

@t�+ u

@

@x�+ v

@�

@y+ �(

@u

@x+@v

@y) = 0 (3)

where u and v are horizontal velocities in the x and y directions respectively and � is the

geopotential � = gh, H is the height of free surface from basic state, h is a deviation of

height from H. To get the linearized system, we de�ne the perturbations:

u = u0 + U

v = v0 + V

� = �0 + �

6

where U = Umean and V = Vmean are constants and � is independent of t. Then the

equations can be written as:

@

@tu+ (u+ U)

@

@xu+ (v + V )

@

@yu+

@

@x(�+ �)� f(v + V ) = 0 (4)

@

@tv + (u+ U)

@

@xv + (v + V )

@

@yv +

@

@y(�+ �) + f(u+ U) = 0 (5)

@

@t�+ (u+ U)

@(� + �)

@x+ (v + V )

@(� + �)

@y+ (�+ �)(

@u

@x+@v

@y) = 0 (6)

where we omitted the primes, so that u; v; � are the perturbation. Using the geostrophic

relations

fU = � @

@y� (7)

fV =@

@x�: (8)

the 2-D linearized shallow water model on an f- plane is then of the form

@

@tu+ U

@

@xu+ V

@

@yu+

@

@x�� fv = 0 (9)

@

@tv + U

@

@xv + V

@

@yv +

@

@y�+ fu = 0 (10)

@

@t�+ U

@

@x(�+ �) + V

@

@y(�+ �) + c2(

@u

@x+@v

@y) + u

@�

@x+ v

@�

@y= 0 (11)

where c is the phase speed of surface gravity waves, i.e. c =pgH. If we assume that � is

a function of y only, then the geostrophic relations yield V = 0 and the equations become

If we assume that � is a function of y only, then the geostrophic relations yield V = 0

and the equations become

@

@tu+ U

@

@xu+

@

@x�� fv = 0 (12)

@

@tv + U

@

@xv +

@

@y�+ fu = 0 (13)

@

@t�+ U

@

@x�+ c2(

@u

@x+@v

@y) + v

@�

@y= 0 (14)

In the case we also exclude Coriolis, we get the form

@

@tu+ U

@

@xu+

@

@x� = 0 (15)

7

@

@tv + U

@

@xv +

@

@y� = 0 (16)

@

@t�+ U

@

@x�+ c2(

@u

@x+@v

@y) + v

@�

@y= 0 (17)

Here (x; y) are scaled with respect to L (characteristic length of the domain), velocity

components (u; v) are scaled with respect to c (gravity wave speed), the time t is scaled

with respect to L=c and � with respect to c2. These equations are identical to the linearized

Euler equations used by Hu (1996) with p replaced by �. Thus the results of Hu (1996)

are applicable to this form of the linearized S-W equations (not including Coriolis).

2.2 Analysis of the linearized S-W equations

The 2-D linearized shallow water model, about U constant, V = 0, and �(y), assumes

the form (12)-(14). This system can be written in matrix form as

@W

@t+ A

@W

@x+B

@W

@y+ CW = 0 (18)

where the vector W is 264 uv1c�

375

and the matrices A;B and C are

A =

264 U 0 c0 U 0c 0 U

375

B =

264 0 0 00 0 c0 c 0

375

C =

2640 �f 0f 0 00 1

c@�@y

0

375

The system (18) is said to be hyperbolic at a point in (x; t) if the eigenvalues of A are

all real and distinct. The same can be said of the behavior of the system in (y; t) with

respect to the eigenvalues of the matrix B. The notion of strongly hyperbolic comes when

the eigenvalues are real and there exists a complete system of eigenvectors, i.e. for an

eigenvalue of multiplicity m we have m linearly independent eigenvectors. The weakly

8

hyperbolic system allows multiple eigenvalues as before but here we are not necessarily

with a complete system of eigenvectors. The initial value problem is not well posed

for a weakly hyperbolic system which is not strongly hyperbolic. It is well posed for a

strongly hyperbolic system. Lower order terms do not destroy the well posedness. Since

A and B are symmetric and the equation is hyperbolic, the system is strongly well-posed

(Gustafson and Sundstrom, 1978).

2.3 The split-PML linearized shallow water equations on the f

plane

The inclusion of the Coriolis factor in the linearized shallow-water equations about (U; V )

requires the following modi�cation of the PML split form:

@u1@t

+ U@u

@x+@�

@x= ��xu1

@u2@t

+ V@u

@y= ��yu2

@u3@t

� fv = 0

@v1@t

+ U@v

@x= ��xv1

@v2@t

+ V@v

@y+@�

@y= ��yv2

@v3@t

+ fu = 0

@�1@t

+ �(y)@u

@x+ U

@�

@x= ��x�1

@�2@t

+ �(y)@v

@y+ V

@(� + �(y))

@y+ v

@�(y)

@y= ��y�2

(19)

In the above the coeÆcients �x and �y have been introduced for the absorption of

waves in the PML. We will refer to them as absorption coeÆcients in this work and they

will be assumed to be non negative. We notice that when

�x = �y = 0 (20)

9

we are reduced to the original linearized 2-D shallow-water equations with

u = u1 + u2 + u3 (21)

v = v1 + v2 + v3 (22)

� = �1 + �2 (23)

The spatial derivatives involve only the total �elds of u, v and � which are assumed to

be continuous at the interface between the interior domain and the PML. Two types of

interfaces are being created, namely, the interfaces between the interior domain and the

PML domain and those between two adjacent PML domains, see the following diagram.

PML (0; �y) (�x; �y)(0; 0) (�x; 0)interior PML

A similar approach was used for the linearized shallow-water equations in oceanogra-

phy by Darblade et al. (1997). If we use the split suggested by Hu (1996), the linearized

system will have a solution u2 = 0 and thus as if we didn't split the u variable in the

PML.

The dispersion relation can be obtained (using MAPLE symbolic manipulator) in a

similar fashion to Darblade (1997)

�!2W 3xW

2y

(�@�(y)

@y(iY Z +XF ) +WyZ

h�(X2 + Y 2)� F 2 + Z2

i)= 0 (24)

where

Z = 1 + iUX (25)

Wx = �x � i!; Wy = �y � i! (26)

X =kxWx

; Y =kyWy

; F =f

!(27)

Our result is somewhat di�erent, because we are allowing the reference geopotential height

to depend on y. The eigenfunctions (up to a multiplicative constant C) are

u = CniFZ � �XY + i@�=@y

WyXo

v = C fZ + iUX + (�� U2)X2g

� = C�nFX �

�iY + @�=@y

�Wy

�Zo

(28)

10

In the case we include the terms with V , the dispersion relation will be the same except

for Z which will depend on V as follows

Z = 1 + iUX + iV Y (29)

2.4 Analysis of unstable solutions of PML equations

Now we turn to the solution of the dispersion equation (24) and see if the imaginary part

of ! is never positive, which means that the solution is not growing exponentially in time.

The general case is very complicated. We started with the special case, @�(y)@y

= 0. In this

case we have Z [�(X2 + Y 2)� F 2 + Z2] = 0. The product is zero if one of the factors

vanish. Suppose Z = 0, then

!2 � [Ukx + V ky + i(�x + �y)]! � i(Ukx�y + V ky�x)� �x�y = 0 (30)

The solution is complex, let ! = �+ i�, then we have 2 equations for � and �. The �rst

equation is quadratic in � and has no dependence on �, i.e.

�2 � (Ukx + V ky)�� �x�y = 0 (31)

The solutions are

�1 = Ukx + V ky + Æ; �2 = ��

where Æ and � are small positive numbers. We substituted these values in the second

equation

2�� � (Ukx + V ky)� � (�x + �y)� + (Ukx�y + V ky�x) = 0 (32)

This gives 2 values for �, one for each �, both negative. By the way, the case where the

mean ow is in the x-direction, yields

! = Ukx � i�x

for which the imaginary part of ! is negative. We now check the other factor, corre-

sponding to the case where the Gaussian bell propagates at an angle to the PML layer

i.e.

�(X2 + Y 2)� F 2 + Z2 = 0

11

This gives a sixth order polynomial in ! with complex coeÆcients,

!6 � 2 (Ukx � i�x + V ky � i�y)!5 +

h(Ukx � i�x)

2 + (V ky � i�y)2 � f 2

��(k2x + k2y) + 2UkxV ky � 4 (�x�y + iV ky�x + iUkx�y)i!4

+ [2iV ky�x(V ky + Ukx � i�x � 2i�y) + 2iUkx�y(V ky + Ukx � i�y � 2i�x)

�2i(�x + �y)(f2 + �x�y)� 2i�(k2x�y + k2y�x)

i!3

+h�((kx�y)

2 + (ky�x)2) + f 2(�2x + �2y + 4�x�y)� (Ukx�y + V ky�x)

2

+2i�x�y(Ukx�y + V ky�x � i�x�y=2)]!2

+2i(�x + �y)�x�yf2! � f 2�2x�

2y = 0

(33)

Again the solution is complex, but it was not possible to �nd it with MAPLE software.

We have decided to look for solutions in the case we experimented with, i.e. U = V = 50,

� = 50000, f = :0001, and �x = �y = :0036. Now we �nd that the sixth order polynomial

for �

�6 � 100(kx + ky)�5 +

��47500(k2x + k2y) + 5000kxky

��4 + :003888(kx + ky)�

3

+��:0648kxky + :6156(k2x + k2y)

��2 � (:1679616)10�17 = 0

is having 1 or 3 positive roots (using Descartes' rule of signs) and the others are negative.

The sum of all roots is positive and equals 100(kx + ky) (coeÆcient of ��5) and the

product is of course negative because the constant term is negative and because there are

odd number of negative roots. The product is small and there are 2 small roots (since

there is no linear term in �). The second equation is linear in �,

D(�)� = �:0036N(�)

where

D(�) = (:9375)10�5�4 � (:78125)10�3(kx + ky)�3 + [:03125kxky � :296875(k2x + k2y)]�

2

+ (:18225)10�7(kx + ky)� + (:192375)10�5(k2x + k2y)� (:2025)10�6kxky

N(�) = (:625)10�5�4 � (:46875)10�2(kx + ky)�3 + [:015625kxky � :1484375(k2x + k2y)]�

2

+ (:2025)10�8(kx + ky)�

so upon substituting the values of � we have found that one of the �s is positive.

12

2.5 Plane Waves in a perfectly matched layer

We now consider a plane wave in the PML domain0BBBBBBBBBBBBB@

u1u2u3v1v2v3�1�2

1CCCCCCCCCCCCCA

=

0BBBBBBBBBBBBB@

u10u20u30v10v20v30�10�20

1CCCCCCCCCCCCCAei(kxx+kyy�!t) (34)

for the simpler case when � constant and U = 0 in (19). By substituting (34) into the

simpli�ed (19), we get(! + i�x)u10 = kx(�10 + �20)u20 = 0!u30 = if(v20 + v30)v10 = 0(! + i�y)v20 = ky(�10 + �20)!v30 = �if(u10 + u30)(! + i�x)�10 = kx�(u10 + u30)(! + i�y)�20 = ky�(v20 + v30)

(35)

Notice that if we divide the �rst equation of (35) by the �fth one, we get

kxky

=! + i�x! + i�y

u10v20

(36)

The solution of the other 6 equations in terms of u10 and v20 yields

u20 = 0

u30 =f

!2 � f 2(fu10 + i!v20)

v10 = 0

v30 = � f

!2 � f 2(i!u10 � fv20)

�10 =kx�!

(! + i�x)(!2 � f 2)(!u10 + ifv20)

�20 = � ky�!

(! + i�y)(!2 � f 2)(ifu10 � !v20)

(37)

Using this solution in the �rst equation and the �fth one, we can solve for kx and ky

kx = �s!2 � f 2

�!2(! + i�x)

u10qu210 + v220

(38)

13

ky = �s!2 � f 2

�!2(! + i�y)

v20qu210 + v220

(39)

The positive and negative signs indicate the direction of wave propagation. We will use

the positive sign here. It can be shown that u10=v20 is real and thus we can express u10

and v20 as

u10 = A cos �

v20 = A sin �

where A is a complex number and � is real. Substituting this in (38)-(39) we have

kx =�

!(! + i�x) cos � (40)

ky =�

!(! + i�y) sin � (41)

where

� =

s!2 � f 2

�:

As a result, we can write the plane wave solution as0BBBBBBBBBBBBB@

u1u2u3v1v2v3�1�2

1CCCCCCCCCCCCCA

= A

0BBBBBBBBBBBBB@

cos �0F1(f cos � + i! sin �)0sin �F1(f sin � � i! cos �)��1 cos �(! cos � + if sin �)��1 sin �(! sin � � if cos �)

1CCCCCCCCCCCCCAei!(�x cos �+�y sin ��t) e��x cos ��x��y sin ��y

(42)

where

F1 =f

!2 � f 2:

Note that when �x or �y is not zero, the magnitude of the wave decreases exponentially

as it propagates in the x or y direction, respectively.

2.6 Analysis of the split-PML linearized Shallow water equa-

tions: Weak well-posedness

The linearized version of the above equations (19) can be written as

@W s

@t+ As@W

s

@x+Bs@W

s

@y+ CsW s = 0

14

where the vector W s is de�ned as

�u1 u2 u3 v1 v2 v3

1

c�1

1

c�2

�T

and the matrices As; Bs and Cs are

As =

266666666666664

U U U 0 0 0 c c0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 U U U 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0c c c 0 0 0 U U0 0 0 0 0 0 0 0

377777777777775

Bs =

266666666666664

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 c c0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 c c c 0 0

377777777777775

Cs =

266666666666664

�x 0 0 0 0 0 0 00 �y 0 0 0 0 0 00 0 0 �f �f �f 0 00 0 0 �x 0 0 0 00 0 0 0 �y 0 0 0f f f 0 0 0 0 00 0 0 0 0 0 �x 00 0 0 @�

@y@�@y

@�@y

0 �y

377777777777775

The matrix S that diagonalizes As is

S =

266666666666664

�1 �1 0 0 0 �1 1 01 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 �1 0 �1 0 0 10 0 1 0 0 0 0 00 0 0 0 1 0 0 00 0 0 �1 0 1 1 00 0 0 1 0 0 0 0

377777777777775

15

It is easy to show that

S�1AsS =

266666666666664

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 U � c 0 00 0 0 0 0 0 U + c 00 0 0 0 0 0 0 U

377777777777775

Note that S�1AsS is a diagonal matrix with 5 zero eigenvalues being introduced as a

result of the splitting. It is straightforward to show that these �ve additional eigenvalues

imply that S and S�1 cannot transform Bs into a matrix that can be made symmetric

by multiplication with a positive de�nite diagonal matrix (see Hesthaven 1998). Thus

we have only a weakly well-posed case as discussed by Hestehaven (1998) .The most

general diagonalizer of As is S = TR (Abarbanel 1999) where the columns of T are the

eigenvectors of As and R is a matrix such that the columns of S are the most general

representation of the eigenvectors of As.

2.7 Re ection and transmission at an interface between two do-

mains

In order to obtain the necessary and suÆcient conditions for perfect transmission, we need

the existence of solutions of a polynomial equation of degree 7. In general the polynomial

cannot be factored, thus it is impossible to draw conclusions. We therefore limit ourselves

to obtain the necessary conditions for perfect transmission of plane waves at the interface

between 2 distinct domains, D1, and D2. This includes the interface between the interior

limited area domain and the PML domain.

The linearized S-W equations can be viewed as the split �eld PML linearized S-W

equations with both absorption coeÆcients being zero across an interface normal to x and

y between an interior domain and a PML domain.

Following Darblade (1997), we let the interface � between the two domains D1 and

D2 be a vertical line x = x0. Let �xi ; �yi ; i = 1; 2 be the absorption coeÆcients in the

x; y directions in Di. The variables u; v; � are continuous across the interface �.We now

�nd the necessary conditions for the perfect transmission at the interface of plane wave

16

in the form

= 0 ei(kxx+kyy�!t)

where = (u1; u2; u3; v1; v2; v3; �1; �2) is the solution of (19) if

1. The triplet (!; kx; ky) satis�es the dispersion equation (24)

2. The amplitudes 0 are the solution of the linear homogeneous system for which the

determinant is the dispersion equation (24).

The rank of the coeÆcient matrix is 7 for all (!; kx; ky) solutions of the dispersion equation.

The components of 0 can be found by solving the system, but for the transmission

conditions we only need (u; v; �) which are given by (28). For a �xed (!; kx; ky) and given

10, the given plane wave 1 which is the solution of (19) in D1, the linear combination of

plane waves which satisfy the condition of transmission across � will satisfy the necessary

and suÆcient conditions

!j = !j0

= !; kjy = kj0

y = ky; 1 � j � nj; 1 � j 0 � nj0

u10eikxx0 +

Pnjj=1 u

j0e

ikjxx0 =Pnj0

j0=1 uj0e

ikj0

x x0

v10eikxx0 +

Pnjj=1 v

j0e

ikjxx0 =Pnj0

j0=1 vj0e

ikj0

x x0

�10eikxx0 +

Pnjj=1 �

j0e

ikjxx0 =Pnj0

j0=1 �j0e

ikj0

x x0

(43)

where the re ected waves have a superscript j and the transmitted ones have a superscript

j 0. For (!; ky) �xed the dispersion equation is of degree 3 in X or kx and therefore there

exist only 2 triplets (!j; kjx; kjy) de�ning the re ected waves of 1 (denoted by 2; 3)

and 3 triplets (!j0

; kj0

x ; kj0

y ) de�ning the transmitted waves (denoted by 4; 5; 6) in D2.

These j; 1 � j � 6 satisfy

�1eikxx0�1u + �2eik2xx0�2u + �3eik

3xx0�3u = �4eik

4xx0�4u + �5eik

5xx0�5u + �6eik

6xx0�6u (44)

�1eikxx0�1v + �2eik2xx0�2v + �3eik

3xx0�3v = �4eik

4xx0�4v + �5eik

5xx0�5v + �6eik

6xx0�6v (45)

�1eikxx0�1� + �2eik2xx0�2� + �3eik

3xx0�3� = �4eik

4xx0�4� + �5eik

5xx0�5� + �6eik

6xx0�6� (46)

17

where�ju = iFZj � �XjY j + i@�=@y

W jy

Xj

�jv = Zj + iUXj + (�� U2) (Xj)2

�j� = FXj ��iY j + @�=@y

�W jy

�Zj

Xj =

8><>:kjx=W

1x 1 � j � 3

kj0

x =W2x 4 � j � 6

Y j =

8><>:kjy=W

1y 1 � j � 3

kj0

y =W2y 4 � j � 6

(47)

The only unknowns are �j; 2 � j � 6 or equivalently �j0

= �jeikjxx0 . Without loss of

generality, one can locate the interface at x = x0 = 0. It was shown by Darblade (1997)

that for �xed (!; ky), the dimension of the space of linear combinations of plane waves in

D1 and D2 satisfying continuity across the interface � is 3. This is clear since we have 3

equations (46) relating these �. So for a �xed plane wave with given triplet (!; kx; ky) and

given � it is suÆcient to know 3 linearly independent modes. The transmission is perfect

if and only if the following conditions are satis�ed for all ! positive and all ky complex

�i�iu = �j�ju�i�iv = �j�jv�i�i� = �j�j�

(48)

where i (respectively j) determines one of the 3 roots of the dispersion equation in D1

(respectively D2). The suÆcient condition for perfect transmission is �y1 = �y2 . Note

that if the interface is parallel to the y axis, the condition becomes �x1 = �x2 .

3 Numerical testing

A 2-D linearized shallow-water equations solver based on the explicit Miller-Pearce �nite

di�erence scheme is used (Miller and Pearce 1974). This scheme is implemented on a

non-staggered grid but provides a fair comparison since all methods are tested using the

same discretization. The scheme has a CFL stability condition

18

�t �q(�x)2 + (�y)2

cp2

where c is the speed of external gravity waves. Spatial di�erencing of the linearized

shallow water equations was carried out on a rectangular domain of 141�141 grid points,

with a uniform spatial horizontal grid length of �x = �y = 100km: We used values of

H = hav = 5000m and a time step of �t = 120sec. At the outer boundary of the PML

domain we apply characteristic boundary conditions.

We compared the results with a control simulation computed on a much larger domain

i.e. a domain of [�200; 200 ]� [�200; 200 ] gridpoints which is not a�ected by the bound-

ary conditions for the integration time span. The interior domain where the unaltered

linearized shallow-water equations are applied is [�50; 50 ]� [�50; 50 ] gridpoints.The depth of the PML layer is 20 gridpoints.

3.1 Testing Adjustment case

We investigated the permeability of the boundaries for the PML case for the linearized

shallow-water equations. The initial case considered is with r�(x; y) 6= 0 and u(x; y; 0) =

v(x; y; 0) = 0. Since the system is not in geostrophic balance, the system radiates adjust-

ment gravity waves and it will adjust to a balanced state given by the �eld (x; y) which

satis�es the equation

@2

@x2+

@2

@y2� f 2

�0

! (x; y) = �f

2

�0�(x; y; 0) (49)

See McDonald (2002), and Gill (1982). For all our numerical tests we used �x = �y =

100km and Lx = Ly = 10; 000km. In all the tests conducted, we used �0 = (5000m) g,

and �̂ = (500m) g. The experiment starts with a bell-shaped Gaussian at the center of

the domain

�(x; y; 0) = �0 + �̂ exp

8<:�

"x� Lx=2

Lx=10

#29=; exp

8<:�

"y � Ly=2

Ly=10

#29=; (50)

with u(x; y; 0) = v(x; y; 0) = 0 and the advecting velocities U; V are also set to zero. The

adjustment process radiates away gravity waves from the center of the domain.

19

In the implementation of PML as an absorbing boundary condition, we let � vary

spatially in the form

� = �m

d

D

!

(51)

where D is the thickness of the PML, d is the distance of the interface with the inte-

rior domain and is a constant. A PML of thickness of only 20�x, was used where

the parameters governing the spatial variation of � for the absorbing layer were = 3

and �m = �x = �y = 0:0018. The asymptotic state arrived at by solving the balance

equation (49) is compared as in McDonald (2002) with the 48-hour forecasts to assess the

transparency of the boundaries.

We display in Figs 1-3 various stages of the adjustment process of the unbalanced

height �eld which consists of adjustment waves radiating away radially from the center of

the domain and Fig. 4 for the solution of the asymptotic state which is undistinguishable

from that obtained after 42 hours for the height �eld. The rms di�erences between a

48-hour forecast and the asymptotic solution given by the balanced state are provided in

the following Table.

rms for h rms for u rms for v0.6378 0.0126 0.0126

Table 1: Root-mean squared di�erences between a 48-hour forecast and the asymptoticsolution given by the balanced state

These results show that the boundaries are almost transparent to the adjustment

waves. The forecast is almost identical to the the asymptotic balanced state described in

McDonald (2002).

A graph of the mean absolute divergence multiplied by 108 is displayed for the propa-

gation of the bell-shaped Gaussian for the case of adjustment is provided in Fig. 5. The

absolute value of divergence is displayed for PML case , small domain without PML and

large domain. The results of PML region and the large domain are practically undistin-

guishable.

20

3.2 Testing Advection with PML

The linearized shallow water equations are given by (9)-(11) and have an analytic solution

describing advection of a bell-shaped Gaussian with constant velocity (U; V ) starting from

center of domain at position (xc; yc). Thus we have an analytical solution to compare

propagation of the Gaussian using the PML approach. The analytical solution for the

bell-shaped Gaussian takes the form (50) for � and

v(x; y; t) =�2(x� xc � Ut)

f(Lx=10)2�(x; y; t) (52)

u(x; y; t) =2(y � yc � V t)

f(Ly=10)2�(x; y; t) (53)

The system is in geostrophic balance while the analytic divergence is zero. As mentioned

by McDonald (2002) this provides an additional test of the eÆcacy of the scheme used.

We tested the split PML for 2 cases, one is a wave propagating parallel to the x-axis

and the other at an angle of 45Æ with it.

1. Propagation parallel to the PML x axis

PML of 20 grid points, U = 50m=sec, x = �y = :0018; = 3; f plane and � = 30Æ

where � is the latitude at which the Coriolis factor is calculated. A 3-D plot of

the evolution of the advection of the bell-shaped Gaussian out of the area using

the PML is provided. We display it for a period of integration of 48 hours in Figs

6-9. We see that the PML layer is very e�ective and performs well as an absorbing

boundary condition.

The root-mean squared di�erence between the 48 hour forecast and the analytical

solution is provided in Table 2.

rms for h rms for u rms for v2.2693 0.3041 0.4978

Table 2: Root-mean squared di�erence between the 48 hour forecast and the analyticalsolution

A graph of the mean absolute divergence multiplied by 108 is displayed for the

21

propagation of the bell-shaped Gaussian for the case of propagation parallel to the

PML. is presented in Fig. 10. A very good geostrophic balance is obtained at the

end of the 48 hours forecast and the results of the PML coincide with those of

the large domain whereas the results in the small region without PML show a big

increase as the bell reaches the boundary (Fig. 10).

2. Propagation of bell-shaped Gaussian at an angle of 45Æ

We start with the bell-shaped Gaussian at the center of the domain and advect it

so that it exits through a corner, i.e. (xc; yc) = (Lx=2; Ly=2) and (U; V ) = (50; 50)

thus ensuring that the Gaussian exits at the corner de�ned by (Lx; Ly). As shown

theoretically in Section 2.4 the split PML equations support unstable solutions.

In the implementation of the PML as an absorbing boundary condition we used a

sigma curve very similar to that used by Tam et al. (1998)

The sigma curve starts with a value of sigma=0 at the �fth mesh point from the

interface between the computational domain and the PML

It is then followed by 8 mesh points where a cubic spline curve is used until the full

value of �x = �y = �m is attained . This was important for the case where arti�cial

damping was used in the case where the bell was propagating at an angle (See Fig.

17)

We present graphically the generation of instabilities in the PML for this case. As

a cure to instabilities manifested (con�ned) primarily to short waves, we applied a

9-point smoother in the PML

uij = uij +1

8(ui�1 j + ui+1 j + ui j�1 + ui j+1 � 4uij)

+1

16(ui�1 j+1 + ui�1 j�1 + ui+1 j+1 + ui+1 j�1 � 4uij)

(54)

Figs 11-12 show the 3-D propagation of the Gaussian bell at an angle of 45Æ.

Figs 14-16 show the 2-D plot of the Gaussian bell propagating at an angle with and

without a 9-point smoother after 60 hours of forecast. Without the smoother the growth

of the excited unstable solution spreads back into the interior computational domain

(Fig. 16) whereas using arti�cial damping provided by the 9 point Laplacian is e�ective

22

in suppressing the instabilities of the PML equations. This �lter continuously damps the

instabilities once they propagate into the PML (Fig. 15).

A graph of the mean absolute divergence multiplied by 108 is displayed for the prop-

agation of the bell-shaped Gaussian for the case of propagation at an angle of 45Æ to the

PML is presented in Fig. 13.

4 Summary and conclusions

In this paper we have described and implemented the PML split equations approach for

the linearized shallow water equations on an f-plane based on an explicit Miller-Pearce

scheme �nite di�erence discretization.

A theoretical analysis of the split PML linearized shallow water equations was derived

in Section 2.6 showing that they are only weakly well-posed.

The split PML approach was tested for its eÆciency as an absorbing boundary condi-

tion for the linearized shallow water equations on an f-plane using three di�erent scenarios.

First we tested permeability of the PML absorbing boundary conditions to adjustment

waves. Measured against an asymptotic balanced state we found small rms errors for the

48-hour forecast in the same range as those found by Mc Donald (2002). The rms between

the 48h forecast and the asymptotic solution are only slightly larger than those obtained

in the best case of McDonald (2002). This may be attributed to our use of a grid-A model.

In a second scenario we tested the split-PML for advection of the bell shape - viewed

as a geostrophically balanced sharp pseudometeorological feature( McDonald 2002) for 2

separate cases, both starting with a bell-shaped Gaussian at the center of the computa-

tional domain.

a) the mean ow is parallel to the PML layer.

b) propagation at an angle of 45Æ exiting through a corner.

Achievement of geostrophical balance is measured by considering the mean absolute

divergence at the end of the 48 hours forecast.

A detailed analysis carried out in the second part of Section 2.4 similar to that of Tam

et al. (1998) �nds that in such case (i.e. propagation at an angle, when the mean ow is

not unidirectional ) the split PML for linearized shallow water equations supports unsta-

ble solutions. Application of a 9-point Laplacian �lter stabilizes the PML. Our numerical

23

experiments show that the stabilized PML performs well as an absorbing boundary con-

dition for the linearized shallow water equations including the Coriolis factor.

Our results compare well with those obtained by Mc Donald (2002) for the same case.

The research carried out here has a natural extension to the formulation of boundary

conditions for advanced mesoscale models, such as the MM5 and the new MRF models,

and promises to improve upon the combination of nudging and sponge layer presently

used in such models. Work with PML in the framework of mesoscale models will mean

that gravity waves can not only leave the domain but also enter it without hindrance. The

�elds imposed by the PML must be well balanced - since while the host �elds are well-

balanced on their own grid - subsequent interpolation to the guest model may upset the

balance (McDonald 2002b, personal communication and paper submitted for publication).

Our results are encouraging and constitute a step towards using the PML absorbing

boundary conditions for full 3D atmospheric and ocean models. One avenue to achieve

this goal is to implement the PML boundary conditions to a 3D multi-layer shallow water

equations model as a way to proceed towards full 3D models. This can be done for the

linearized hydrostatic equations by carrying out a normal mode decomposition yielding a

shallow water equation for each vertical mode.

Development of a non-split version of both the linearized and the nonlinear version of

the shallow water equations based on ideas put forward by Abarbanel and Gottlieb (1998),

Abarbanel et al. (1999) and Hesthaven (1998) is presently also being investigated.

Acknowledgments

The �rst author would like to express his gratitude for the support extended to him by

the Center of Excellence (COE) at Florida State University during part of the write-up of

this paper. This is a contribution from the climate institute, a center of excellence funded

by the FSU research foundation.He also acknowledges support from NSF grant ATM-

97B1472 managed by Dr. Pamela Stephens. He would also like to express his gratitude

for the support and encouragement provided to him during his short stay at NCAR, fall

1999 made possible by DR Ying-Hwa Kuo, Head Mesoscale Prediction Group and by Dr

Joseph Klemp, Head Mesoscale Dynamics Group both at NCAR/MMM.We would like to

thank Prof. Alain-Yves Leroux from the University of Bordeaux for making available the

doctoral Thesis of Dr Giles Darblade.The expert programming help provided by Dr Zhuo

24

Liu during the implementation stages at C.S.I.T. is gratefully acknowledged. The useful

discussion with Dr Sajal K. Kar is acknowledged. The secretarial support and the expert

librarian support at NCAR is also acknowledged.

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29

Captions

Fig. 1 The adjustment of a bell shape using PML absorbing boundary condition: 0 hour

forecast.

Fig. 2 The adjustment of a bell shape using PML absorbing boundary condition: 6 hours

forecast.

Fig. 3 The adjustment of a bell shape using PML absorbing boundary condition: 42

hour forecast.

Fig. 4 Asymptotic balanced state as described by Equ (49)

Fig. 5 Graph of the mean absolute divergence for the adjustment case multiplied by 108.

Case of no PML, large computational domain and a PML of 20 gridpoints thickness

are displayed (in red, green and blue respectively).

Fig. 6 The advection of a bell shape out of computational domain moving parallel to

PML x-direction: 0 hour forecast.

Fig. 7 The advection of a bell shape out of computational domain moving parallel to

PML x-direction: 24 hour forecast.

Fig. 8 The advection of a bell shape out of computational domain moving parallel to

PML x-direction: 36 hour forecast.

Fig. 9 The advection of a bell shape out of computational domain moving parallel to

PML x-direction: 39 hour forecast.

Fig. 10 Graph of the mean absolute divergence for the case of advection of a bell shape

out of computational domain moving parallel to PML multiplied by 108. Case of

no PML, large computational domain and a PML of 20 gridpoints thickness are

displayed (in red, green and blue respectively).

Fig. 11 The advection of a bell shape out of computational domain moving at angle of

45 degrees to PML x-direction: 24 hour forecast.

30

Fig. 12 The advection of a bell shape out of computational domain moving at angle of

45 degrees to PML x-direction: 36 hour forecast.

Fig. 13 Graph of the mean absolute divergence for the case of advection of a bell shape

out of computational domain moving at an angle of 45Æ to PML multiplied by 108.

Case of no PML, large computational domain and a PML of 20 gridpoints thickness

are displayed (in red, green and blue respectively).

Fig. 14 The advection of the bell shape (2-D) out of the computational domain moving

at an angle of 45Æ to PML : 20 hour forecast using 9 point �lter.

Fig. 15 The advection of the bell shape (2-D) out of the computational domain moving

at an angle of 45Æ to PML : 60 hour forecast using 9 point �lter.Simulation showing

damping of unstable waves.

Fig. 16 The advection of the bell shape (2-D) out of the computational domain moving

at an angle of 45Æ to PML : 60 hour forecast without using the 9 point �lter.

Simulation showing propagation of unstable waves in the PML.

Fig. 17 Distribution of sigma within PML layer for the case of bell propagation at an

angle of 45Æ.

31

−50−40

−30−20

−100

1020

3040

50

−50 −40 −30 −20 −10 0 10 20 30 40 50

−100

0

100

200

300

400

500

h at time 0 Hours

Figure 1: The adjustment of a bell shape using PML absorbing boundary condition: 0hour forecast.

32

−50−40

−30−20

−100

1020

3040

50

−50 −40 −30 −20 −10 0 10 20 30 40 50

−100

0

100

200

300

400

500

h at time 6 Hours

Figure 2: The adjustment of a bell shape using PML absorbing boundary condition: 6hours forecast.

33

−50−40

−30−20

−100

1020

3040

50

−50 −40 −30 −20 −10 0 10 20 30 40 50

−100

0

100

200

300

400

500

h at time 42 Hours

Figure 3: The adjustment of a bell shape using PML absorbing boundary condition: 42hour forecast.

34

−50

−30

−10

10

30

50

−50−30

−1010

3050

−100

0

100

200

300

400

500

Figure 4: Asymptotic balanced state as described by Equ (49)

35

Figure 5: Graph of the mean absolute divergence for the adjustment case multiplied by108. Case of no PML, large computational domain and a PML of 20 gridpoints thicknessare displayed (in red, green and blue respectively).

36

−60−40

−200

2040

60

−60−40

−200

2040

60

−100

0

100

200

300

400

500

h at time 0 Hours

Figure 6: The advection of a bell shape out of computational domain moving parallel toPML x-direction: 0 hour forecast.

37

−60−40

−200

2040

60

−60−40

−200

2040

60

−100

0

100

200

300

400

500

h at time 24 Hours

Figure 7: The advection of a bell shape out of computational domain moving parallel toPML x-direction: 24 hour forecast.

38

−60−40

−200

2040

60

−60−40

−200

2040

60

−100

0

100

200

300

400

500

h at time 36 Hours

Figure 8: The advection of a bell shape out of computational domain moving parallel toPML x-direction: 36 hour forecast.

39

−60−40

−200

2040

60

−60−40

−200

2040

60

−100

0

100

200

300

400

500

h at time 39 Hours

Figure 9: The advection of a bell shape out of computational domain moving parallel toPML x-direction: 39 hour forecast.

40

Figure 10: Graph of the mean absolute divergence for the case of advection of a bell shapeout of computational domain moving parallel to PML multiplied by 108. Case of no PML,large computational domain and a PML of 20 gridpoints thickness are displayed (in red,green and blue respectively).

41

−60−40

−200

2040

60

−60−40

−200

2040

60

−100

0

100

200

300

400

500

h at time 24 Hours

Figure 11: The advection of a bell shape out of computational domain moving at angleof 45 degrees to PML x-direction: 24 hour forecast.

42

−60−40

−200

2040

60

−60−40

−200

2040

60

−100

0

100

200

300

400

500

h at time 36 Hours

Figure 12: The advection of a bell shape out of computational domain moving at angleof 45 degrees to PML x-direction: 36 hour forecast.

43

Figure 13: Graph of the mean absolute divergence for the case of advection of a bellshape out of computational domain moving at an angle of 45Æ to PML multiplied by 108.Case of no PML, large computational domain and a PML of 20 gridpoints thickness aredisplayed (in red, green and blue respectively).

44

Figure 14: The advection of the bell shape (2-D) out of the computational domain movingat an angle of 45Æ to PML: 20 hour forecast using 9 point �lter.

45

Figure 15: The advection of the bell shape (2-D) out of the computational domain movingat an angle of 45Æ to PML: 60 hour forecast using 9 point �lter. Simulation showingdamping of unstable waves.

46

Figure 16: The advection of the bell shape (2-D) out of the computational domain movingat an angle of 45Æ to PML: 60 hour forecast without using the 9 point �lter. Simulationshowing propagation of unstable waves in the PML.

47

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−3

Number of grid points from boundary into PML

Val

ue o

f sig

ma

used

sigmam

=

Figure 17: Distribution of sigma within PML layer for the case of bell propagation at anangle of 45Æ

48