A Model for Nonlinear Gravity Waves
in Stratified Flows
by
Ivan Skopovi
B.S., Mechanical Engineering (2000)
University of Maryland, Baltimore County
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 2002
@2002 Massachusetts Institute of Technology
All rights reserved
.....
A uthor ...........................
Department of Mechanical Engineering
August 26, 2002
Certified by...................
Trian a~llos R. Akylas
Professor of Mechanical Engineering
Thesis Supervisor
Accepted by................
MASSACHUSETTS INSTIClY
OFTECHNOLOGY
OCT 2 5 2002
LIBRARIES
BARKER
Ain A. Sonin
Protessor o1 Mechanical Engineering
Chairman, Graduate Committee
A Model for Nonlinear Gravity Waves
in Stratified Flows
by
Ivan Skopovi
B.S., Mechanical Engineering (2000)
University of Maryland, Baltimore County
Submitted to the Department of Mechanical Engineering
on August 26, 2002, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
Internal gravity waves are buoyancy-induced disturbances that arise when a stably
stratified fluid is forced to rise over a topographic barrier. Owing their existence to
buoyancy, they predominantly occur in the ocean and the atmosphere where they profoundly influence various engineering disciplines ranging from deep-water drilling and
pollution disposal strategies to meteorology and aviation. In terms of their capacity
to transport momentum and energy, the most significant internal gravity waves are
associated with the flow of stratified fluid over large amplitude topography for which
the Navier-Stokes equations become strongly nonlinear and therefore analytically intractable in general. Moreover, buoyancy frequency varies rapidly in the atmospheric
tropopause and the oceanic thermocline, which even further complicates the analysis. At this point, numerical modeling becomes an attractive approach to studying
the bahaviour of this natural phenomenon. Accordingly, this manuscript presents a
nonlinear model for simulating the dynamics of internal gravity waves. Based on the
results of the order of magnitude analysis, it is determined that modeling the media in
which buoyancy induced disturbances take place as two-dimensional, incompressible,
inviscid and Boussinesq significantly simplifies the mathematical description of the
physical model while still capturing essential physics of the geophysical waves. The
Boussinesq form of the Euler's equations is solved using the Second-Order Projection
Method for variable density flows. The algorithm is implemented for the localized
topography of arbitrary structure and the flow conditions characterized by arbitrary
variations of the Brunt-Viisdid frequency. It is further demonstrated that the numerical scheme correctly simulates the nonlinear properties of the gravity-wave dynamics
in all but the wave breaking regime. Ultimately, the constructed model is utilized to
numerically confirm predictions of the moderately nonlinear-weakly dispersive theory
2
of Prasad and Akylas, which suggests the presence of upstream propagating shelves
in a non-uniformly stratified flow through a horizontal wave guide.
Thesis Supervisor: Triantaphyllos R. Akylas
Title: Professor of Mechanical Engineering
3
Acknowledgments
I would foremost like to express my deepest gratitude to Professor Akylas for his
patience, encouragement, and guidance over the past two years. I am extremely
grateful to him for providing me with the opportunity to work on this project. It has
truly been an honor and indeed a privilege to have him as an advisor and teacher.
Furthermore, I am indebted to Professor Kevin G. Lamb of the University of
Waterloo for the assistance and support with the implementation of the numerical
algorithm. I would also like to acknowledge contributions of my officemate Ali Tabaei
and my longtime friend David Willis who have provided assistance through numerous
discussions. A special thanks to Hayder Salman for the help with typesetting. Financial support from the Air Force Office of Scientific Research (Grant F49620-01-1-001)
and the National Science Foundation (Grant DMS-0072145) is greatly appreciated.
On a personal note, I would like to thank my parents and my brother for their
support and levity during difficult times. This work is dedicated to my grandmother
Marija. Her persistence and emotional strength in gloomy moments of her strenuous
life have been, and continue to be, a true inspiration in my own ambition to live up
to extraordinary but exciting standards. I feel fortunate that she is able to witness
my longtime dreams finally come true. Thank you for all grandma.
4
Contents
10
1 Internal Gravity Waves
10
1.1
General Introduction . . . . . . . . . . . . . . . . . . . .... ...
1.2
General Properties of Internal Gravity Waves in the Ocean and the
. . . .. .
12
.
.
Atm osphere . . . . . . . . . . . . . . . . . . . . .
15
2 Physical Model
4
2.2
Simplifications and Governing Equations . . . . . . . . . . . . . . .
15
2.3
Past Undertakings
.
15
.
. . . . . . . . . . . . . . . . . . .. .. ...
18
20
Numerical Model
.. . .
Introduction ...............
3.2
Benefits of the Utilized Numerical Approach . . . . . . . . . . . . . .
3.3
Temporal Discretization.....
. . . . . . . . . . . . . . . . . - . . 22
3.4
Spatial Discretization . . . . . .
. ... ... ... ... .. .. .. .
24
3.5
Projection . . . . . . . . . . . .
... ... ... ... ... .. .. .
31
.
3.1
.
.. ..-..20
Introduction . . . . . . . . . . . . . . . . . . . .
21
38
Testing of the Numerical Scheme
38
4.1
Introduction . . . . . . . . . . .
.. ... ... ... .. ... . - - -
4.2
Uniformly Stratified Flow
. . . . . . . . . . . . . . . . . . . . 39
4.3
Nonuniformly Stratified Flow
. .. ... .... .. ... .. .. .
45
4.4
Interaction of Two Solitary Waves . . . . . . . . . . . . . . . . . . . .
55
.
3
2.1
.
.
.. . .. ......
5
5
6
60
Upstream Influence in a Horizontal Waveguide
. ..
......
. . ..
..
. . . . . . ..
. ..
. . . . .
5.1
Prelim inaries
5.2
Comparison with the Second-Order Projection Method . . . . . . . .
60
61
69
Concluding Remarks
A Linear Solution for the Uniformly Stratified Flow Over Localized
72
Topography
6
List of Figures
. . . . . .
14
. . . . . . . . . . . . . . . . . . . . . . . . . .
26
1-1
Oceanic Brunt-Viisijh frequency variation
3-1
Staggered grid system
4-1
A schematic diagram of the flow under consideration
4-2
Mode-one horizontal velocities upstream of the Witch of Agnesi topog-
(Adopted from [12])
. . . . . . . . .
raphy profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3
42
The amplitude function of the resonant mode for the cases with Ka=
0.1, D=2.0 and different values of K . . . . . . . . . . . . . . . . . .
4-4
39
44
Contour plots of the density field for the case with Ka = 0.1, D = 2.0
and K = 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4-5
Velocity modes for s=0, n=1, a=0.01, and D=20 . . . . . . . . . .
51
4-6
Velocity modes for s=3, n=1, a=0.002, and D=20 . . . . . . . . . .
52
4-7
Velocity modes for s=0, n=1, a=0.1, and D=20 . . . . . . . . . . .
53
4-8
Velocity modes for s=0, n =1, a=0.01, and D=2 . . . . . . . . . . .
54
4-9
Comparison between the moderately nonlinear-weakly dispersive theory (solid line) and the numerical model (dashed line) for s =3, n =1,
a= 0.002, and D = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4-10 Interaction of two solitary waves. Model solution (solid line), KDV
equation (dashed line), superposition (dotted line) . . . . . . . . . . .
5-1
Horizontal velocity modes at t = 260 for s = 0, n = 2, a = 0.02, and
D = 2. .. . . . . . .
. . . . . . . . ...................63
7
59
5-2
Horizontal velocity modes at t = 260 for s = 3, n = 2, a = 0.03, and
D = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3
Horizontal velocity modes at t = 520 for s = 3, n = 2, a = 0.03, and
D = 2. . . . . .....
5-4
5-6
. . .....
65
. . . . . . . . . . . . . . . . . . . ...
Horizontal velocity modes at t = 260 for s = 3, n = 3, a = 0.01, and
D = 2. . . . ...
5-5
64
..... .
..
......
...
. . ..
. ...
...
66
.. ..
Horizontal velocity modes at t = 260 for s = 3, n = 2, a = 0.03, and
D = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
Stream function at t = 260 for s = 3, n = 2, a = 0.03, and D = 2.
68
8
.
.
List of Tables
4.1
Wave breaking data for Spectral-Collocation Method (SC), Grimshaw-Yi
theory (GY), and Second-Order Projection Method. The symbol * indicates
that no wave breaking has occurred. Results presented for Ka = 0.1 and D = 2 45
9
Chapter 1
Internal Gravity Waves
1.1
General Introduction
In contrast to surface waves that propagate along a perturbed boundary separating
two distinct fluids, for instance water and air, internal waves reside within a single
body of fluid. Internal waves sustained by the restoring force of gravity are termed
internal gravity waves. Buoyancy is a physical mechanism that gives rise to internal
gravity waves. As a result, they may occur in any body of fluid stratified by either
temperature or its mineral content. In nature, they predominantly occur in the ocean
and the atmosphere where they range from small-scale shallow water micro-structures
to air-born disturbances with amplitudes of several hundred meters and velocities in
excess of 50'/. Atmospheric buoyancy induced waves are often associated with
regions of air turbulence that pose hazard to aviation. They are also accountable for
strong surface winds that blow down a mountain along its lee slope. In the ocean,
on the other hand, turbulence caused by gravity waves in the saline water flow over
uneven bathymetry may exert volatile forces on submarines and other autonomous
under-sea devices. Dynamics of oceanic gravity waves also profoundly affects pollution
disposal strategies and deep-water drilling techniques. Owing to its influence on such
a wide range of engineering disciplines, comprehensive understanding of the internal
gravity-wave dynamics can hardly be overemphasized.
In regard to the current level of understanding of this natural phenomenon, it is im10
portant to highlight that only properties of the buoyancy-wave dynamics corresponding to the flow of uniformly stratified fluid (constant Brunt-Viislii frequency) over
small and moderate size topography have been thoroughly explored. For uniformly
stratified flow over small and moderate size topography, analytical advancements were
made possible by treating the Navier-Stokes equations as linear and weakly nonlinear
respectively. Frequently in nature, however, gravity waves are induced by the flow of
uniformly stratified fluid over topography of large amplitude for which the governing
equations become strongly nonlinear and therefore analytically intractable in general.
In addition, Brunt-ViisdTh frequency varies rapidly near the atmospheric tropopause
and oceanic thermocline, which even further complicates the analysis. At this point,
numerical modeling becomes an attractive approach to studying the physical behavior
of internal gravity waves.
This document describes a nonlinear model for simulating the evolution of buoyancy induced disturbances characterized by arbitrary density stratification. In that
regard, the following section depicts typical values of fluid properties corresponding
to the propagation of internal gravity waves in the ocean and the atmosphere. The
subsequent chapter then utilizes this information to build a physical archetype for
simulating the gravity-wave dynamics. Constitutive relationships guiding the behavior of the constructed model are nonlinear partial differential equations; hence, their
solution must be determined numerically. Consequently, chapter 3 elucidates details
of the computational scheme utilized to approximate the solution of the governing
equations. Ultimately, chapter 4 presents the results associated with the testing of
the numerical algorithm while the last segment of the manuscript exploits the gravitywave simulator to investigate the occurrence of upstream influence - disturbances of
permanent form propagating against the basic flow - in a horizontal wave guide.
11
1.2
General Properties of Internal Gravity Waves
in the Ocean and the Atmosphere
A significant feature that distinguishes geophysical flows from other terrestrial areas
of fluid mechanics is the density stratification of the medium. Owing to the fact that
both dynamics of the atmosphere and the ocean are highly dominated by the density
stratification, it is possible to study them together. Accordingly, the present section
identifies typical values of fluid properties corresponding to the evolution of buoyancy
induced disturbances in the ocean and the atmosphere. This information is utilized
in the subsequent chapter to construct a physical archetype for simulating the nature
of internal gravity waves.
An important quantity related to the behavior of continuously stratified media is
the Brunt-Vdisidl
(or buoyancy) frequency. For incompressible fluids it is defined as
P
In this expression,
N denotes
the buoyancy frequency, g represents the local accelera-
tion of gravity directed in the negative
direction, and P( ) designates the hydrostatic
density. There are three physical implications of the Brunt-Vdisdli frequency:
" In order for the gravity waves to exist it must be a real quantity
* It represents the frequency of unforced small amplitude vertical particle oscillations
* It denotes the greatest possible frequency of local oscillations.
The first and second property of the buoyancy frequency may be easily portrayed
by considering a vertical perturbation of a fluid particle of density 3, initially positioned at elevation
, in a body of continuously stratified hydrostatic fluid. A small
vertical displacement,
, in the positive
where the local density is p +
p.
direction will bring the particle into a region
Hence, the parcel experiences the buoyancy force
12
gpj
oriented in the positive 2 direction. Equating this expression to the particle's
acceleration yields
pi - g P_ = 0
or utilizing (1.1)
u + N 25= 0.
(1.2)
Based on the theory of ordinary linear differential equations, it is apparent from this
result that
N2
must be positive in order for buoyancy induced oscillations to exist.
This implies that the Brunt-ViisdIk frequency must be a real quantity, which in turn
requires hydrostatic density gradient in (1.1) to be negative. Incompressible media in
which hydrostatic density decreases with height are termed stably stratified. Figure
1-1 illustrates the buoyancy frequency variation in the ocean. Atmospheric BruntVdisdId variation may be found in the work by Lim [13]. It is evident form these
3
results that in both media N is real with the magnitude of the order of 10- Hz and
the maximum value of approximately 0.01Hz.
Furthermore, if the fluid parcel is forced to oscillate along a path inclined from
the vertical axis at an angle 9, buoyancy driven forces will generate an oscillatory
motion at the reduced frequency
Ncos 9.
Analytical derivations of this result have are
portrayed by Gill [6] and Lighthill [121. This result indicates that if some oscillatory
forcing effect had a frequency cZ, it could excite oscillations only in the layer where
N(2) < cZ. The oscillations would then be trapped within the layer and could only
propagate horizontally.
In regard to the other fluid properties associated with the propagation of oceanic
internal gravity waves, it is important to emphasize that the flow velocities typically
range from few tenths of a meter per second to 2'/, for the shock waves in the
Straits of Gibraltar. Furthermore, the oceanic buoyancy waves possess wavelengths
of several kilometers while the amplitudes as high as 180m have been measured in
the Andaman Sea. The oceanic density and temperature variations are typically very
small. In particular, the total change of the hydrostatic density in the ocean is about
13
ioz6
0
1027
-
0
0.01
A
N/s-1
I
-
, /kg m-
0.005
1.5
-
0.5-
Figure 1-1: Oceanic Brunt-Viisili frequency variation
(Adopted from [12])
0.4% while the temperature varies by approximately 9%; that is, from 276K at the
surface to 300K at the depth of 5km.
In the atmosphere, on the other hand, fluid velocities are larger by a factor of
100 and are typically of the order of 10'/,. Moreover, atmospheric internal gravity
waves possess massive scales; amplitudes of several hundred meters are not uncommon while the wavelengths are frequently of the order of several hundred kilometers.
Ultimately, it is important to emphasize that the total change of the temperature
in the atmosphere is approximately 13%; that is from 288K at the surface to about
335K in the tropopause.
14
Chapter 2
Physical Model
2.1
Introduction
The previous chapter depicted typical values of fluid properties associated with the
propagation of internal gravity waves in the ocean and the atmosphere. The objective
of this portion of the document is to portray the manner in which this data was utilized
to construct a physical archetype for simulating the evolution of buoyancy induced
disturbances.
In that regard, section 2.2 foremost identifies physical mechanisms
that play a principal role in defining the character of this natural phenomenon. It
then exploits this information to formulate mathematical expressions for modeling
the behavior of gravity-wave dynamics. Ultimately, section 2.3 provides an overview
of the past undertakings in field of simulating the dynamics of internal gravity waves
2.2
Simplifications and Governing Equations
It is foremost evident from the information presented in the previous chapter that
the Mach number - defined as the ratio of the magnitude of the fluid velocity and the
local speed of sound - is much smaller than unity. In particular, the speed of sound in
the atmosphere is approximately 3. 102m/ while typical fluid velocities, as illustrated
in the previous chapter, are of the order of 1Om/,. Hence, the Mach number is in the
neighborhood of 0.03. In a similar fashion, the speed of sound in the ocean is about
15
1.5. 10 3 m/s while commonly measured buoyancy-wave flow speeds are of the order of
few meters per second. Thus, the Mach number is about 0.001. Owing to the fact
that the Mach number corresponding to the propagation of the internal gravity waves
in the ocean and the atmosphere is much less than 0.3, say, it is reasonable to model
these media as incompressible.
Furthermore, it can be deduced from the data presented in section 1.2 that due to
large length scales the Reynolds number associated with the evolution of buoyancy
induced disturbances is very high. The Reynolds number for gravity-wave flows is
customarily defined as
Rex =
(2.1)
where ,o and A^denote representative values of fluid density and viscosity respectively,
U designates the characteristic value of the velocity, and ^ depicts typical value
of the wavelength.
Based on the data presented in section 1.2, it can be shown
that the Reynolds numbers corresponding to the gravity-wave flows in the ocean and
the atmosphere are roughly of the order of 10' and 106 respectively. As a result,
the viscous effects are confined to thin boundary layers; hence, both media may be
perceived as inviscid.
Moreover, it is important to emphasize that due to large horizontal length scales
and small fluid velocities, the Rossby number defined as
U
Ro = U
(2.2)
is much smaller than unity. Q in (2.2) denotes the angular velocity of Earth and it
equals 13,757 rd/,.
Consequently, effects of the Coriolis acceleration may be neglected
in examining the character of the gravity-wave dynamics.
Next, owing to the fact that the nature of topographic barriers is often such
that one of the dimensions is much larger than the others, the buoyancy-induced
disturbances may be regarded as two-dimensional. However, it is significant to point
out that this is an assumption valid far away from the barriers' ends. The structure
of the flow near the ends is highly three-dimensional.
16
Ultimately, because density changes associated with the gravity-wave dynamics
are typically small, one may model both ocean and atmosphere as Boussinesq fluids.
In other words, one may neglect the change in all fluid properties except density.
Moreover, density changes are neglected everywhere except where they give rise to
buoyancy forces.
Substituting the aforementioned simplifications into the governing equations, namely,
Navier-Stokes equations and the continuity equation, yields
yo(ft; + ftf2! + T66) = -
,O(z; + fil0 + lit)
In these expressions, X and
=_-i
+i 130
(2.3a)
P)-
(2.3b)
-
Pi + fip. + W2 = 0
(2.3c)
fl! + Tb2 = 0.
(2.3d)
denote lengths in the horizontal and vertical directions
respectively. In a similar fashion, U^and ti represent components of the velocity vector
in the , and
directions. Moreover, I depicts time while p represents perturbation
pressure. Ultimately, P and b signify the fluid density and the body force per unit
volume of the fluid respectively while PO denotes the characteristic value of density.
Making all lengths in (2.3) dimensionless with a characteristic height,
f,
time
with a representative value of the buoyancy frequency, N 0 , and all velocities with the
product ftko yields
ut + uux+ wuz =-px + b
Wt
+
UWX + WWZ =
Pz - r
(2.4a)
(2.4b)
rt + ur + wrz = wN 2
(2.4c)
uX + wZ = 0
(2.4d)
p = P(1 + #r)
(2.5)
where
17
and
N#2=
(2.6)
9
Lastly, the values of the characteristic parameters employed to make the governing
equations dimensionless will be clearly emphasized for each problem considered in the
following developments.
2.3
Past Undertakings
From the historical standpoint, gravity-wave dynamics of inviscid incompressible
Boussinesq fluid over two-dimensional landscape has been the subject of extensive analytical study since the ingenious work of Long [14]. Specifically, in 1955 Long demonstrated that nonlinear equations governing the steady uniformly stratified Boussinesq
flow may be transformed into a linear one by assuming that all streamlines originate
upstream. Long portrayed that physically consistent solutions associated with his
model exist if the flow is not close to resonance. Near resonance, closed streamlines
develop, indicating that steady solution cannot occur and that internal wave-breaking
may take place.
In 1972, McIntyre [15] developed a solution for unsteady flow of uniformly stratified fluid as an expansion in powers of the landscape height. Solutions of the linear
initial-value problem revealed that at resonance the topography forces an internal
wave mode that grows indefinitely with time. Hence, small topography gives rise to
large amplitude internal waves and nonlinear behavior.
Grimshaw and Smyth [7] in 1986 derived a forced Korteweg-de Vries (fKDV)
equation that describes the amplitude evolution of the upstream component of the
resonant mode for arbitrary stable density stratification and uniform mean flow. Their
analysis is based on the balance between weak dispersion and weak nonlinearity;
hence, it is valid for the moderately elevated topography. The approach fails, however,
in the limit of uniform stratification because nonlinear terms in their expansion vanish
and therefore cannot balance the dispersion. Consequently, Grimshaw and Yi [8] in
1991 proposed a new evolution equation for the resonant flow in uniformly stratified
18
flows. Their analysis is valid for the landscape of small amplitude and moderate slope.
Grimshaw-Yi (GY) is capable of tracing the evolution of finite amplitude disturbances
up to the onset of wave breaking. Predictions of GY theory have been numerically
confirmed by Rottman, Broutman and Grimshaw [17] in 1996.
Based on the numerical simulations by Lamb [10], Prasad and Akylas [16] in 1997
observed that the GY equation is not uniformly valid far downstream of the topography where multiple fronts or 'shelves' are generated. They also noted that although
these shelves have relatively small amplitude, they are driven by the "nonlinear interactions precipitated by the transience of the main disturbances over topography".
Moreover, they suggested that in the case of nonuniform density stratification shelves
may propagate upstream as well as downstream. Hence, in the last chapter of the
manuscript, the constructed physical model is utilized to numerically verify this hypothesis.
19
Chapter 3
Numerical Model
3.1
Introduction
The solution of nonlinear governing equations, namely (2.4a) through (2.4d), is approximated using a Second-Order Projection Method for the Incompressible NavierStokes Equations, originally developed by Bell et al. [2]. Bell and Marcus [3] extended
the approach to variable density flows while Bell, Solomon, and Szymczak [4] adopted
the formulation to quadrilateral computational grids. In the following developments,
the numerical scheme is further modified to allow specification of the initial density
field in terms of the buoyancy frequency.
The method is a second-order fractional step scheme in which nonlinear convective
terms and reduced density are foremost advanced in time by the amount A without
imposing the incompressibility constraint.
The resulting intermediate vector field
consisted of the sum of these two terms is then projected onto the space of discretely
divergence free vector fields. The projection yields the value of Ut at the intermediate
time level. Velocity field at the time level n+1 is then obtained from the knowledge
of U at the time level n and Ut at the time level n+ }.
This chapter illustrates implementation details of the numerical scheme applied
to the solution of equations (2.4a)-(2.4d). Specifically, section 3.2 addresses benefits
of the utilized numerical approach while section 3.3 provides details of the temporal
discretization. Furthermore, sections 3.4 and 3.5 depict spatial discretization and
20
projection portions of the algorithm respectively.
3.2
Benefits of the Utilized Numerical Approach
From the numerical standpoint, the problem of approximating the solution of governing equations may be classified as two-dimensional, nonlinear, and unsteady. Although in this study the bahaviour of buoyancy induced disturbances is considered in
two-dimensional limit, accurate capturing of the complex gravity-wave structure necessitates large and densely spaced computational grids. Accordingly, it is imperative
that the chosen numerical scheme performs its task with the minimum expenditure of
computational resources. It that regard, the Second-Order Projection Method possesses the following properties that make it a suitable choice for solving equations
(2.4a) through (2.4d).
* In the Boussinesq limit, the projection matrix associated with the linear algebra
problem P -a = S that needs to solved at each time step does not need to be
updated with time.
* The algorithm treats nonlinear convective terms in a non-iterative manner.
* The projection matrix is symmetric, positive-definite, and block-tridiagonal.
These matrix properties maybe utilized to rapidly obtain the solution of the
linear system at each time step.
" The Second-Order Projection Method is a finite difference scheme that is second
order accurate in both space and time.
" Lamb [10] has demonstrated the ability of the method to approximate the solution of a similar problem. In 1994, he has utilized this numerical approach to
simulate the structure of two-dimensional, inviscid, and incompressible flow of
uniformly stratified Boussinesq fluid over smooth topography.
21
3.3
Temporal Discretization
As indicated in section 3.1, the Second-Order Projection Method is a fractional step
scheme. The fractional step method is a technique of approximation of evolution
equations based on a decomposition of the operators. The concept may be readily
illustrated by considering the problem of numerically approximating the solution of
the evolution equation
Ut +AU = 0,
U(x, z, 0) = Uo.
(3.1)
where following the nomenclature of the previous chapter U denotes a two-dimensional
velocity vector with components u(x, z, t) and w(x, z, t) oriented in the i and k directions respectively. Moreover, Uo represents the value of U at time t = 0 while A is a
2 x 2 matrix. The solution of the mathematical problem contained in (3.1) is usually
numerically approximated via standard implicit approach
Un+ 1 _ U + AUn+ 1 = 0.
At
(3.2)
Temam [18], however, points out that the solution of (3.1) may also be computationally approximated by utilizing the fractional step scheme
Un+1q+ - Un.+iq
At
AZUn+! = 0
(3.3)
where q denotes the order of the scheme. Correspondingly, for a second-order fractional step scheme, q = 2. Hence, (3.3) yields
U" + A 1 Un+l = 0
-(.a
At
U n+.!2 -
Un+1 - Uni + A2 Un+ 1 = 0.
At
+Z1
(3.4a)
(3.4b)
As indicated by Temam, this scheme can be adopted to the Navier-Stokes equations
in many ways corresponding to the many possible decompositions of operators. For
22
[(V -U) U]'+2
the Second-Order Projection Method A 1 U'+1 =
-
r+21
A2
while
is
a heuristic operator that involves Vp and the incompressibility condition, V -U = 0.
Before elucidating further details of the time discretization, it is important to
emphasize that the solution of the governing equations (2.4a)-(2.4d) is numerically
approximated in the computational space E = ( , r1) defined via transformation
X
= <
(3.5)
(EE)
from the physical space X = (x, y). In this space, the governing equations have form
U + -[(U-V-) U]=
J
J
TtVp + bi- rk
(3.6a)
12
rt + -J[(U-V=) r] = wN 2
(3.6b)
V= - O = 0
(3.6c)
where V=- and V= denote the divergence and the gradient operators in the computational space while T designates the transformation matrix
T =
z
Xq
(3.7)
Moreover, J represents the Jacobian of the transformation
J
= xCz, -
XqzC
(3.8)
while
V = TU.
(3.9)
In accordance with the second-order fractional step scheme, the value of Un+, is
obtained by first explicitly calculating convective terms in (3.6a) and (3.6b) at time
level n + 1 from the knowledge of U", r", and Vp"--. Details of this computational
procedure are presented in section 3.4. Once, the values of the convective terms at
23
the intermediate time step are known, reduced density at t'+' is obtained by first
calculating the value of r at the time level n + 1 via
r n+1 - r n
At
where
1
1
= -[(U
- V=)r]n+i + wn+! N2
J
(3.10)
is attained from
Wn+
+
3
n
_
1
__
2
2
-
(3.11)
rfl2 is then computed by averaging the values of r" and rn+1. Furthermore, the
projection vector field is constructed as
V-
_[(U.
J
The value of Ut
2
V=)Un+- + bn+i
i
r
-
=U
+ !Tt
J
pn2i.
(3.12)
is next obtained by projecting Vn I onto the space of divergence
free vector fields. This operation can be mathematically expressed as
fl(Vn+-)
(3.13)
= Un+
where fJ denotes the projection operator and its numerical formulation is presented
is known, TtVpn+i is computed as
in section 3.5. Once the value of U
J_
=
Vti
(3.14)
-
!TtVpn+I
Ultimately, the velocity field is updated using the central differencing approach
Un+ 1
-
Un
n+1
=tU=
3.4
2.
(3.15)
Spatial Discretization
This segment of the document provides an overview of the spatial discretization portion of the numerical algorithm. Specifically, the section illustrates the computation
24
of convective terms [(U- V=) U n I and [(U- V=) r],+i. Spatial discretization is performed on a staggered grid depicted in figure 3-1 (a). On this lattice, quantities p and
V- are specified on the primary grid points marked with o while quantities U, V, and
r are defined on the secondary grid points labeled with x. The transformation 4I
in (3.5) is defined such that the grid in the computational space, depicted in figure
3-1(b), is composed of unit squares; hence, A6 = Aq = 1. On the computational
lattice, the primary grid point (i, j) has coordinates (6, q) = (i, j) for i = 0, -- -, I
and j = 0, - --, J. The secondary grid is consisted of the set of interior grid points
and the set of boundary grid points. Interior secondary grid points are located at
cell centers with the point (i, j) having coordinates (i - 1, j - 1) for i = 1,
j=
,I
and
1,--- , J. Boundary secondary grid points, on the other hand, are specified at
the midpoints of cell edges that lie along the boundary of the computational space.
They have coordinates (0, j - 1) and (I, j boundaries respectively, and (i - 1, 0) and (i -
j) for j = 1, },
J) for i =1,
, J at the left and right
, I at the bottom and
top boundaries correspondingly. Moreover, boundary secondary grid points occupy
four corners of both physical and computational space where they coincide with the
primary grid points as illustrated in figure 3-1.
The computational lattice in the physical space is formed by first choosing coordinates of the points associated with the primary lattice. The secondary grid points
are then computed using
Xi' X
1
4,
= (_,_1 +XT-_1,3 +X%,!_1j + X;,%
(31a
XXTy)T
(3.16a)
for the interior points and
1
X
=
(XU + XU,3_1)
(3.16b)
for the left boundary points. Coordinates of the grid points lying along the other
domain boundaries are obtained using analogous formulae. In expressions (3.16), bars
are utilized to emphasize that the grid points under consideration are of the primary
nature.
25
- 11 11 11 111
x
x
x
x
x
x
x
x
x
I
x
xx
x
x
x
x
x
x
x
x
X
X X xx X x
x
x
x
x
x
x
X
x
x
x
x
x
X
X x XX x x
x
x
x
x
x
x
X
x
x
x
x
x
x
x X XX X
x
x
x
x
x
x
x
X
x
x
x
x
x
x x x x
x
x
x
x
x
x
x
x
X
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x X XX X
x rI
x xx x
x x
xx
x
x
x
x
X
x
x
x
x
x
x
K-1 x xx x 1--f x
x
)4 )1 x
x x 4r-, Y'.4
x
x
x
x
x
x
x
x
x X
x
x
x
x
x
x ()I x
x
x
x
x
x
x
x
x
x
x
xx
,
C
x
x
x
x xx x x
x
x
8n
(a) Staggered grid in the physical space
X
x
x
X
x
x
x
x
x
A,
'k
X
x
x
x
x
x
IC
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x X
A,
A,
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
X
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
X
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
X
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
C
A
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x X
Cx
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x (1)
)( E) x
x E) x a x E) ' E) x E) x E) x
X E) X E)
(b) Staggered grid in the computational space
Figure 3-1: Staggered grid system
26
Once the coordinates of the grid points in the physical space are known, components of the transformation matrix (3.7) are obtained numerically from the primary
grid points using
(X
Xt i~i =
X~,-=
1
1
Xi_,
1
+ X
(X_,_
+X,5_
+ X;,)
(3.17a)
X;,
+ X;,)
(3.17b)
-1,
for the interior secondary points and
X 01i = 3XC 1 jX71
=
X6,3
-
X
(3.17c)
2,j
(3.17d)
X63-1
at the left boundary. Equivalent expressions may be readily obtained for the other
domain boundaries.
and [(U. Vs) r
The next step in the computation of [(U1 V) U
is an
extrapolation of Un and r" to predict their values on cell edges at time level n+}.
This task is accomplished using Taylor series expansion. Hence, for the edge with
coordinates ((,q)=(i+ ,j) this gives
uL,
=r , + 2
+
(3.18a)
,
At
(3.18b)
"
rIn
n ,
-
r+ 3'
U" ,,
U i'j +
+
Un+ui+. L= U!,+
2,) +
when extrapolating from the secondary grid point i, j and
S+
n
U"t i,,
(3.19a)
-
+ 1,j
1
n+ - IR
2'
r
in
=
At
+ 2r +,
+
r
At
=R
-
n+ 1
U 2
r
(3.19b)
when extrapolating from the secondary grid point i+1, j. Expressions (3.18) represent
the extrapolation of Un and r" to the left side of edge i+ , j while (3.19) denote the
27
extrapolation to the right side of the edge. Analogous expressions may be formulated
for other cell edges. Utilizing the governing equations (3.6a) and (3.6b) to express
Un+-'L = Un
+
the time derivatives gives for (3.18)
2,
-At
SL
(3.20a)
-n
At
n-1
2J
n+.! L
(1
21~
At
n-
(3.20b)
+ 2 (b" I- rij k
At -n
n At._
2J UiJ
n-2J
2
(3.20c)
U i l
(3.21a)
At
2
,723
and for (3.19)
n+
ui+I~j
=,
At Cn
+At fnU
Uni+1,j - SR
At +
n-f.1
i++1 U
-~j+
2J-
-
+f r
+ At
2 ~J
-
rn+! R
SR
ri+1d k)
+
+
At
2 (b"
'+ i)
-+1
2 +1,jr 77i+,j
(3.21b)
+2
where SL is 1 if sii
> 0 and 0 otherwise while SR is 1 if iii,
< 0 and 0 otherwise.
In these expressions, derivatives normal to the cell edge, namely Un, are evaluated
using central differences
1
1
U U+
2U'_1".
(3.22a)
On the left and right boundaries, (3.22a) is modified as
Un
U 1 ,j
-
~4 U
=
0," + U"n. +
4 ~
Unjrj = 4Un+1d -
U
U
U1. j
(3.22b)
(3.22c)
The transverse derivative is evaluated using an upwind difference approximation. In
28
particular,
UTfl - U(+
n
U1
U2j+
-
if i!l > 0
Ut ,) - U",_
(U
1( -
(3.23a)
2
T-U,-
1 +
U20s1
U"ij)
otherwise
for the interior cells and
Uni,1 = 2(U, 1
-
Uo)
Un";, = 2(Un,+1 - Un)
if
(3.23b)
jgn > 0
if
(3.23c)
fi4j < 0
for the top and bottom boundary cells. Density derivatives are obtained in the
analogous fashion; that is by replacing, U" with rn in (3.22) and (3.23).
At this point, it is important to emphasize that so far we have addressed computing
the values of fluid velocity and reduced density on the vertical cell edges at time level
n+!. Expressions for the horizontal edges may be obtained using analogous formulae.
Furthermore, before evaluating the flux, one must first resolve the ambiguities in
edge values introduced by (3.20) and (3.21). In particular, the characteristic extrapolation has defined double values of U"+2 and rn+- for each interior domain edge corresponding to expansions from either side of the interface. Due to reasons explained in
[4], up-winding of ii is based on the Riemann problem for Burgers' equation, namely,
jUL
if 11L > 0
ui+.
if iiL < 0,
0
2,3
uL + iR > 0
R
(3.24)
0
otherwise.
U and r are then up-winded based on i
UL
U1 .i.j,
ri1 i,3 =
if
rL
if
UR, rR
U(UL +U R
L + if
29
i+,
> 0
fli+I,j < 0
i+I
0.
(3.25)
=-U.
It is important to emphasize that all values in (3.24) and (3.25) are evaluated at
i, j+1!22 and time level n+ !.
The indices are suppressed for clarity. Up-winding, on
the horizontal edges is conducted in the equivalent manner.
Furthermore, the convective terms at the interior secondary grid points are com-
- V=-)U];j
~~ 1-(
i+Ij +6; igg-')(Ui+-1' -ui U
1
+ 1( ij++ 7gj!)(Uig+! - U
_)
(3.26)
where it is understood that all quantities in (3.26) are computed at time level n+
The values of [(-
)U
.
[(
)
puted from the predicted values of u,w, and r at time level n+ . via
I and [(T-VE)r]n i corresponding to the top and
bottom boundaries are evaluated by foremost recognizing that at impermeable boundaries
zo=0.
(3.27)
Furthermore, at top and bottom inviscid boundaries
W= Lt .
(3.28)
and
U
(3.29)
Accordingly, the convective portion of both the fluid acceleration and the continuity
equation are obtained by foremost extrapolating the horizontal component of the
velocity and reduced density to the cell corners via (3.18) through (3.22). In order
to ensure that the the conservation of mass at the top and bottom boundaries is
satisfied, the values of
and
+
are then obtained using (3.28) and (3.29)
respectively. Next, the up-winding of u, U, and r is conducted in accordance with
(3.24) and (3.25). The upper and lower boundary convective terms are then computed
using (3.26).
and [(U-V")
Ultimately, the values of [("Vs) U
30
associated with the
,
inflow and outflow and boundaries are evaluated via extrapolation form the interior;
namely,
Ko, =
Kj+13 =
where K denotes [(2Vs) U
3
3
1
-K2J
1K
Kr'
-
(3.30a)
1
2K-ij
(3.30b)
or [(2.V.) r]n i.
Finally, it is important to emphasize that the Godunov method is an explicit finite
difference scheme; accordingly requires a time step restriction. Accordingly,
(3.31)
max
<
for stability.
3.5
Projection
The numerical problem that must be solved is the following. Given the intermediate
vector field Vn+ ', one must solve
Vn+- =
+Vp"-'-
(3.32)
Un+.!2 such that V-Utn+ 12 =0 and V x Vpn+ 1
for Ut
=0. Following developments of Bell
et al. [2], this task is accomplished by constructing a basis of divergence free vectors,
T, and equating Ut2 to
Ut
=
5 a'ITT'l
(3.33)
where the summation limits remain to be determined. The problem then reduces to
computing coefficients a.
The basis of divergence free vector fields is constructed by foremost noting that for
any scalar field q, the vector field Oi - 0,qk is divergence free. In the computational
31
space, this result may be expressed as
Ao
s.t
#b
_ig_1
(3.34)
Accordingly, let OP' be the scalar field with values
{
=
at scalar grid point k, 1
2
(3.35)
0 elsewhere.
Then
(3.36)
T,) =Tyields a divergence free vector in the computational domain. Values of
#
', and
#',
are computed using second-order central differencing, namely,
1(
,
1
--
(
2~
+
--
_1'j_)
-1
-
(3.37a)
-
_IkI(3.37b)
)
+
--
+
-lk
(
kI
-- I
4-
-
-
--
These expressions are modified in a straight forward fashion along the boundaries.
The basis vectors corresponding to quadrilateral computational grid have been derived
by Lamb in [11]. Accordingly,
(-x
+ X, 1 1 -Zt + z77)
(xt + X77, zt +
Tk ' =
1(-Xt - Xq, -zt
z7)
- Z,7)
(Xt - x,,, zt - z,)
0
for
at vector grid point (k + 1,1 + 1),
at vector grid point (k + 1, l),
(3.38)
at vector grid point (kl+ 1),
at vector grid point (k, l),
elsewhere.
-, J-1. On the left boundary, for
=1, 2, .. , I-1 and 1= 1,2, ..
32
=
and
m
, J- 1
basis vectors have values
/
1=1, 2, ...
(-x, +
J(XC
=
, -z) + Z7)
+ X17, ZC + ZO)
at vector grid point (1,1 + 1),
at vector grid point (1, 1),
(-2xC, - 2zC)
at vector grid point (0, 1 + 1),
(2xC, 2zC)
at vector grid point (0,1),
S0
(3.39)
elsewhere.
Ultimately, on the bottom left corner of the computational domain
4(-x + x,, -z
(2x,, 2z,7)
at vector grid point (1,0),
(-2x , -2zC)
at vector grid point (0, 1),
-
T
+ z,) at vector grid point (1, 1),
(3.40)
elsewhere.
0
The analogous formulae can be constructed for other boundaries and domain corners.
These results are presented in [11].
Furthermore, it is important to emphasize that divergence free vectors constructed
above are not linearly independent. As illustrated by Lamb [11], the basis of divergence free vectors is formed by removing one of the boundary vectors. Hence, the
vector T7' is removed. Consequently, the basis of divergence free vectors is formed
by the remaining collection of (I+1)(J+1)-1 vectors.
It has been pointed out in section 3.2 that in the Boussinesq limit it is possible to
eliminate the boundary dependence of the projection. In that regard, we construct
a divergence free boundary vector VB which satisfies boundary conditions along the
left (inflow), top, and bottom boundaries. Hence, we now have the problem
VD + V
m
(3.41)
- VB
(3.42)
-= V
where
VD = U
33
and
Vm=V-VB
(343)
where it is understood that all quantities are evaluated at time level n + '. At this
point, it is important to emphasize that VD in addition to being divergence free has
homogeneous boundary conditions along all but the right boundary. Correspondingly,
one may write
I J-1
E akfk,,
_
VD
(3.44)
k=1 1=1
Furthermore, in order to motivate the treatment of the right (outflow) boundary
condition following Lamb [11], let us suppose that
J
- X,-X.) d dz =
Xz, -Xx) d dz +ILR X F - tds (3.45)
ILVD
where the integration is performed over the entire physical domain R for all X which
are equal to zero along the top, bottom, and left boundary (denoted by OR). Here,
t is the unit vector in the counter clockwise direction along OR. The application of
the divergence theorem yields
X(V x VD
V x Vm) dx dz=
X(Vm-VD - F) - tds
(3.46)
If this expression is to hold for all X which are zero on the left, top, and bottom
boundaries then it must follow that
V X VD = V X Vm
(3.47)
(VmVD - F) - t = 0
(3.48)
inside R and
along the outflow boundary. Using (3.41) in the latter it follows that
Vp -t = F -t.
34
(3.49)
Due to the fact that the right boundary is a vertical line it follows that
(3.50)
Pz = F -k
Thus, one needs to choose F so that its vertical component is equal to pz. A significant
advantage of this approach is that one does not need to specify Ut at the outflow
boundary through which there are often numerous waves passing through.
Once the outflow boundary condition has been constructed, the stage has been
set for determining the values of the unknowns ao'J. This is accomplished by solving
a collection of I(J -1)
equations that result when (3.44) is substituted into (3.45)
Here, over-
J-1 and V's = (Xz, -Xx).
T =1, 2, ...
with X = 0's" for -=1, 2, .- -I,
line indicates that the summation index is associated with the primary grid points.
Moreover, one should not confuse the summation index w with the vertical component
of the velocity vector. Hence,
T
fI R'5
F -t ds (3.51)
V" -Vdx dz +
ik
k=1 1=1
In this expression, Tki - TV'7= 0 except for the nine values (k,1) = (q
(q-- 1, T), (q
-1, +1),7
(4,
-
)
U,7), (-q, i + 1), (q +- 1, i -1),
-
1, w
-
1),
(q W+')1, ),
(q+1,w + 1). This results in the equation
Aa '*a -1'w-1 + Ab',aq-1,' + Ack'*a ~1,w+1
+B0az'4w'- + Bbz'ua4'T + Bcz'*a4,w+1
+Caf'7a
'w
+ C'Oaq+l,: + Ccku*aq+1,w+1
S4,' + Fw
(3.52)
where as indicated by Lamb [11]
AOi' = f fR Tq~'-1 - Yqu dx dz
35
(3.53a)
I
Abs' =
- T4,'7 dx dz
T
AcI'y = f
Ba4'W =
Cai' =
f
CAW
=
- T'7 dx dz
JR
and
Sq'U =
(3.53e)
- T4'T dx dz
Tl'V
1
JR
J
(3.53f)
(3.53g)
- T' dx dz
TJR
J JR
J
(3.53d)
- T-'- dx dz
I
J
J
Cbq'U =
(3.53c)
- TV'," dx dz
Bbs'u =
BcI'7 =
(3.53b)
(3.53h)
dx dz
-lwA
T'u dx dz
(3.53i)
V - TI'7 dx dz
(3.54)
where
Fw= f
'OR
(3.55)
q$I7WF, dz.
Equation (3.52) may be expressed in the matrix-vector form as
(3.56)
Pa = S
where the projection matrix is tridiagonal with each element being tridiagonal as well.
That is,
B 1 C1
A2
0
0
0
0
---
0
C2
0
0
...
0
A 3 B3
C3
0
...
0
.
.
.
.
B1-2
C1-2
0
B2
P.
(3.57)
0
-
0
A 1-
0
...
0
0
A 1 _1
BI-1
C-11
0
...
0
0
0
A1
BI
2
36
/
with the block matrices of the form
Qci
Qbi
0
Qa2 Qb 2 Qc 2
0
Qa3 Qb 3
Q
for
Q=
0
0
0
0
0
0
Qc 3
0
0
=(3.58)
0
...
0
Qaj-3
Qbj- 3
QcJ- 3
0
...
0
0
Qaj-2
QbJ- 2
QcJ_
0
...
0
0
0
Qaj_1
Qbjl
0
2
A, B, and C.
Lastly, as indicated by Lamb [11], the boundary vector VB that satisfies the
boundary conditions along the top, bottom, and the left boundary can be expressed
in terms of the basis vectors. Thus, one may write
I-1
VB =
I
Z
V
+
q=O
ZJ
_
J-1
_
_
+ E a'T'TO'.
(3.59)
w=1
q=O
The impermeable upper and lower boundary conditions result in
(3.60)
and
=a'
a
=a
=-
=a
'
a
(3.61)
Due to the fact that the right boundary is vertical, one gets
ao'W-1 = au'W + -z Ut - ii.
(3.62)
where it assumed that the value of Ut at the left boundary is known. Ultimately, the
source terms are given by
S4'W =
J
V. -a dx dz -
J
JR
VB
37
- '4" dx dz.
(3.63)
Chapter 4
Testing of the Numerical Scheme
4.1
Introduction
The previous chapter presented details associated with the implementation of the
Second-Order Projection Method for Variable Density Flows, a computational scheme
utilized to approximate the solution of governing equations 2.4. Here we evaluate the
performance of the numerical approach by comparing its predictions to known theoretical results and numerical results attained using other computational techniques.
Accordingly, section 4.2 appraises the ability of the scheme to simulate the evolution
of buoyancy-induced disturbances created when a uniformly stratified fluid is forced to
rise over a topographic barrier. The results of the model are compared to McIntyre's
linear theory, which is valid for topography of small amplitude and flow conditions
that are far away from the resonance of linear modes. The competence of the scheme
in the flow regimes beyond the applicability of the McIntyre's archetype is investigated through the comparison with computational results of Lamb and Rottman et
al. Furthermore, sections 4.3 and 4.4 examine the aptitude of the scheme to simulate
the gravity-wave dynamics in a fluid layer characterized by arbitrary density stratification. Specifically, section 4.3 compares the predictions of the model to the limiting
cases of the long-wave theory while section 4.4 assesses the ability of the simulator to
correctly interpret the interaction of two solitary waves.
38
4.2
Uniformly Stratified Flow
The present section is concerned with assessing the ability of the numerical model
to simulate the evolution of buoyancy-induced disturbances in a flow of uniformly
stratified fluid. In the subsequent developments, the disturbances are caused by the
presence of a stationary topography in a channel of depth H that is bounded above by
a horizontal rigid lid. The topography is presumed to be locally confined, symmetric,
and streamlined with maximum amplitude a and half-width D. It is furthermore
assumed that the fluid is impulsively accelerated from rest via arbitrary body force
(that is only a function of time) such that its velocity far upstream and downstream
of the topography has a uniform value U, as illustrated in figure 4-1.
U
h (x)
H
ND
Figure 4-1: A schematic diagram of the flow under consideration
For the landscape of small amplitude compared to the channel height, McIntyre
hypothesized that that the response of the fluid to the presence of the topography is
proportional to the amplitude of the topography. Accordingly, he developed a perturbation solution of the nonlinear governing equations in the form of a power series
in a small parameter e, defined as the ratio of the maximum topography amplitude
and the channel height. To the leading order in E, the governing equations linearize
and their solution may be expressed as
IF(X, Z, t) =
+0 ekkx x(k, z, t)dk
+ (k, z, t) + F (k, z, t)
T (k, z, t) = V'(k, z) +
39
(4.1)
(4.2)
where T denotes the perturbation stream function defined as
U = U + "F
w
=
(4.3)
(4.4)
-. ,
while U designates the background velocity. Furthermore,
*(k,
sin[
z) = -Uh
](4.5)
sin[VIT
- k2l
00
1 (k, t) sin (nirz)
4-(k, z, t) =
(4.6)
n=1
T (k, t) = -F
n7rcnhUe-ik(U cn)t
h(x)e-ikxdk
(4.8)
1 + k2
(4.9)
h(k) =
C
(4.7)
-/n27r2
where h(x) is the topography profile.
It is apparent from equation (4.2) that McIntyre's linear solution is comprised of
a steady part Vi, and the transient component 5+(k, z, t) + 4- (k, z, t). Moreover,
the transient part consist of a sum over all modes with each mode having two components; namely, one that propagates with the background flow and the another that
propagates against it. For the purpose of comparison with the numerical results of
the Second-Order Projection Method, it is important to highlight that McIntyre's
theory is valid for flow conditions such that the Froude number
K = U7.
NOH
(4.10)
in not an integer as illustrated in [15]. Details associated with the derivation of the
McIntyre's theory are presented in Appendix A.
40
In figure 4-2(a), numerical results using a 'Witch of Agnesi' topography profile
1+
()2(4.11)
for K = 1.5 and D = 1.6 are compared with the predictions of the linear theory for
different obstacle amplitudes. Following Lamb [10], the two responses are contrasted
in terms of the first horizontal velocity mode, U1 , obtained by decomposing the xcomponent of the velocity vector as
U(,Zt
+01t
U(x, z, t)=1 - h(x)
m11
o
(x,
z - h(x)
m7r
(.2
- h(x))
Far away from the obstacle where h is effectively zero, this expression reduces to
00
U(x, z, t) = 1 + E a Um(x, t) cos(mirz).
(4.13)
m=1
It is apparent form the figure that for topography amplitudes of 0.002 and 0.01, the
results of the numerical model closely correspond to the predictions of the linear theory. Nevertheless, as the value of a is increased to 0.05, the computed amplitude of U1
is larger than the linear theory predicts. Behavior of that nature is expected because
as the topography amplitude enlarges, the fluid response becomes more nonlinear;
correspondingly, McIntyre's linear model becomes inadequate tool for replicating the
intricacies of the gravity-wave dynamics. Due to mathematical complexity of Euler's
equations, there is no analytical theory that describes the gravity-wave dynamics of
strongly nonlinear geophysical flows which take place, for instance, when stratified
fluid is forced to rise over a large amplitude topographic barrier. Accordingly, the
performance of the numerical scheme in this flow regime is evaluated through the
comparison with numerical results of Lamb who employed the same computational
approach for his simulations of uniformly stratified flows over smooth topography
[10]. Visual comparison of U1 profiles for weakly nonlinear case (a=0.05) and the
strongly nonlinear response (a=0.13) in figure 4-2(a) to their counterparts presented
in Lamb's figure 2, reveals that the results are virtually identical. Furthermore, fig41
irI
I
2
1
N
I..'
'
I
~
/
-,
-1
-2
-3
-7 0
-60
-50
-40
-30
-10
-20
(a) Mode-one horizontal velocities at Lamb's time tL = 100 for
K = 1.5 and D = 1.6 for the following responses: theory (solid
line), a = 0.13 (bold solid line), a = 0.05 (dashed line), a = 0.01
(dotted line) and a=0.002 (dot-dashed line)
2.5
2
-
1.5
1
0.51F
0
-
-0.5
-1
1.5
-2'
-400
-350
-300
-250
-200
-150
-100
-50
0
x
(b) Mode-one horizontal velocity at time tL = 650 for K = 1.5,
D=1.6, and a=0.13
Figure 4-2: Mode-one horizontal velocities upstream of the Witch of Agnesi topography profile
42
ure 4-2(b) depicts the first horizontal velocity mode for the strongly nonlinear case
at Lamb's time tL = 650. The contrast with Lamb's figure 3 again reveals that the
results are practically identical far upstream of the obstacle. However, in the vicinity
of the barrier - where (4.13) does not hold - there is a slight discrepancy, which is
attributed to difference in numerical implementation of the modal decomposition.
The second flow regime that is outside the validity range of the McIntyre's model
takes place when the Froude number becomes an integer. For the flow conditions of
this nature, one of the modes in (4.7) becomes resonant and the upstream propagating
portion of (4.2) grows with time. Ultimately, the amplitude of this mode becomes
large enough to invalidate the assumption that the flow response is proportional to
the topography amplitude.
The competence of the numerical scheme to simulate the gravity-wave dynamics
of the resonant flow is examined though the comparison with the numerical results
of Rottman et al. [17] who modeled the evolution of topography induced buoyancy
waves using the Spectral Collocation Method. Figure 4-3 presents the time evolution
of the first amplitude mode, A 1 , for Ka=0.1, D=2 and three different values of K,
namely, 0.95, 1.00, and 1.05 respectively. The first amplitude mode may be readily
deduced from the expression for the amplitude function, which following Rottman et
al. is defined as
00
r(x, z, t)
-
r(x, 0, t)(1 - z) =
E
Ak(x, t) sin(kirz)
(4.14)
k=1
Comparing these figures to their counterparts in Rottman's figure 8, it is evident that
results of the Second-Order Projection Method closely correspond to their Spectral
Collocation counterparts for x < 37. The discrepancy for larger values of x is due to
the fact that Rottman et al. placed a sponge layer next to their outflow boundary
(located at x = 40) in order to absorb waves that are reflected form the boundary.
As a next test of the Second-Order Projection Method's performance, contour
plots of the density filed computed by the numerical model are compared to those
generated by the Spectral-Collocation approach for the case with K = 1.2. Con-
43
100
80
60
40
A
2
0
5
10
15
20
25
30
35
40
25
30
35
40
30
35
x
(a) K=0.95
80
-
60
40
20
0'
0
5
10
15
20
x
(b) K=1.00
An.
0
5
10
15
20
25
40
a;
(c) K = 1.20
Figure 4-3: The amplitude function of the resonant mode for the cases with Ka= 0.1,
D = 2.0 and different values of K
44
Wave Breaking Location
SOP
SC GY
*
*
*
*
*
79
35
31
27
59
28
24
21
66
37
29
23
4.4
2.8
2.2
1.4
4.7
3.2
2.5
1.9
*
K
0.95
1.00
1.05
1.10
1.20
Wave Breaking Times
SC GY
SOP
4.5
3.7
2.9
2.3
Table 4.1: Wave breaking data for Spectral-Collocation Method (SC), Grimshaw-Yi theory
(GY), and Second-Order Projection Method. The symbol * indicates that no wave breaking
has occurred. Results presented for Ka = 0.1 and D =2
trasting results in figure 4-4 to those in Rottman's' figure 12 reveals close agreement
before the wave breaking take place. Discrepancy in the wave breaking regime may be
attributed to the inability of the current implementation of the Second-Order Projection Method to handle discontinuities in the streamline structure that wave breaking
represents.
In regard to wave breaking, it is important to emphasize that wave breaking times
predicted by the Second-Order Projection are in the vicinity of those computed by
Rottman et al., as illustrated in Table 4.1. The inconsistency may be credited to the
difference in numerical implementations of the wave breaking condition. Moreover,
similar to the Spectral-Collocation Scheme, the Second-Order Projection overestimates the wave breaking times compared to the Grimshaw-Yi theory as indicated in
Table 4.1.
4.3
Nonuniformly Stratified Flow
The previous section examined the ability of the Second-Order Projection Method to
simulate the evolution of topography forced internal buoyancy waves in a body of uniformly stratified fluid. It is the aim of the present section to extend the assessment to
the flows characterized by arbitrary variations of the Brunt-ViiisdId frequency. In that
regard, this section begins with an overview of both linear and moderately nonlinearweakly dispersive long-wave theories of gravity-wave dynamics. It is important to
emphasize, that in order to reduce algebraic impediments caused by the presence of
45
(a) tL=0
(c) tL =
2 5 .5
(e) tL = 3 0.0
(b) tL
7. 5
(d) tL
27
(f) tL =
.0
4 9 .5
D=2.0 and
Figure 4-4: Contour plots of the density field for the case with Ka= 0.1,
K=1.2
46
the topography, the long-wave theory for arbitrary stratified flows is developed for
a channel bounded from above and below by rigid walls. Physically, the problem
under consideration corresponds to the propagation of buoyancy waves in a stratum
separating the fluid of lesser and greater density. Once the fundamental aspects of
the aforementioned theories are introduced, their predictions are compared with the
results of the numerical model.
Following developments of Prasad and Akylas [16], basic features of the long-wave
theory may be gamely demonstrated by considering the propagation of a buoyancy
induced disturbance in a layer of arbitrarily stratified fluid of depth H. Taking L to
denote horizontal length scale of the disturbance, one may define a non-dimensional
parameter
(4.15)
A= H
L
whose importance is readily elucidated by considering the dynamics of infinitesimal
long waves, which, in the jargon of the present discussion, are characterized by small
amplitudes and large lengths compared to the height of the channel. Accordingly,
their physics may be modeled by foremost nondimensionalizing the horizontal and
the vertical coordinate of equations (2.3a) though (2.3d) with L and H respectively
and time with p-iN-j to obtain
pt + J(p, T) = 0
(P' 1 zt)z + [p J([z, T)z - -1 px = -p
2
{[p J(',
(4.16)
F)] + (ppxt )x}
(4.17)
where 3 denotes Boussinesq parameter defined by (2.6), T represents the stream
function defined by (4.3) and (4.4) while J designates the Jacobian
J(a, b) = axbz - a bx.
(4.18)
Furthermore, linearizing (4.16) and (4.17) to the leading order in /-t yields linear
hydrostatic equations
pt +pN
2
47
Tx
=0
(4.19)
1
(pizt)z
PX =
-
(4.20)
0
which subject to impermeable boundary conditions
z=0 and z=1
at
TX = 0
(4.21)
admit a separable solution of the form
T = A(x
Pp
-
ct)q(z)
(4.22)
N29
(4.23)
Cn
where c is an arbitrary constant. Expressions (4.22) and (4.23) solve equations (4.19)
and (4.20) provided that O(z) is a solution of the eigenvalue problem
(pqZ)z +
pN 2 ~
= 0
(4.24)
z=0 and z=1
(4.25)
2
C2
0
with boundary conditions
at
=0
There is an infinite number eigensolutions, eigenvalue (cn) and eigenfunction (#n)
pairings, that solve (4.24). Moreover, eigenfunctions form an orthogonal and complete
set. In general, the solution of the eigenvalue problem must be obtained numerically;
nevertheless, in the special case of uniformly stratified Boussinesq fluid (4.24) takes
form
1
(4.26)
C+
and it possesses a closed form solution
#n(z)
=
z
sin-,
Cn
1
cn = - ,
n7r
n =1,2, ...
(4.27)
Furthermore, inspection of (4.22) reveals that any infinitesimal long disturbance in
48
a layer of arbitrarily stratified fluid propagates though the layer without change in
form with constant speed c. Due to the fact that linear-long wave eignmodes form
orthogonal and complete set, any fluid property associated with the evolution of
the disturbance may be studied as a superposition of linear long-wave modes. For
instance, horizontal component of the velocity vector may be expressed as
00
u(x, z, t) = $u
(x,t)#!,(z)
(4.28)
j=1
where uj denotes the jth horizontal velocity long-wave mode. As a result, one would
expect the first horizontal velocity mode of a disturbance that initially has the form
TI(x, z, to) = A(x + cito)q1 (z)
(4.29)
to evolve without change in form at linear long-wave speed ci as long as the length and
amplitude of the profile compile with the provisions of the linear long-wave theory.
For the purpose of examining the ability of the Second-Order Projection Method to
correctly simulate the evolution of infinitesimal long waves, the disturbance initial
stream function is the product of the Gaussian amplitude profile
A(x) = a exp -
(4.30)
and the first linear long-wave mode.
Figure 4-5 compares the predictions of the long-wave theory with results of the
computational model for uniformly stratified fluid layer with a = 0.01 and D = 20. In
particular, the first portion of the figure depicts the time evolution of the dominant
velocity mode while figure 4-5(b) portrays profiles of higher modes at t=260. As
expected, in the frame of reference moving to the left with the linear long wave speed
=1, the profile of the principal mode does not change while amplitudes of the
higher modes are virtually zero. The same response is observed for the propagation of
an identical disturbance through the fluid layer of non-uniform buoyancy frequency
variation, N = 1 + 3z. In other words, the profile of the dominant velocity mode
49
does not change with time, as illustrated in figure 4-6(a), while the amplitude of the
higher modes is of the order of 10' times smaller as shown in 4-6(b).
Ultimately, it is important to emphasize that for constant buoyancy frequency,
linear hydrostatic equations are coincidently solutions of full nonlinear governing
equations 2.4. This provides an additional opportunity to test the competence of
the numerical scheme. In other words, one would expect the first velocity mode of
the highly nonlinear long-wave disturbance not to change in form as the waveform
evolves through the uniformly stratified fluid layer. This type of behaviors is correctly
interpreted by the model as illustrated in figure 4-7 for a = 0.1 and D = 20. Owing
to the fact that wave breaking takes place for a = 0.103, the disturbance in figure 4-7
may be regarded as highly nonlinear.
When the structure of the disturbance does not conform to the requirements of
the long wave theory, the waveform associated with the principal mode disperses, as
illustrated in figure 4-8. For a weakly nonlinear-moderately dispersive disturbance,
Prasad and Akylas [16] have derived an evolution equation for the amplitude of the
principal mode. This result may be cast in the form
aT =
(4.31)
2raa. + s
where
r
p03
dz
(4.32)
0#
dz
(4.33)
s = -
dz
,lqz
I=j
T = et
(4.34)
(4.35)
In (4.35), e is a nondimensional parameter that denotes the ratio of the disturbance
amplitude and the channel height. Figure 4-9 compares the predictions of the weaklynonlinear moderately dispersive theory to the results of the computational model for
N = 1+ 3z, D = 2, and a = 0.002. It is evident from the figure that results of
50
r
7-
-
-
w
350
300
250
:200
150
100
50
U'
-80
-40
-60
-20
60
40
20
0
80
x
(a) Time evolution of the principal velocity mode
x 10
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.8
-1a
-100
I
-50
I
I
I
I
0
50
x
100
150
200
(b) Higher velocity modes at t = 260. U2 (solid line), U3 (dotdashed line), U 3 (dotted line)
Figure 4-5: Velocity modes for s=0, n=1, a=0.01, and D=20
51
AM.l
350300
250
S200
150
100
50
0
-40
-60
-80
-20
0
40
20
60
80
x
(a) Time evolution of the principal velocity mode
X10
I
I
6
4
2
-/
-
~ ~
...-... ..
..~~~
...
. ........
-2
-4
-6
-100
-50
0
50
100
150
200
(b) Higher velocity modes at t = 260. U2 (solid line), U3 (dotdashed line), U 3 (dotted line)
Figure 4-6: Velocity modes for s=3, n=1, a=0.002, and D=20
52
400
350300250
S200
150
100
50
-20
-4
-60
-80
0
x
40
20
60
80
(a) Time evolution of the principal velocity mode
1x 10F
I.
0.51
0
-0.5
-1
-1.5
-2
-2.5
-1c )0
-50
0
50
100
150
200
x
(b) Higher velocity modes at t = 260. U 2 (solid line), U 3 (dotdashed line), U 3 (dotted line)
Figure 4-7: Velocity modes for s=0, n=1, a=0.1, and D= 20
53
4C )0
35 030)025 0
S20)0
15
0
10)0
-10
-15
-5
5
0
10
x
15
20
30
25
3A
(a) Time evolution of the principal velocity mode
x 10~
4
2
'I
0
-2
b
-4
-6
-8
-10
-12
-14
-1c
0
-50
0
50
x
100
150
200
(b) Higher velocity modes at t = 260. U2 (solid line), U3 (dotdashed line), U3 (dotted line)
Figure 4-8: Velocity modes for s=0, n=1, a=0.01, and D =2
54
simulation and the small amplitude weakly-nonlinear moderately dispersive theory
agree well.
I
I
I
I
15
10
5
0
-5
-2 0
-10
0
20
10
30
40
50
Figure 4-9: Comparison between the moderately nonlinear-weakly dispersive theory
0.002, and
(solid line) and the numerical model (dashed line) for s = 3, n = 1, a =
D=2
4.4
Interaction of Two Solitary Waves
As a final test of the Second-Order Projection Method for the Variable Density Flows,
section 4.3 examines the ability of the computational scheme to simulate the interaction of two solitary waves in a fluid layer of arbitrary stratification. Based on the
assumption that the reader may not be familiar with the concept of solitons, the section begins with a brief overview of elementary aspects of this natural phenomenon.
A solitary wave or a soliton, as the name indicates, is a disturbance with a single
peak that travels through a medium without change in form. Due to the fact that
all waveforms are subject to the law of dispersion it is reasonable to inquire if this
rule is repealed for solitary waves. While the thorough mathematical answer to this
55
question is rather intricate, the phenomenon can be explained qualitatively in rather
simple terms through the following line of reasoning.
Most media in nature are dispersive. In a dispersive medium, sinusoidal waves of
different frequencies travel at different speeds. As a rule, the higher the frequency, the
slower the wave. Consequently, if two initially aligned sinusoidal waves of different
frequency propagate through a dispersive medium, the higher frequency harmonic will
fall steadily further behind. Majority of the waveforms in nature, however, are not
consisted of a single sinusoidal wave but they are rather complex structures, which
can be viewed as Fourier sum of infinitely many harmonics.
For complex waves,
dispersion manifests itself in a slightly different fashion. A complex waveform that
is initially tall and narrow gradually spreads as its higher frequency harmonics lag
behind their lower frequency counterparts.
Moreover, the amplitude diminishes as
the disturbance broadens. The process continues until the waveform disperses; that
is, its amplitude becomes unnoticeably small.
A soliton, according to its definition, is a waveform that is zero everywhere except
in a small region where it describes either a peak or a valley of an arbitrary shape.
It is apparent that a disturbance of this structure is not sinusoidal; nevertheless, as
any waveform it can be represented as a Fourier sum of infinitely many harmonics.
Consequently, one may conclude that every solitary wave is a complex waveform
and as such it must be subject to dispersion. Solitions, however, do not scatter as
they propagate through a dispersive media because they posses a mechanism that
balances the effect of dispersion. The compensating effect in solitary waves is the
coupling between the amplitude and speed; namely, the higher the amplitude, the
faster the wave. Therefore, as a soliton disperses, the waveform becomes smaller in
amplitude. However, lower amplitude components travel more slowly, which means
that the peak of the wave begins to overtake the lower amplitude leading edge. The
amplitude effect steepens the wave. The two competing phenomena - flattening from
frequency dependence and stepening from amplitude dependence - quickly reach a
point of stable equilibrium where the waveform remains constant.
Solitary waves were first observed in the mid-19th century, but it was 50 years
56
before mathematicians could properly explain their existence. For nearly 70 years
thereafter, solitons were regarded as unimportant curiosities in the field of nonlinear
dynamics.
Then in 1965 came a completely unexpected discovery.
Zabusky and
Kruskal [19] reported on the outcome of a computer simulation modeling the collision
of two solitary waves. At the time, it was generally believed that a clash of two
solitons would result in a strong nonlinear interaction that would ultimately destroy
both waveforms.
Instead, the solitary waves passed through each retaining their
shapes and propagation velocities after the collision. The discovery of Zabusky and
Kruskal triggered a renewed interest in properties of solitary waves. Nowadays, more
than hundred equations are known to possess solitary wave solutions. One of them
is the Korteweg-deVries equation discussed in the previous chapter. Its solitary wave
solution may be expressed as
A(x, T) =
sech2 L
(
T - 6)] ,
(x E R, t > 0)
(4.36)
where o and 6 are arbitrary constants such that 6 E R, o > 0. Details associated
with the derivation of this result may be found in [5]. It is important to note two
properties of solitary waves embedded in 4.4 that play significant role in the forthcoming discussion. Foremost, the velocity of the soliton depends upon its height; and
second, larger the amplitude narrower the waveform.
As a final test of the Second-Order Projection Method for the Variable Density
Flows, the remaining portion of this section examines the model's ability to simulate
the interaction of two solitary waves in a fluid layer characterized with linear buoyancy
frequency variation N = 1 + 3z. The flow field is initialized with the product of the
first linear long wave mode and the superposition of two solitary waves
V)(x, z, T) ={
+{
osech2 [
sech2
1
a(x +
iT
a (x +
a2T
- 61)]}
-
(z)
(4.37)
J2)
with ai = 3.5, o 2 = 0.35 and values of r and s as in Figure 4.9. The initial principal
57
velocity mode profile is depicted in Figure 4.10-a. It is apparent form the figure that
the smaller wave is positioned at x = 6 = 25 while the lagging wave is stationed at
x
= 62
= 0. Owing to the fact that the initial amplitude of the leading wave is 10 times
larger than the amplitude of its lagging counterpart, the taller wave at first travels
10 times faster. Its velocity is expected to reduce, however, because its amplitude
is - as illustrated in Figure 4.10 - outside the realm of the moderately nonlinearweakly dispersive theory; consequently, the waveform is subject to dispersion which
flattens it and in turn reduces its speed. In attempt to highlight this effect, Figure
4.10 contrasts the solution of the model to that of the KdV equation. Moreover, in
order to illustrate a nonlinear nature of the solitary wave interaction the results of
the model are additionally compared to the superposition of two solitary waves.
Initially, crests of all three aforementioned responses overlap, as indicated in figure 4.10-a. As time goes on, however, the lagging wave disperses; consequently, the
numerical solution deviates form both the solution of the KDV equation and the
superposition (figure 4.10-b). The latter two responses are in accord until the interaction of the solitons takes place. In particular, both solutions correctly predict that
solitons retain their shapes following the intercourse; nevertheless, the superposition
(as a sole property of the linear systems) does not account for the nonlinear character of the interaction. During the intercourse the energy transfer takes place form
the taller wave to the smaller one, which in turn propels the lower soliton forward
and retracts its taller counterpart (figure 4.10-d). Lastly, it is significant to mention
that the Second-Order Projection method correctly interpreted the interaction of two
solitary waves in a sense that it properly interpreted both phenomena: waveforms
retained their shapes following the interaction and the shift of energy took place form
the larger to the smaller wave, as depicted in figure 4.10-d.
58
0.1
-t=0
0.050-
-80
-60
-40
-20
20
0
t=300
0.05-
$
-80
-60
-40
-20
20
0
40
_
40
0.1t=1000
060-80
-
0.05 -60
-40
-20
0
20
40
t=2000
-
0.05
.
0.1-
-80
-60
-40
-20
0
20
40
x
Figure 4-10: Interaction of two solitary waves. Model solution (solid line), KDV
equation (dashed line), superposition (dotted line)
59
Chapter 5
Upstream Influence in a Horizontal
Waveguide
5.1
Preliminaries
The theory of horizontal waveguides provides a valuable analytical tool for modeling a number of internal gravity wave phenomena such as propagation of buoyancy
induced disturbances in the oceanic thermocline and the atmospheric tropopause.
An important and yet novel aspect of the theory is the recently recognized ability
of transient nonlinear interactions to give birth to shelves, that is, finite-amplitude
structures of large extent that remain coherent over vast distances and times. Specifically, in 1997 Prasad and Akylas [16} analytically demonstrated that the presence of
moderately nonlinear-weakly dispersive disturbance in a body of unsteady uniformly
stratified fluid generates shelves purely downstream of the principal wave. In contrast to this special case of constant buoyancy frequency, they also found that the
transience of identical disturbance in nonuniformly stratified fluid produces shelves
that generally extend both upstream and downstream of the main surge. Lastly,
they mathematically illustrated that the wave whose initial stream function is the
product of an amplitude profile, A(x), and the nth linear-long wave mode generates
upstream propagating shelves exclusively in lower long-wave modes, namely, n-m
where m = 1, 2...n-1.
60
Prior to the work of Prasad and Akylas, the problem of upstream influence has
generally been associated with stratified flows over topography. In particular, the
question of what influence the presence of an obstacle exerts far upstream has been
a contentious issue for many years. The question has been investigated theoretically
by, among others, McIntyre [15] who developed the solution for the non-resonant
flow of finite depth over weakly nonlinear topography as an expansion in terms of
the obstacle amplitude. Accordingly, he demonstrated that upstream-propagating
disturbances of 0(E) 2 may be created by nonlinear intercourse of unsteady lee-wave
tails. Shelves examined by Prasad and Akylas, on the other hand, are of O(E) 4 and
are due to the transience of the weakly nonlinear-moderately dispersive disturbance.
Ultimately, it is important to emphasize that in the case of uniform stratification no
upstream influence is predicted by the theory of Prasad and Akylas.
5.2
Comparison with the Second-Order Projection
Method
In attempt to verify the premises of the moderately nonlinear-weakly dispersive formulation, the present section compares the predictions of the analytical model to the
numerical results generated by the Second-Order Projection Method. Owing to the
fact that disturbances under consideration are of 0(E) 4 , the comparison is conducted
is qualitative terms. Moreover, this section attempts to numerically illustrate the
following theoretical postulates.
" In the case of uniform stratification, buoyancy induced moderately nonlinearweakly dispersive disturbance comprised of the product of the amplitude function and the second long-wave mode generates shelves in the higher long-wave
modes, i.e. third, fourth etc., which extend only downstream of the main disturbance.
" In the case of nonuniform stratification, buoyancy induced moderately nonlinearweakly dispersive disturbance - comprised of the product of the amplitude profile
61
and the second long-wave mode - generates shelves in both higher and lower
modes. Shelves corresponding to the higher modes propagate only downstream
of the main disturbance while those associated with the lower modes, namely
first, extend both upstream and downstream of the main wave.
The physical mechanism responsible for the upstream shelf generation possesses
nonlinear nature.
These premises are numerically confirmed by considering the evolution of buoyancy induced disturbance propagating through a horizontal wave guide characterized
by the linear Brunt-Viisdii
frequency variation
(5.1)
N = 1+ s-z
In the following developments, it is presumed that the moderately nonlinear-weakly
dispersive disturbance initially possesses form that can mathematically be represented
as
V)(x, z, to) = A(x, to)0,,(z)
A(x, to) = a exp[(X
Ct)2
(5.2)
(5.3)
where a and d represent the height and half-width of the amplitude function, A.
Moreover, b denotes the perturbation stream function while c, and 0,/
represent the
nth linear long-wave eigenvalue and eigenmode respectively.
In attempt to numerically justify the first of the three aforementioned hypothesis
of the weakly nonlinear-weakly dispersive theory, Figure 5-1 depicts the transient
self-interaction of a disturbance in a uniformly stratified fluid at time t = 260. The
initial structure of the disturbance is described by (5.2) and (5.3) with a = 0.02,
D = 2, and n = 2. Owing to the fact that the stream function is initially product of
the amplitude profile A and the second long-wave mode, the amplitude of the second
horizontal velocity mode decreases with time as the waveform disperses. Moreover,
in accordance with the predictions of the theory, downstream propagating shelves
are associated only with the higher velocity long-wave modes as shown in figures 562
1(c) and 5-1(d) while the first mode is zero. Observing figures 5-1(c) and 5-1(d) it
furthermore becomes apparent that shelves are are indeed finite-amplitude structures
of large extent compared to the width of the main disturbance.
0 x10-8
20r
1
9
15
0
10
-1
-2
5
-3
0
-
-4
-100
-50
0
50
100
150
200
5
-50 -40 -30 -20 -10
10
20
30 40
50
(b)
(a)
3 X10
0
8
5 x10-
5
4231
2
0
-2-
-2-
-3-100
-
10__
2-
-3-50
0
50
100
150
-100
200
-50
0
50
100
150
200
3;
3;
(d)
(c)
Figure 5-1: Horizontal velocity modes at t = 260 for s = 0, n = 2, a = 0.02, and
D = 2.
Furthermore, figure 5-2 portrays the transient self interaction of a disturbance in a
linearly stratified fluid with s = 3. The initial structure of the disturbance is similar to
that of figure 5-1. Nevertheless, due to non-uniform stratification of the fluid layer,
a different physical response is observed. Namely, in accordance with the second
aforementioned premise, downstream propagating shelves are associated with both
63
higher and lower modes as indicated in figures 5-2(a), 5-2(c), and 5-2(d). Additionally,
x10-
3
3
0.025
2
0.02
1
0.015
0
0.01
-1
0.005
-2
0
-3
-1i 00
-50
50
0
x
100
150
200
-0-005
-3 0 -20
20
30
40
50
(b)
(a)
1
10
a;
0
-10
2.5
. ;x10-3
x10-4
21
1.50.5
1
0.5
-0
0
-0.5
-0.5
-1
-100
-1
-50
0
50
100
150
200
-1.5
00
-50
0
a;
50
100
150
200
a;
(d)
(c)
Figure 5-2: Horizontal velocity modes at t =260 for s
D = 2.
=
3, n = 2, a = 0.03, and
the first mode contains upstream propagating shelve as illustrated in figure 5-2(a).
An additional aspect of the first mode that is important to emphasize is the presence
of oscillating tails, which have been previously observed by, among others, Akylas and
Grimshaw [1] in association with solitary waves. Moreover, figure 5-3 depicts the same
results as figure 5-2 but at a later time, namely t = 520. Comparing the waveforms in
this figure to those in figure 5-2, it becomes apparent that shelves are formations that
remain coherent over large distances and times. Furthermore, figure 5-4 presents the
64
transient self-interaction of a disturbance characterized with s = 3, a = 0.01, D = 2,
and n = 3. In agreement with the moderately nonlinear-weakly dispersive theory,
upstream propagating shelves are again present in modes one and two while higher
modes contain only downstream propagating waves.
3
0.035
4 x10-
0.03
3-
0.025
2
0.02
1
0.015
0
0.01
1
0.005
2-
0
_n nn-r
200 -100
100
0
200
300
-30
400
-20
10
0
-10
20
30
40
50
x
x
(b)
(a)
x10-3
2.5
1. I
X 10-4
2
1
1.5
0.5
1
3 0
S0.5
0
-0.5
-0.5
-1
-1
-1.5
-200
-100
0
100
x
200
300
-1.5
-2 00 -100
400
0
100
x
200
300
400
(d)
(c)
Figure 5-3: Horizontal velocity modes at t = 520 for s = 3, n = 2, a = 0.03, and
D = 2.
Lastly, in attempt to illustrate that the formation of shelves is a result of nonlinear
interactions, figure 5-5 depicts the upstream propagating shelves corresponding to
the disturbance with parameters s = 3, a = 0.03, D = 2, and n = 2 at time t = 260.
In contrast to figure 5-2, however, these results were obtained by solving equations
65
attained by eliminating the convective terms from the governing equations (2.4). The
fact that the amplitude of the upstream propagating shelve in figure 5-5(a) is larger by
the order of magnitude from the shelve in figure 5-5(a) indicates that the mechanism
associated with the shelve generation is nonlinear in nature. The effect can also be
observed by considering stream function of the two responses as indicated in figure
5-6.
4
x10-4
x10
5
3-
4-
2-
3-
1
2-
0
1-
4
0
-1-
-3-
-2-
-4-
-3-
-150 -100
0
-50
50
150
100
200
250
-
-2
150 -100
-50
50
0
100
150
200
2 30
x
x
(b)
(a)
4
X 102
.
12 x10-3
1.5-
1u
1 -8-
0.5
6-
S0
-0
-
-0.5
4
2
-1.5
0
-1 5
-10
0
-5
5
10
-2
-150 -100
15
x
-50
0
50
100
150
200
21 50
x
(d)
(C)
Figure 5-4: Horizontal velocity modes at t = 260 for s = 3, n = 3, a = 0.01, and
D = 2.
66
4 x10-
4
0.0252
0.02-
0
0.015
0.01
-2
0.0051
-4
0
-005
-100
-50
50
x
0
100
150
-30
200
-20
0
-10
10
20
30
40
50
x
(b)
(a)
2.5rx10-
x10-4
5
,
4r'i
2
1.5
2
1
)y
S0.5
A
-
0
-21
0.5
-1
1.5
-100
-50
0
50
100
150
-21
200
-100
x
-50
0
50
x
100
150
200
(d)
(C)
Figure 5-5: Horizontal velocity modes at t = 260 for s = 3, n = 2, a = 0.03, and
D = 2.
67
1
0.9
0.8
A
0.7
0.6
N0.5
-Th
0.4
0.3
0.2
0.1
0
-50
100
50
100
150
x
(a) Nonlinear model
1
0.9
0.8
0.7
0.6
0.5
0.4
--
0.3
-
\
0.2
0.1
0'
-10
I
I
0
-50
0
50
100
150
(b) Linear model
Figure 5-6: Stream function at t = 260 for s = 3, n = 2, a = 0.03, and D = 2.
68
Chapter 6
Concluding Remarks
This manuscript systematically documented the development of a nonlinear model for
simulating the evolution of internal gravity waves in the ocean and the atmosphere.
The model was constructed by foremost identifying physical mechanisms that play a
subordinate role in defining the character of this natural phenomenon. Correspondingly, based on the results of the order of magnitude analysis, it was determined
that modeling the media in which buoyancy induced disturbances are taking place
as two-dimensional, incompressible, inviscid and Boussinesq significantly simplifies
mathematical description of the physical model while still capturing essential physics
of the gravity-wave dynamics. Implementation of the abovementioned simplification
revealed that the nature of the physical model is governed by the Boussinesq form of
Euler's' equations. Owing to the fact that Euler's equations possess nonlinear structure, their solution had to be approximated numerically. This task was accomplished
using the Second-Order Projection Method for variable density flows. The following
benefits of the scheme made it an attractive choice for modeling the evolution of buoyancy induced disturbances. Foremost, the utilized numerical scheme treats nonlinear
convective terms in a non-iterative manner. Second, the symmetric, positive definite,
and block tridiagonal matrix A, corresponding to the linear algebra problem Ax=b,
in the Boussinesq limit is independent of time; hence, it only needs to be constructed
once. Lastly, the numerical scheme is second-order accurate in both space and time.
While the focus of Chapters 1,2 and 3 was on the development of the model, Chap69
ter 4 presented results associated with the testing of the numerical algorithm. In that
regard, it is important to emphasize that the predictions of the model closely agreed
with McIntyre's linear theory, which is valid for the topography of small amplitude
and flow conditions that are far away from the resonance of linear modes. The competence of the scheme in the flow regimes beyond the applicability of the McIntyre's
archetype was tested through the comparison with computational results of Lamb
and Rottman et al. The analysis revealed that in the presence of wave breaking,
predictions of the Second-Order Projection Method disagreed with the numerical results of Rottman et al. The discrepancy was attributed to the inability of the current
implementation of the Second-Order Projection Method to handle discontinuities in
the streamline structure that wave breaking represents.
Ultimately, it is significant to point out that the scheme successfully simulated the
gravity-wave dynamics of the moderately nonlinear-weakly dispersive disturbance in
waveguide as well as the interaction of two solitary waves in a fluid layer bounded
from above and below by impermeable solid walls.
Ultimately, in the last chap-
ter of the document, the model was employed to numerically verify premises of the
moderately nonlinear-weakly dispersive theory of Prasad and Akylas. Specifically,
the archetype successfully verified the presence of the upstream propagating shelves
caused by the self-interaction of the main disturbance in a nonuniformly stratified
horizontal waveguide. The successful numerical confirmation of this result naturally
raises the following question.
Is there a form of nonuniform density stratification
that yields exclusively downstream propagating shelves in a horizontal waveguide,
i.e., no upstream influence? Furthermore, it is also of particular practical interest
to examine the properties of upstream influence, due to the transience of the moderately nonlinear-weakly dispersive disturbance, in nonuniformly stratified flow over
topography.
Lastly, in regard to possible future improvements to the model, it is important to
mention that it would be desirable to extend the performance of the Second-Order
projection method to the flows characterized by the wave breaking. Moreover, in
order to increase the range of applicability of the physical archetype it would be also
70
beneficial to implement a different type of the upper boundary condition; namely,
the radiation condition for modeling the gravity-wave dynamics of the atmosphere
and the atmosphere-ocean boundary condition simulating the evolution of oceanic
buoyancy induced disturbances.
71
Appendix A
Linear Solution for the Uniformly
Stratified Flow Over Localized
Topography
It has been illustrated in chapter 2 that equations governing the the two-dimensional
flow of an inviscid, incompressible, uniformly stratified, Boussinesq fluid may be expressed as
ut + uUx + wuz = -px + b
(A.1a)
Wt + UWX + WWZ
(A.1b)
=
-Pz - r
rt + ur + wrz = N 2w
Ux
+
Wz =
0.
(A.1c)
(A.1d)
Assuming that the fluid flows past a stationary locally confined topography
z = h(x)
(A.2)
with a uniform background velocity U oriented in the positive x direction, far away
from the topography one may write
72
u-+U and w,p,r=O
as
|xI -
oo.
(A.3)
Moreover, assuming that the top and bottom boundaries are rigid, it follows that
w = -U
dh
at
z = h(x)
(A.4a)
and
w=O
at
z = 1.
(A.4b)
For a topography of small amplitude compared to the height of the channel, it is
reasonable to expect that the effect of the landscape on the flow structure is going to
be small. Accordingly, one may define a disturbance stream function as
U = HU+ Tz
w = -XF
(A.5a)
(A.5b)
where H denotes the Heaviside step function with respect to time. In addition, it will
be assumed that the flow is accelerated impulsively; hence, one may write
XF,p,r = 0
at
t = 0.
(A.6)
Substituting (A.5a) and (A.5b) into the governing equations (A.1a) though (A.1d)
and eliminating pressure yields
V 2 XPt + HUV 2 X'
-
rX = J(XI', V 2 X)
rt + HUrx + Tx = J(I, r).
(A.7a)
(A.7b)
Following McIntyre [15], the solution of (A.7) with the boundary conditions given
by (A.3) and (A.4) and initial conditions depicted by (A.6) is sought by formally
assuming that in the limit of the small topography compared to the channel height
73
the solution is proportional to the topography amplitude.
Hence, expanding the
governing equations in terms of the powers of the small parameter epsilon, which
represents the ratio of the topography amplitude to the channel height, yields
+ CWp
=
+O)
~()...
r = r( 0 ) + er(l) +
(A.8a)
(A.8b)
r2 + ...
To the leading order in E, equations (A.7) may be expressed as
V2,J0) + HUV 2 4) - r)
r
0
0 ) + T(O) = 0.
+ HUr(
(A.9a)
(A.9b)
At this point, one may define a Fourier transform of an arbitrary function f(x) as
f (k) = 27r -o
f(x)eikxdx
(A.10a)
and its inverse as
f(x)
=
f(k)eikxdk.
(A.10b)
Transforming the horizontal coordinate of the governing equations to the Fourier
space yields
-k2g
I
+ VO)) - HUik1o) + HUikI'O) - ik( 0 ) = 0
-(0)
rt
+ HUikf(0) + ik
_
= 0.
-(O)
(A.lla)
(A.11b)
Furthermore, one may define Laplace transform and its inverse counterpart of an
arbitrary function g(t) as
= Z(s) g(t)e-tdt
(A.12a)
and its inverse as
c+iOO
g
g9t) = .2rf-o g(s)e'tds.
74
(A.12b)
= - k2.
sn (A.
16)
Taking Laplace transform of the governing equations and substituting the density
equation into the momentum one reduces the problem at hand to an ordinary differential equations with respect to z, namely,
Z(0)
=(0)
(=
'ZZ +
where
(A.14)
(A.13)
0
V
(s + ikU) 2
Furthermore, linearizing (A.4a) and taking first Fourier then Laplace transform of
(A.4a) and (A.4b) yields
(0)
Uht
=
at z = 0
-V
(A.15a)
8
(0)
(
=0
(A.15b)
at z = 1.
The solution of (A.13) with boundary conditions (A.15) may be expressed as
T
(0)
=
Uh sin[#(1 - z)]
sin/i
s
.
The inverse Laplace transform may be performed analytically via contour integration.
The result may be expressed as
TI(k, z, t) = 's(k, z) + x'-(k, z, t) + '~ (k, z, t)
T'(k, z) = -Uh
i
(
2
sin[ V
2-
Z)]
(A.17)
(A.18)
k2
00
N (k, t) sin(nrz)
(k, z, t) =
(A.19)
n=1
T' (k, nt) =T
nrcanhUe-ik(U cn)t
U t
C
-2Vn2
1 + k2
Cn
(A.20)
(A.21)
In contrast to the inverse Laplace transform, the inverse Fourier transform must be
75
performed numerically.
76
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78