SUB-MESOSCALE MODELING OF ENVIRONMENTAL
VARIABILITY IN A SHELF-SLOPE REGION AND THE
EFFECT ON ACOUSTIC FLUCTUATIONS
STEVEN FINETTE*, THOMAS EVANS** AND COLIN SHEN**
*Acoustics Division, **Remote Sensing Division,
Naval Research Laboratory, Washington DC 20375, USA
E-mail: [email protected]
A coupled oceanographic/acoustic simulation model is under development for studying
the relationship between acoustic field variability and dynamic oceanographic processes
in a continental shelf/slope environment. The oceanographic component of the model
involves numerical integration of the non-linear hydrodynamic equations of motion
describing density, temperature and salinity distributions as a function of space and
time. This component includes sub-mesoscale dynamics, allowing for the generation
and propagation of non-hydrostatically generated phenomena such as tidally driven
internal tides and solitary waves. Results are mapped into the corresponding sound
speed distribution, and the resulting set of time evolved sound speed fields is used as
input to a wide-angle parabolic equation that computes the acoustic field propagating
through the environment. The general approach is discussed, and an illustrative result is
presented that links acoustic field variability to specific oceanographic features.
1
Introduction
Modeling acoustic propagation through littoral regions is a difficult problem because of
the complex dynamic structure of the temperature and salinity distributions found in
such environments. A significant source of this structure is tidal forcing of stratified
water at the shelf break which, through buoyancy effects, can induce baroclinic motions
in the form of internal tides and solitary waves. In order to properly model acoustic
system performance in littoral areas, it is important to develop synthetic sound speed
fields that "faithfully" reproduce the proper space-time characteristics of the sound
speed variations in the real ocean. The reason is that the sound speed distribution, in
conjunction with acoustic source properties, source geometry and waveguide boundary
conditions uniquely determines the space-time coherence of the acoustic field and
coherence is the fundamental physical property that places restrictions on phasesensitive sonar processing. For the purpose of studying volumetric contributions to
space-time coherence in shallow water, we are developing a hydrodynamic model for
the evolution of the sound speed field in a continental shelf-slope environment. An
example of acoustic propagation through this shallow water waveguide is presented to
illustrate the effect of tidal forcing on transmission loss.
Mesoscale hydrodynamic models are not appropriate in the littoral because they do
not allow for small spatial scale (horizontal) ocean dynamics on the order of 50–500 m
and below, thus filtering out internal tides, solitary wave production and propagation.
401
N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 401-408.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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S. FINETTE ET AL.
We discuss below a non-hydrostatic (sub-mesoscale) model of the ocean environment.
This model is forced by the M2 tide over a stratified water column with variable
bathymetry, generating an internal tide and solitary waves originating at the shelf-break
region. As an illustration, a wide-angle parabolic equation is used to propagate the
acoustic field emitted by a point source through a set of time-evolved 2-D sound speed
snapshots. Internal tides and solitary wave packets are important examples of submesoscale dynamics that play a significant acoustic role in these environments, altering
the amplitude and phase of acoustic waves propagating in the waveguide. A number of
attempts to model the acoustic wave/internal wave interaction have been made recently
[1–7]. The influence of azimuthally anisotropic solitary waves on horizontal array
performance has been considered using a data constrained oceanographic model [8,9].
2
Oceanographic and acoustic models
The sub-mesoscale model is based on a vorticity dynamics formulation of the
hydrodynamic equations of motion [10]. The model allows for a free ocean surface,
stratified density distribution, variable bathymetry, coriolis and tidal forcing. The
equations of motion in this formulation are based on the Boussinesq approximation for
the density variations and are given by the following expressions.
Sea surface height evolution equation:
where
η
∂η !
+ U η ⋅ ∇η = w η ,
∂t
(1)
is the vertical displacement of the ocean surface from its resting water level,
!
w η is the surface vertical velocity, Uη ≡ (uη , vη ) represents the horizontal surface
velocity vector and ∇ ≡ (
∂ ∂
, ).
∂x ∂y
Surface momentum equation:
!
!
2
!
  Dw 
 ρη


 DU ! ! 
U
∂
2
+ f × U  = − 
+ g  ∇η +  µ H ∇ U + µ Z 2  ,


  Dt η
ρ
∂z η

 Dt
η

 o
(2)
!
!
D ∂
= + U * ⋅ ∇ * , U * = (u, v, w ) is the total velocity vector,
Dt ∂t
!
U = (u, v) the horizontal velocity vector, g is the gravitational acceleration,
∂2
∂2
∂2 !
, f = (0,0, f ) describes the Coriolis rotational frequency
∇ *2 ≡ 2 + 2 +
∂x
∂y
∂z
where
vector,
ρη
the sea surface density,
ρo
a constant reference density and
represents the horizontal and vertical eddy viscosities.
µ H ,Z
SUB-MESOSCALE MODELING OF ENVIRONMENTAL VARIABILITY
403
The interior evolution equation for the horizontal vorticity vector:
!
!
! !
!
!
Dζ
∇ρ
∂ 2ζ
2
ˆ
= (ζ * + f ) ⋅ ∇*U + k ×
+ µ H ∇ ζ + µZ 2 ,
Dt
∂z
ρo
!
with
ζ = (ζ x , ζ y )
(3)
!
the horizontal vorticity vector,
vorticity vector, (ζ x , ζ y ) = (
ζ * = (ζ x , ζ y , ζ z ) the
total
!
∂w ∂v ∂u ∂w
∂v ∂u
, ∇* ⋅ ζ * = 0 ,
− ,
−
−
), ζ z =
∂y ∂z ∂z ∂x
∂x ∂y
and k̂ is a vertical unit vector.
The continuity equation:
!
∇* ⋅ U * = 0
(4)
Temperature and salinity equations:
T 
D 
2
2
2
 S  = ν  ∂ + ∂ T  + ν ∂ T 
H
Z
2
Dt
∂y 2  S 
∂z 2  S 
 ∂x
(5)
where T , S represent temperature and salinity respectively, and ν H ,ν z are the
horizontal and vertical diffusivities. The latter terms are used to stabilize the
temperature and salinity fields against high frequency fluctuations in these quantities.
Equation of state (IES 80):
ρ ( x, y, z ) = Θ(T , S , P )
(6)
where P is the pressure. The functional form of Eq. (6) is given in [11].
A split-time semi-Lagrangian technique is used for the integration of the above
equations, to achieve both efficiency and accuracy in the modeling of the nonlinear flow
effect and the free surface motion. In essence, this technique computes the flow
!
variables, U η ,
!
ζ , T and S at fluid particles’ positions from the free surface to the
bottom and then interpolates the calculated quantities back to a fixed reference grid. The
interior velocity field is then calculated from the kinematic relations,
∇2w +
!
∂2w
ˆ × ζ , ∂w − ∂v = ζ , and ∂u − ∂w = ζ ,
k
=
∇
⋅
x
y
∂z ∂x
∂y ∂z
∂z 2
(7 )
and ρ from the equation of state. These new velocities and density are used to start the
next Lagrangian time step calculation. In the split-time integration, the faster evolving
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S. FINETTE ET AL.
surface motion governed by the surface equations (1) and (2) is integrated at small time
steps set by the CFL condition for the surface gravity waves, while the slower evolving
interior flow whose time scale is of the order of the buoyancy period is updated only
after many small surface time steps. The spatial derivatives are evaluated using higher
order finite difference schemes. The solution of the elliptic equation for w uses the
second order finite-difference MUDPACK library from NCAR. The ocean model has
been validated by comparing its accuracy in simulating flow instabilities in channels
against the highly accurate pseudo-spectral calculations as well as testing it against
known analytical solutions for waves and currents. The details are to be reported
elsewhere (Shen and Evans, in preparation).
The sound speed field C ( x, y, z, t ) is considered as environmental input data for a
wide-angle, split-step parabolic equation algorithm [12] describing acoustic propagation
from a point source. Sound speed is computed through its functional dependence on
temperature, salinity and pressure [13], with the latter quantities determined throughout
the water column from the solution of Eq. (1–6). We invoke the frozen ocean
assumption, by which temporal variations in the ocean are considered negligible during
the passage of the acoustic wave from source to receiver. This is well satisfied for
acoustic propagation over 30 km for the interior of the ocean, where typical internal
waves have speeds less than 1 m/s. The long wave assumption is made for surface
waves: C =
gH ≅ 30 m/s for a maximum water depth H of 100 m, and the
assumption is marginally satisfied there. However, the long M2 tide wavelength implies
that it will have only a small vertical component of the surface elevation gradient over
the 30 km of acoustic propagation and this surface variation is assumed to not
significantly violate the condition.
3
Results
An example of the evolution of the sound speed field through a shelf-break region and
acoustic propagation in this environment are presented below. The 2-D simulation of a
shelf/shelf-break environment for acoustic modeling is carried out in a 100 km wide
domain with the shelf-break modeled by h = -25m[3+tanh((x-50km)/1.25km)], where h
is the depth of the bottom below the mean sea level, z=0, and x is the horizontal distance
measured from the left boundary, x=0, which is a vertical wall assumed to be the
location of the coast. The right boundary at x=100 km is assumed to be the open ocean,
and the flow there responds to tidal forcing which is applied at the right boundary by
varying the sea surface height sinusoidally with an amplitude of 2 m and period of
12.4 h. No wind forcing is applied, and the surface boundary condition is thus stressfree, u/ z=0= v/ z. The same stress-free condition is also used for the bottom and
coast line to eliminate the frictional influence from these boundaries, since the focus of
this simulation is the generation and propagation of internal waves in the presence of
minimal frictional influence. In follow-up studies, viscous effects from the boundaries
are to be considered. The domain is resolved horizontally with 24.4 m grid spacing. A
surface and bottom following vertical coordinate system is employed, and so the
number of vertical grid points is fixed at 33 both on and off the shelf. A 30 km subsection of the 100 km oceanographic domain and simulated internal wave structure is
SUB-MESOSCALE MODELING OF ENVIRONMENTAL VARIABILITY
405
used for acoustic computations and is shown in Fig. (1), with the starting range
coordinate renumbered to zero. The parameters used in the governing equations are
f=2π/12.4 h, g=9.8 m/s2 , µH =νH =0.5 m2/s, µz =0. 05 m2/s m, νz =10-4 m2/s.
A depth dependent temperature field describing a summer thermocline is used as a
starting environment, with salinity chosen as a constant 35 ppt. The profile was obtained
from the SWARM95 data set [2]; for simplicity, it is assumed to be range independent
at the beginning of the simulation, when the model ocean is in a ”resting” state. An
acoustic source of frequency 400 Hz is placed at a depth of 30 m at the range origin and
the acoustic field is propagated over a 30 km range from the shelf region downslope
through the shelf-break area.
The figure below gives an example of the evolving ocean sound speed environment
covering a 7.5 h period, along with the corresponding acoustic transmission loss for the
selected environmental snapshots. Geometric spreading has been removed to emphasize
environmental variability. Note that white in the transmission loss figures represents
losss greater than 70 dB. Time is measured from the starting point (zero hours) as
indicated on the plots. The initial off-shore flow causes a significant variation of the
thermocline to appear over the shelf break at t=5 h which, through the corresponding
density perturbations, generates a baroclinic tide that propagates outward in both
directions (t=6 h) from the shelf-break. The internal tide transfers energy to higher
spatial frequencies in the form of two solitary wave packets (t=7.5 h), propagating away
from their generation site at the shelf-break.
In this simulation, the solitary wave packets are dominated by the first internal
wave mode. Both adiabatic propagation and mode coupling can play a role in acoustic
transmission for this environment. There are two potential sources of acoustic mode
coupling. One is associated with the range dependent bathymetry and may be causing
conversion of higher order modes to lower order modes starting around 20 km from the
acoustic source, resulting in a high loss region in the upper 20–25 m of the water
column at ranges greater than 22 km. However, the relative importance of the mode
coupling and adiabatic terms in this region would have to be assesesed by modal
decomposition of the field; that analysis is beyond the scope of this paper.
The second source of acoustic mode coupling involves the range dependent
variations of the thermocline induced by tidal forcing. Transmission loss variability
beyond a range of 22 km is evident in the modal interference pattern between the 5 h
and 6 h snapshots and is caused by the transformation of the depression into an internal
tide propagating on and off the shelf.
The most significant variation occurs between 6 and 7.5 h and is induced by the
solitary wave packet propagating on the shelf, causing a mean drop in transmission loss
of about 8 dB across the shelf break. The seawardc propagating packet does not
contibute to this enhancement because it resides at depths less than about 25 m, where
only weak acoustic energy levels are available for interacting with the wave packet.
Acknowledgment
This research was supported with funds from the Office of Naval Research.
406
S. FINETTE ET AL.
depth [m]
20
40
.5 hours
60
80
100
120
0
6000
12000
18000
24000
30000
range [m]
m/sec
1470
1482
1495
1507
1520
0
depth [m]
20
40
.5 hours
60
80
100
120
0
6000
12000
18000
24000
30000
range [m]
dB
10
20
30
40
50
60
70
depth [m]
20
40
60
5 hours
80
100
120
0
6000
12000
18000
24000
30000
range [m]
m/sec
1470
1483
1495
1508
1520
0
depth [m]
20
40
5 hours
60
80
100
120
0
6000
12000
18000
24000
30000
range [m]
dB
10
20
30
40
50
60
70
407
SUB-MESOSCALE MODELING OF ENVIRONMENTAL VARIABILITY
depth [m]
20
40
6 hours
60
80
100
120
0
6000
12000
18000
24000
30000
range [m]
m/sec
1470
1483
1495
1508
1520
0
depth [m]
20
40
6 hours
60
80
100
120
0
6000
12000
18000
24000
30000
range [m]
dB
10
20
30
40
50
60
70
depth [m]
20
40
7.5 hours
60
80
100
120
0
6000
12000
18000
24000
30000
range [m]
m/sec
1470
1482
1495
1507
1520
0
depth [m]
20
40
7.5 hours
60
80
100
120
0
6000
12000
18000
24000
30000
range [m]
dB
10
20
30
40
50
60
70
Figure 1. Sound speed fields and corresponding acoustic transmission loss for a 400 Hz point
source placed at 30 m depth and zero range, for selected environmental snapshots. A 30 km subsection of the sound speed field was used in the acoustic computations; the range axis is relabeled.
408
S. FINETTE ET AL.
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