Introduction
Geostrophic balance, namely the balance between the pressure gradient and the
Coriolis pseudoforce, is observed to hold to a good approximation for many
large-scale motions in the atmosphere and the ocean. The process through
which some disturbed state reaches this balance is called geostrophic
adjustment. The linear problem was first considered by .
Using conservation of momentum and mass, he derived the geostrophic steady
state corresponding to an initial perturbation. In the original publication,
Rossby noted that the final state of the system possessed less energy than
the initial state. The cause of this difference was identified by
, who showed that the end state is reached via inertial
oscillations, which disperse energy through waves. Since then, numerous
papers have used a variety of methods such as asymptotic expansions and
numerical integration to solve this linear problem. There has been a great
deal of published work on the linear problem
, but little on the
fully nonlinear one. This is partly because nonlinear problems rarely yield
analytical solutions in closed form, and partly because numerical methods
applied to the problem must accurately resolve multiple length scales.
, a key paper in the study of the nonlinear problem,
considered the adjustment problem in the context of the single-layer
shallow-water equations in one dimension. The authors built on the results of
and performed a numerical analysis of the fully nonlinear
problem with “dam break” initial conditions (see ). The
authors found that the nonlinearity and rotation led to bore generation, with
the bores dissipating energy as they propagated away from the geostrophic
state. Since the nonlinear shallow-water equations neglect non-hydrostatic
dispersion, these bores manifested as shock-like fronts. This is in contrast
to the non-rotating stratified adjustment problem which leads to the
generation of either a rank-ordered train of internal solitary waves or an
undular bore. Indeed, in this dispersive system, for the majority of
parameter space, breaking is not observed. The authors also found that the
inertial oscillations within the geostrophic state can persist for long times
and are highly dependent on the initial conditions. In their analysis of the
energy within the geostrophic state, the authors found that the ratio of
change in kinetic energy (ΔKE) to change in potential energy
(ΔPE) tended to 13, which is the theoretically predicted
value for both the linear and nonlinear problems (see ).
However, showed that, even for late times, the energy
ratio fluctuated by up to 30 % around the 13 value.
While primarily viewed from a theoretical framework, rotation-modified
adjustment has been shown to arise naturally in the ocean. Examples of this
are upwelling fronts which can create the initial density anomaly that then
must adjust; see for a more complete discussion. More
recently, and , as part of the Coastal
Mixing and Optics experiment, showed the presence of patches of well-mixed
regions (density anomalies) throughout a background of stable fluid along the
New England shelf. The specifics of this adjustment were investigated in
. The authors performed fully 3-D numerical
simulations of moderate resolution, using the nonhydrostatic equations under
the Boussinesq approximation, of the adjustment process resulting from one of
these density anomaly patches. To allow for more reasonable computation, the
authors followed the procedure outlined in , and reduced
the physical ratio for Nf (by varying f and holding N
constant), where N is the buoyancy frequency and f is the Coriolis
parameter. For analysis, they separated the energy into kinetic and
potential, and separated the domain into two regions, an inner region
associated with the geostrophic state and an outer region associated with the
waves. Since the initial conditions are static, the initial energy of the
system is contained solely as potential energy. By comparing the energy
within different areas of the simulation to the initial energy, the authors
found that initial conditions with RorL=1,
where Ror is the Rossby radius of deformation, and L
is the half-width of the initial state, were the most effective at generating
kinetic energy in the geostrophic state. This is in contrast to cases with
RorL<1, where rotation effects dominate and
little potential energy is converted to kinetic, or cases with
RorL>1, where the potential energy is primarily
converted to wave energy. In all cases considered by these authors the
radiated wave train was weak, composed of long waves and well approximated by
linear theory.
The nonlinear effects on the rotating adjustment problem have been
investigated analytically using multiple-scale perturbation analysis of the
shallow-water and fully stratified equations. In part one of a two-part paper
series, perturb the rotating shallow-water
equations using the Rossby number as their small parameter. The authors
proceed to confirm that a slow–fast splitting is possible, with the slow
state largely remaining in geostrophic balance and largely unaffected by the
fast state. In the waves that are generated,
observe shock formation and present a semi-quantitative criterion for this,
based on the initial conditions. In the second paper,
, the authors generalize their results to the case of
continuous stratification and also consider two-layer and quasi-two-layer
stratifications. perform a number of asymptotic
expansions for different initial isopycnal deviation regimes. They conclude
that for large deviations the model strongly depends on the ratio of the
layer depths and that the waves produced from the initialization obey a
Schrödinger-type modulation equation. For small deviations the waves
generated are not impacted by the geostrophic state, which is left to evolve
according to the standard quasi-geostrophic (QG) equations.
Rotation-influenced nonlinear waves have also been considered using a model
nonlinear wave equation, in this case a member of the Korteweg–de Vries
(KdV) family of equations. The KdV equation is the simplest model equation
that allows for a balance between nonlinear and dispersive effects, with a
rich mathematical structure which makes predictions of the evolution of an
initial state that are remarkably robust in both laboratory and field
settings (see ). A rotation-modified version of the KdV
equation was first derived by Ostrovsky (see
for an in-depth discussion of the equation properties and references to the
Russian literature). This new equation was subsequently analysed both through
theoretical solutions found by asymptotic expansions and through numerical
solutions. Investigation of the model equations revealed that the precise
balance between nonlinearity and dispersion that leads to the traditional
soliton solution of the KdV equation is destroyed by the addition of
rotation, and that over time the soliton breaks down into a nonlinear wave
packet . This hypothesis was later supported by
experimental results . From a theoretical point
of view, also found that the extended nonlinear
Schrödinger (NLS) equation provides a good qualitative description of the
wave packet. While the mathematical developments of the rotation-modified
theory are substantial, it is also true that this theory has a number of
pathologies not observed in the non-rotating KdV-based theory, which in
itself has been shown to misrepresent aspects of large amplitude solitary
waves (, is one of many papers to discuss some of these
discrepancies).
Work has also been performed using models with higher-order nonlinearity
but weak nonhydrostatic effects, as well as with the full
set of stratified Euler equations . Both of these
studies considered the breakdown of an initial solitary wave in the presence
of rotation. Helfrich suggested that the initial solitary wave breaks down
into a coherent leading wave packet with a trailing tail of waves.
suggested that for large amplitude, exact internal
solitary waves that are solutions to the Dubreil–Jacotin–Long (DJL)
equation, Helfrich's result was observed for artificially high rotation
rates, while rotation rates typical of mid-latitudes led to a disturbance
that never fully separated from the trailing tail. Despite differences in
details, the qualitative features observed in both studies were quite
similar. Additionally, they also performed collision experiments, finding
that the packets that emerge from the initial solitary waves can merge during
collisions and hence do not interact as classical solitons. Finally,
also found that by increasing the width of a
flat-crested wave, more energy was deposited into the tail. It remains to
reconcile the two sets of results in detail, likely by systematically
reducing the solitary wave amplitude used as an initial condition.
In this paper, we present the results of high-resolution simulations of the
geostrophic adjustment of a stratified fluid with a single pycnocline on an
experimental scale. Our simulations consider the full set of stratified Euler
equations using a pseudo-spectral collocation method. We begin by providing
and reviewing the non-rotating case and the changes that arise when the
polarity of the initial condition is changed. Next we present the general
evolution of the rotating case using classical theory and two “base” cases,
one of which is comparable to one of the cases presented in
. We subsequently identify the manner in which
nonlinearity is exhibited in the problem, focusing on both the wave train and
the geostrophic state and its inertial oscillations. We are able to clearly
show the generation of a leftward propagating wave from the initial condition
(especially evident for wider initial perturbations) and its subsequent
reflection from the left wall. This wave train interacts with the geostrophic
state, before and after reflecting off the left wall of the domain, and then
continues to propagate rightward across the tank. This is of potential
interest to future experiments. We then focus on the geostrophic state in
detail, specifically examining the change in kinetic energy and the change in
potential energy for different initial widths, as well as the changes in the
kinetic energy in the geostrophic state and the propagating wave train as the
Rossby number varies. These results make the closest contact with the work of
. Finally we draw a number of conclusions based on
our findings and identify directions for future work.
Methods
For the following numerical simulations, the full set of stratified
Navier–Stokes equations for an incompressible fluid were used, though no
span-wise variations were considered. Rotation was incorporated using an
f plane approximation and the non-traditional Coriolis terms were dropped.
For a review of the effects of the non-traditional Coriolis terms, see
. The x axis is taken as parallel to the flat
bottom with the z axis pointing upward (k^ is the upward
directed unit vector). The origin is placed in the bottom left corner so that
both axes are positive. The incompressible Navier–Stokes equations for
velocity u=[u(x,z,t),v(x,z,t),w(x,z,t)], density ρ(x,z,t), and
pressure P(x,z,t) are
DuDt+-fv,fu,0=-∇P-ρ′gk^+ν∇2u,∇×u=0,DρDt=0,
where f is the constant Coriolis parameter, g is acceleration due to
gravity, and ν is the kinematic viscosity. In accordance with convention,
we have divided the momentum equation by the constant reference density
ρ0 and absorbed the hydrostatic pressure into the pressure P. We make
the Boussinesq approximation for density and write ρ=ρ0(1+ρ′(x,z,t)), where ρ′ is considered a small perturbation. Due to
our interest in the wave dynamics in the main water column, as opposed to
details of the boundary-layer dynamics, we impose free slip boundary
conditions at the top and bottom of our domain. This will also ensure that
the boundary layer does not play a significant role in the simulations on
which we report. The walls allow us to mimic a lock–release set-up that is
used to create waves in many laboratory set-ups
. We have chosen to
neglect the span-wise dimension (y), as the lab results in
were performed away from any side boundaries and
the authors elected to neglect any curvature from the waves created. Another
change from is that we have a rigid lid as
opposed to their free surface; this is due to the computational difficulty of
a moving boundary.
In the following set of experiments the dominant dimensionless number is the
Rossby number. This number is defined as Ro=UfL, where
U is the typical wave speed, L is the typical length scale, and f is
the Coriolis parameter. This reflects a ratio of the inertia term to the
Coriolis pseudoforce term (henceforth just force). When the Coriolis force
dominates, the fluid can reach a balance between the rotation and pressure
terms, i.e., geostrophic balance. Since the full equations contain the
diffusion term ν∇2, it can be used to form the dimensionless
Reynolds number which is given by Re=ULν. U and L
are the same as for Ro and ν is the kinematic viscosity. The
other relevant number considered is the Froude number Fr=Uc, which compares the typical wave speed U to the theoretical
wave speed c. In addition to these traditional dimensionless numbers we
also define a nonlinearity parameter α. Following from
this is defined as α=ηH1,
where η is the height of the displacement in isopycnals and H1 is the
height of the undisturbed fluid interface. This parameter can be used to
modify the strength of the nonlinearity, and is well suited for shallow-water
equations. However, for the full set of incompressible Navier–Stokes
equations some ambiguity is introduced by the vertical structure of the
stratification and the initial perturbation. Nevertheless, we have found
α to be a useful parameter, likely since the disturbances in our
simulations are dominated by mode-1 waves.
The numerical simulations presented here were performed using an
incompressible Navier–Stokes equation solver which implements a
pseudo-spectral collocation method (SPINS), presented in
. The solver uses spectral methods resulting in the order
of accuracy scaling with the number of grid points. To deal with the build-up
of energy in the high wavenumbers, an exponential filter is used after a
specific wavenumber cut-off.
We computed a series of 2-D lab-scale numerical simulations on a similar
scale to the physical experiments presented in ,
which were performed using the 13 m diameter rotating platform at the
LEGI-Coriolis Laboratory in Grenoble. Motivated by the results presented in
, a domain 4 times longer than the physical tank
(Lx=52 m) was used, as 13 m is an insufficient length when considering
lower (closer to physical) rotation rates. In addition to this it was decided
to change the tank depth to a more evenly divisible 0.4 m (from a laboratory
value of 0.36 m). The density difference was set to 1 % to match
. The different physical parameters related to the
initial set-up are illustrated in Fig. .
A schematic of the tank simulation set-up which illustrates the
different parameters. The dotted line represents the isopycnal found at the
centre of the pycnocline on the far right of the domain. The largest
deflection (both polarities are shown in the figure) occurs at the left end
point of the domain. H1 and H2 represent the depth of fluid below and
above the centre of the undisturbed pycnocline respectively. H0 is the
maximum or minimum height of the pycnocline created by the initial
conditions. η is the isopycnal displacement. w is the width of the
initial condition defined from the left-hand wall to where the pycnocline
reaches within 1 % of the undisturbed height.
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In total, 8192 grid
points were used to resolve the 52 m length of the tank and 192 points were
used for the 0.4 m height, providing a 0.006 m horizontal resolution and a
0.002 m vertical resolution. To easily compare these numerical results to
the experimental values in , our Coriolis
parameter was based on their lowest presented value of f, which had a value
of 0.105 s-1. It was also decided to base the initial perturbation
width w0 on twice the Rossby radius of deformation
(Ror=Uf) so as to allow for a neater
examination of the parameter space. We used the same change in density as
, 1 % between the upper and lower fluids. In
each of the simulations, the initial conditions were given by a quiescent
fluid, and a density field defined via the isopycnal displacement η,
ρ′(x,z,t=0)=-0.005tanhz-η-0.30.01,η=±0.05exp-xw8,
where w is the half-width of the initial perturbation and the sign changes
correspond to changes in perturbation polarity.
Results
In this section we present the results of multiple numerical simulations.
Parameters were primarily modified by changing either the initial width of
the perturbation or by changing the rotation rate. Using the initial width as
the typical length scale, L=w, we are thus varying the Rossby number. The
resulting values of Ro are shown in Table . Across all
the cases, the depth does not change, and hence neither does the two-layer
linear longwave speed U=gΔρρ0H1H2H1+H2=0.0858 m s-1, where g=9.81 m s-2,
Δρ=10 kg m-3, ρ0=1000 kg m-3, H1=0.3 m, and
H2=0.1 m. Using the same wave speed and length scales as for the Rossby
number, the corresponding Reynolds numbers can be calculated; however, since
we are primarily concerned with internal waves, viscosity is negligible until
the waves disperse to scales where viscosity is dominant .
The kinematic viscosity was the same for all simulations,
ν=1×10-6 m2 s-1. Several additional experiments were
carried out by changing the initial wave amplitude, which results in
different “nonlinearity” parameters. For the initial amplitude of
η=0.05 m and undisturbed pycnocline height H1=0.3 m we have
α=0.1667. For the cases where amplitude is halved and quartered,
corresponding alpha values are α=0.0833 and α=0.0416. The
initial value of α=0.1667 allows for an easy comparison to many of the
figures in which are based on a value of 0.1. In
addition to the simulations above, another set of simulations was performed
using the opposite polarity of the initial disturbance. These opposite
polarity simulations correspond exactly to the cases seen in
Table , the only difference being the sign in the isopycnal
displacement used in the initial conditions.
Rossby number of each simulation, where f0=0.105 s-1 and
w0=1.63 m.
Ro
14w0
12w0
w0
2w0
4w0
f0
2
1
12
14
18
12f0
4
2
1
12
14
14f0
8
4
2
1
12
Several simulations were also performed on an extra-long tank to investigate
the long-time results of adjustment. For these simulations the tank length
was L=260 m and the number of horizontal grid points was increased
to 16 384, providing a 0.0158 m resolution. The vertical height and
grid points were kept the same from the smaller case.
Unless otherwise stated the following scaling is used for all figures:
T=1f, Lz=Lz, and Lx=Ror, with Lz=0.4 m
corresponding to the depth of the tank. For kinetic energy, we scale by the
maximum kinetic energy for all spaces and times to show relative changes.
The non-rotating case
We begin by reproducing the results of the adjustment problem without
rotation. The solution to this problem is well known, though we are not aware
of any references that present the result in detail. We thus state the
result, with a numerical example, and briefly outline the weakly nonlinear
theory behind it. Non-rotating adjustment yields either a rank-ordered train
of solitary waves or an undular bore forming from the initial disturbance,
depending upon the polarity of the initial disturbance. Examples of these two
cases are shown in Fig. . Since there is no rotation, the
advective timescale was chosen, T=LU, to nondimensionalize time,
with the initial width w=12w0 chosen for the typical length
scale. The stark difference between these cases is readily apparent in both
types of plots.
A space–time filled contour plot of vertically integrated kinetic
energy and density isocontours at t=4275 showing the differences between
the positive and negative polarity initial conditions with w=12w0. Panels (a) and (c) correspond to the positive
polarity case, and panels (b) and (d) to the negative
case.
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The result may be understood in terms of KdV theory. Using the notation of
, separation of variables is applied to the streamfunction so
that
ψ(x,z,t)=B(x,t)ϕ(z).
The vertical structure is determined from a linear eigenvalue problem, while
to first order in amplitude and aspect ratio B(x,t) is governed by a KdV
equation for waves propagating in each direction. The KdV equation
corresponding to rightward propagating waves reads as
Bt=-cBx+2cr10BBx+r01Bxxx,
where c is the linear longwave speed, r10 is the nonlinearity
coefficient, and r01 is the dispersive coefficient. The numerical value
for c is computed from the linear longwave eigenvalue problem (Lamb's
Eq. 8a), while r10 and r01 are computed from the integral
expressions involving the eigenfunctions of the same problem (Lamb's Eqs. 10a
and b). The dispersive coefficient, r01, is always negative, while the
nonlinear coefficient, r10, switches sign depending on the functional
form of the stratification. In the case of a two-layer flow exact expressions
can be derived. Solitary wave solutions of Eq. () are of the
classical sech2 form. The propagation speed equals the linear longwave
speed to leading order, with a nonlinear correction that is proportional to
amplitude and r10 (Lamb's Eq. 17). Thus the sign of r10 also
determines solitary wave polarity. In the absence of background shear
currents this implies that stratifications centered above (below) the
mid-depth yield solitary waves of depression (elevation). All numerical
experiments performed with exact solitary waves computed using the DJL
equation that we are aware of match the predictions of the KdV theory
presented above, as far as solitary wave polarity is concerned. Of course,
KdV theory is not necessarily a quantitatively accurate predictor of the
structure of large solitary waves (, is one of many papers
to discuss some of the discrepancies).
Rotation-modified evolution
As discussed in the introduction, a variety of model equations have been
derived that account for the effects of rotation, with
providing a relatively recent summary. The
essential aspects of the role of rotation can be gleaned from linear theory.
In this case the streamfunction is governed by
μ2ψxx+ψzztt+1Ro2ψzz+1Fr2N(z)2ψxx=0,
where μ=HL is the aspect ratio. When the assumption of a linear
stratification is made, the vertical structure of ψ is sinusoidal, so
that for the first vertical mode a separation of variables like
Eq. () yields
μ2Bxx-π2H2Btt-π2H21Ro2B+1Fr2Bxx=0.
The well-known dispersion relation of rotation-modified internal waves in a
channel is readily recovered by assuming a travelling wave solution
. In the hydrostatic limit this equation reduces to the
simplest example of a partial differential equation that is both hyperbolic
and dispersive, the classical Klein–Gordon equation of mathematical physics,
Btt+1Ro2B=H2π2Fr2Bxx.
By using the plane wave ansatz, it can
immediately be seen that the dispersion relation yields a lower bound on
frequency in the longwave limit, and hence the phase speed is unbounded in
this same limit. This is the central problem that model equation theories
such as the Ostrovsky equation face when implemented numerically. It is also
readily apparent from Eq. () that a non-trivial, time
independent state is possible, and this state corresponds to the geostrophic
state of classical geophysical fluid dynamics .
Furthermore, it is clear that a spatially independent inertial oscillation is
a possible solution to the equation. However, for a given initial condition
it is not immediately obvious what the precise split is between the portion
of the initial state that propagates away and the portion left behind. While
the case in which the disturbance that emanates from the initial condition is
small enough to be well described by linear wave theory has been studied in
detail by Lelong and Sundermeyer , the significant
amount of literature on the combined effects of nonlinearity, dispersion, and
rotation, and especially the experimental results in
, suggest that the initial value problem should
be reconsidered without a priori approximations.
Using our definition of the Rossby number, we find that the experiments
presented in , with a Coriolis parameter of
f=0.105 s-1, had a corresponding Rossby number of 0.667. Therefore,
we consider the f=f0 and w=w0 case (Ro=0.5) as our baseline.
We also pick a negative initial condition to match their configuration. With
this in mind, Fig. compares this baseline case with one where the
only difference is that the rotation rate has been quartered
(f=14f0, Ro=2). Figure a and c correspond
to the baseline case, while Fig. b and d correspond to the reduced
rotation case. Figure a and b show vertically integrated kinetic
energy space–time plots, while Fig. c and d show density isolines
at t=47.25 and t=11.81 respectively. The vertical lines correspond to the
locations of the waves that emanate from the initial disturbance as described
by linear theory. We computed the spectrum of the horizontal velocities to
extract the dominant wavenumbers (k≈0.84 and k≈0.48
respectively) and used the algorithm outlined in to
calculate the speeds. We have presented both the linear phase and linear
group speeds.
Panels (a) and (b) show a space–time plot of
vertically integrated kinetic energy, while panels (c) and
(d) show three density isolines at t=47.25 and t=11.81
respectively. Panels (a) and (c) correspond to the
Ro=0.5, f=f0, and w=w0 case, and panels (b) and
(d) to the Ro=2, f=14f0, and w=w0 case.
The vertical lines in panels (c) and (d) represent the
distance the waves would have travelled according to the linear phase and
group speed.
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Comparing the results seen in Fig. with Fig. b and c
(since they both began with a negative polarity initial condition), there are
immediate differences in both styles of plots. The most striking of these
differences are the retention of energy in the geostrophic state, and the
spreading of the ejected waves. The geostrophic state is visible in all plots
along the left-hand side of the tank (near the wall). Comparing the two
columns in Fig. , the case with a stronger rotation rate traps
more energy in the geostrophic state. We will investigate differences in the
geostrophic state in Sect. 3.4. The wave spreading is visible in the
space–time plots as the waves propagate and within the new structure of the
density isolines. At both rotation rates, the solitary wave, which is
produced in the non-rotating case, has broken down into a series of smaller
waves. A transition also appears to occur in the wave speed as the rotation
rate changes. In Fig. d the wave front roughly corresponds to the
linear phase speed (which is, in turn, a good approximation of the solitary
wave propagation speed), while in Fig. c the front appears to have
shifted to the linear group speed. This suggests that the low rotation case
develops a wave train that can be interpreted as a rotation-modified solitary
wave (at least on the timescales considered), while the high rotation case
develops a wave train that can be interpreted as a wave packet. The
importance of nonlinearity for both of these cases, and indeed for the
geostrophic state, remains to be assessed.
Observing the structure that appears throughout the figure, we argue that
these waves closely resemble a modulated wave packet as presented in
, instead of the rotation-modified bore seen in
. When comparing to the work done by
, we first note that for our simulations the nonlinear
parameter is quite small at α≈0.166. However, their work
suggests that, even for this small value and smooth initial condition,
breaking will still occur. The present simulations were carried out with the
full set of incompressible Navier–Stokes equations. As such, the dispersion
that is neglected in the shallow-water equations, used by
, becomes important when the wave front steepens.
Dispersion breaks the front down into a train of smaller waves and eliminates
shock formation. In addition to the change in steepening dynamics, the
initially localized waves disperse over time, yet are observed to remain
bound together (corresponding to the width of the packet envelope). For this
reason we find that the modulated wave packet is a better description for
these dynamics, though we note that in all our simulations the wave packet
never completely separates from the trailing waves. This description is also
supported by the shift in propagation speed to the linear group speed, as
this is the first-order estimate of the speed which a wave packet would
propagate at.
Nonlinear and polarity effects
Since the majority of the classical literature on the geostrophic adjustment
problem considers the linear problem, it is important to clearly identify
those aspects of our simulations that are nonlinear in nature. One way to
investigate the nonlinear effects in the evolution, shown in
Fig. , is to consider how the spectrum evolves in time, since (in
the absence of dissipation) linear dispersive waves maintain the spectrum of
the initial conditions for all times. Figure shows the spectrum
of the horizontal velocity profile at the surface (the results at other
depths, and indeed for other fields, yielded qualitatively unchanged results)
at various times for both cases shown in Fig. . Respectively,
these correspond to t=15.75 and t=3.94, t=31.50 and t=7.87, t=47.25
and t=11.81, and t=63 and t=15.75. The spectral power density was
scaled by the maximum power for all profiles shown in order to highlight the
differences. It is readily apparent from Fig. a that in the
Ro=0.5, f=f0, and w=w0 case there is little change in the
spectrum as time evolves. As time increases there appears to be a slow decay
in the power at the excited wavenumbers. There is no shift in wavenumber for
the various peaks, or indeed any other major change evident in the spectrum.
This is not the case in Fig. b for the Ro=2,
f=14f0, and w=w0 case. The spectrum in this case contains
large fluctuations (more than 25 % for the peak value) in power and
shifts in the excited wavenumbers. These changes in the spectra as time
evolves are hallmark effects of nonlinearity, and hence indicate that, while
the emanating wavetrain in the Ro=0.5 case appears to be well
described by linear theory, if we consider weaker rotation effects such as in
the Ro=2 case, linear theory is no longer a useful description.
A comparison of how the horizontal velocity spectra for the
Ro=0.5, f=f0, and w=w0 case (a) and the
Ro=2, f=14f0, and w=w0 case (b) change
over time. The spectral power has been scaled by the maximum between the
cases to highlight the differences. The green line corresponds to t=15.75
and t=3.94, the black line to t=31.50 and t=7.87, the blue
line to t=47.25 and t=11.81, and the red line to t=63 and t=15.75.
There are clear changes in the spectra for the weakly rotating case as the
waves evolve. In the strong rotation case the only temporal differences are a
slow decay. The differences seen in the weakly rotating case highlight the
nonlinear effects which are present in this regime.
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A comparison of the 1-D KE for several different cases to outline
the effects of nonlinearity. All plots are taken at t=1140 for the
non-rotating case, t=25.2 for the f0 case (w=12w0,
Ro=1), and t=6.3 for the 14f0 case
(w=12w0, Ro=4). The reduced amplitude nearly linear
cases have been scaled by the change in amplitude squared. The kinetic energy
has also been normalized by the maximum of the non-rotating negative polarity
case. Panel (a) highlights the differences that arise from halving
the initial amplitude and by changing the polarity of the non-rotating case.
Panel (b) compares the same changes as the panel above; however, we
have included rotation, and also consider a “nearly linear” case with an
initial amplitude of 1200η0. Panel (c) is a
comparison of different negative polarity cases, one with no rotation and the
standard amplitude, and two others at 14f0 with standard and half
amplitude.
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In order to investigate these effects in a more systematic manner, we started
from the case with w=12w0 (which was the smallest width that
still produced a solitary wave) and ran several simulations where we varied
the amplitude (by halving it), varied the rotation rate (which was quartered
to show clear differences), compared the different polarities, and considered
an extremely small amplitude “nearly linear” case. To compare with known
nonlinear wave results we computed the non-rotating version of a number of
the cases. A comparison of the 1-D-averaged KE for a number of the cases is
presented in Fig. , where we have scaled any reduced amplitude
cases so that, were linear theory to apply, the curves would collapse onto a
single profile. Kinetic energy was chosen as the variable shown, since it
provides information about both the structure of the dynamics and the
magnitude of the velocities. Figure a shows how the non-rotating
adjustment yields waves that are profoundly affected by changes in polarity
and to a lesser degree by changes in amplitude. A solitary wave train is
observed for the negative polarity case and an undular bore for the positive
polarity case. The change in amplitude results in a phase shift; however, the
amplitude of the solitary wave remains nearly constant. These results are a
clear indication of nonlinear behaviour for the non-rotating case. In
Fig. b we compare several cases with rotation following a similar
methodology to Fig. a; however, we have included our “nearly
linear” case where the amplitude has been reduced by a factor of 200. The
change in polarity does not significantly change the dynamics of the ejected
waves, with the largest change between these cases being that the positive
polarity case has a higher amplitude both within the wave packet and in the
geostrophic state. The effect of changing the amplitude does not
significantly change the wave packet, since the wave packet is quite small in
this case, and hence to leading order can be understood from the point of
view of linear dispersive wave theory, similarly to what was observed based
on the spectrum in the discussion above (Fig. and the related
discussion). For the geostrophic state, the changes in the initial
disturbance amplitude result in changes to the amplitude and the location of
the peak in kinetic energy, with the reduction in amplitude yielding a
greater than linear response in the amplitude of the geostrophic state.
Again, changing polarity yields the most significant changes. Linear theory,
as exemplified by the green curve, provides a reasonable prediction, though
details are amplitude dependent. For Fig. c we kept the polarity
of the initial disturbance negative and compared the change in amplitude for
a smaller rotation rate as well as the non-rotating case. The lower rotation
rate allows for more energy to be deposited into the wave train. The primary
change for the reduction in amplitude is that the individual waves within the
wave packet of the scaled reduced case (in blue) appear to be larger in
amplitude compared to the base case (in black). There also appears to be a
slight phase shift between the cases (consistent with a packet that travels
at a slightly different speed). Comparing these two cases to the non-rotating
case shows that, while the peak in energy has been shifted back, the wave
front of the solitary wave and the wave packets are at roughly the same
location. For this lower rotation case there is very little difference in the
geostrophic state as a result of amplitude reduction, implying that for low
rotation rates the geostrophic state can be well described by linear theory.
To investigate the nonlinear effects that arise from changes in polarity in
the geostrophic state (Fig. b), long-time simulations with a
rotation rate set to f=2f0 and an initial width of w=12w0 were
computed, resulting in a Rossby number of 12. These results are
presented in Fig. . Figure a and b show the vertically
integrated kinetic energy at the location of the maximum induced by the
geostrophic state. Figure c and d show the total spatial
distribution of the normalized kinetic energy in the region around the
geostrophic state at 720 s (151.2), along with three contours of constant
density, for a negative and positive initial polarity respectively.
Figure a clearly shows the energy difference that was seen in
Fig. b, indicating that the positive polarity case appears more
efficient at keeping energy in the geostrophic state. Figure b
shows the time series of the logarithm of kinetic energy after the packet has
been ejected. If we ignore the inertial oscillations, which appear to be
rapid on the timescale shown, we can see a clear decay. From this panel it is
also possible to note that the oscillations appear to persist significantly
longer in the positive case. By computing the logarithm of the time series
(Fig. b) we are able to show that the decay is nearly exponential,
with the positive polarity case decaying roughly 5 % faster. The decay
rate decreases over time. From the bottom two panels, Fig. c and
d, it is clear that the polarity of the geostrophic state strongly modifies
the vertical distribution of the kinetic energy. Note in particular the
difference in strength of kinetic energy below the pycnocline and the tilt of
the high kinetic region that follows the deformed pycnocline.
Long-time simulations comparing the differences in the geostrophic
state for negative and positive initializations. For both simulations
f=2f0, w=12w0 and Ro=12.
Panel (a) presents the long-time time series of vertically
integrated kinetic energy in the geostrophic state for both cases.
Panel (b) shows the base-10 logarithm of the geostrophic state
kinetic energy for both cases after the packet has been ejected.
Panels (c) and (d) show the shaded distribution of kinetic
energy in the geostrophic state along with contours of constant density, for
a negative and positive initial polarity respectively.
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To quantify the nonlinear behaviour of the geostrophic state and the inertial
oscillations that accompany it, Fig. shows the differences in
total kinetic energy within the geosotrophic state between our original
amplitude cases and cases with a 10-fold reduction in amplitude. As in
Fig. we have scaled up the reduced amplitude cases, and thus for
a purely linear problem there should be no differences between the two curves
shown. Figure shows five cases of differing initial width and the
same Coriolis parameter f=f0, Fig. a to w=14w0
(Ro=2), Fig. b to w=12w0
(Ro=1), Fig. c to w=w0 (Ro=0.5),
Fig. d to w=2w0 (Ro=0.25), and Fig. e to
w=4w0 (Ro=0.125). Though the nearly linear case does behave in
a qualitatively similar manner to the original cases, there are key
differences. For one, in Fig. a–c there are clear magnitude
differences between the two cases. For almost all times the scaled kinetic
energy curve for the amplitude reduced case lies below the corresponding
curve for the original cases. This remains true for the later cases
(Fig. d and e), though not to the same extent. Second, closely
examining the inertial oscillations reveals that they appear to decay more
rapidly for the amplitude reduced cases compared to the original cases; this
can especially be seen in Fig. c and d. Only for the two rightmost
panels (Ro≤0.25) could it be said that the two curves are
nearly coincident. Thus even the geostrophic state exhibits clear evidence of
nonlinearity.
The difference in total inner kinetic energy between our original
amplitude cases and cases where the amplitude has been reduced by a factor of
10. The reduced cases have then been linearly scaled to account for this
amplitude change. The original energies are shown in red, while the reduced
ones are in blue. Discrepancies between the two cases are due to nonlinear
effects. (a) w=14w0 (Ro=2),
(b) w=12w0 (Ro=1), (c) w=w0
(Ro=0.5), (d) w=2w0 (Ro=0.25), and
(e) w=4w0 (Ro=0.125).
![]()
A space–time plot of kinetic energy. The different columns
correspond to different combinations of f and w used to form a value of
Ro=1. (a) f=f0 and w=12w0,
(b) f=12f0 and w=w0, and (c) f=14f0 and w=2w0. The second row corresponds to the same case as the first
columns, but the aspect ratio has been scaled by the change in rotation
rate.
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The primary dynamic variable for these simulations is the Rossby number since
both changes to rotation rate and changes to the initial width are both just
modifications to this dimensionless parameter. A different manner in which
the effects of nonlinearity may be investigated is by asking whether the
dynamics collapse onto a single case for the same Rossby number;
Fig. a–c show the space–time plots of vertically integrated
kinetic energy for three cases with the same Rossby number but different
combinations of parameters. The Froude number, Fr=Uc, is
computed dynamically, with U set by the maximum horizontal velocity from
the simulation at a given time and c given by the linear group speed
calculated using the algorithm outlined in using the
dominant wavenumbers (k≈0.84 and k≈0.48 respectively) for
each rotation rate. Fig. a corresponds to f=f0 and
w=12w0, Fig. b to f=12f0 and w=w0,
and Fig. c to f=14f0 and w=2w0. The axis of
Fig. b and c has been scaled by the corresponding change in
Coriolis parameter (12 and 14 respectively).
Figure d shows the time series of the Froude number,
Fr=Uc. While the oscillations of the geostrophic state
near x=1 in the space–time plots are quantitatively similar, the number
and shape of the waves that are produced in the wave train are slightly
different. The reason for these differences is highlighted in
Fig. d, where the Froude numbers match for early times, but begin
to drift rapidly. These results again illustrate the importance of nonlinear
effects within this system and the necessity to include such effects when
modeling the system.
During the analysis of the numerical experiments that varied the width of the
initial condition, an interesting observation about multiple wave trains was
made. The initial condition yields both rightward and leftward propagating
waves. For narrow initial conditions the leftward propagating waves reflect
from the left wall early in the simulation and are difficult to disentangle
from the initially rightward propagating wave train. However, for wider
initial conditions the leftward propagating waves must travel a longer
distance before reflecting off the wall, allowing for them to appear separate
from rightward propagating waves. This interaction is shown in
Fig. using the potential energy field because the amount of
span-wise velocity created in the geostrophic state is so great that it
drowns out this reflection signal in the kinetic energy field. Both cases
maintain the same Coriolis parameter f=f0, but Fig. a
corresponds to w=0.5w0, while Fig. b corresponds to w=2w0.
In Fig. a it is difficult to distinguish the two wave trains
(though we have superimposed coloured arrows in order to accentuate the
pattern for the reader). This distinction is much clearer in
Fig. b, where the leftward travelling wave takes roughly twice as
long to reach the left wall. In this case it is possible to distinguish the
wave trains within the pattern of waves that are produced. Thus a natural
method of generation of waves in a tank will create waves in both directions
which must be accounted for in the interpretation of physical experiments.
Space–time pseudocolour plots of the change in potential energy for
the Ro=1, f=f0, and w=0.5w0 case (a), and the
Ro=0.25, f=f0, and w=2w0 case (b). As for the
kinetic energy, it has been scaled to the maximum value. Visible in both
cases (though significantly easier to see in panel b), there are
both rightward (black arrow) and leftward (white arrow) propagating wave
trains created by the initial conditions. The leftward travelling wave train
will eventually reflect off the close left-hand wall and propagate
rightwards. If the initial conditions are quite narrow, the leftward
propagating wave train reflects quickly off the wall and is difficult to
disentangle from the rightward propagating wave train.
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The geostrophic state
In the rotation-modified adjustment problem there are two dominant features,
the geostrophic state that is left over from the initial conditions and the
train of Poincaré waves that carries energy away from it. For this
section we will focus on the dynamics, and changes, of the geostrophic state.
We will primarily be comparing our results with those from
, who performed 3-D numerical simulations of the
adjustment problem. focused on the
energetics of the geostrophic state that are generated by a well-mixed region
of intermediate density fluid. Though not exactly the same case,
performed their simulations using the full set of
equations and thus provide an apt point of comparison. To facilitate this
comparison between our work and theirs, Table provides a
translation of our notation to that of .
A major difference between the two sets of experiments is the background
stratification and density anomaly. As given explicitly in
Sect. , we have a two-layer stratification given by the tanh
function with an anomaly also given by the tanh function.
use a localized anomaly diffusivity to create a
two-lobed axisymmetric lens density perturbation, with a linear background
stratification.
A comparison of the notation in our work and that of
.
Our notation
Ror
R
Ro
R/L
w
L
Geostrophic state KE / PE
KEv / PEv
Outside geostrophic state KE / PE
KEw / PEw
Figure shows space–time plots of vertically integrated kinetic
energy within the geostrophic state for five cases of different initial
widths where the rotation rate has been held constant at f=f0. These cases
have Rossby numbers 2, 1, 12, 14, and 18
for Fig. a, b, c, d, and e respectively. The figure has been
saturated by the maximum kinetic energy across all cases. Once
Ro≥1 (Fig. b–e), the geostrophic state shows clear
oscillations within the kinetic energy. These spikes in kinetic energy occur
during the vertical oscillations of the geostrophic state. It is also
possible to see from this figure that all the cases initially spike with
roughly the same magnitude of kinetic energy, but then differ greatly
depending on the Rossby number. In Fig. d and e it is possible to
identify the reflected wave interfering with the oscillations of the
geostrophic state, matching the features seen in Fig. .
A space–time plot of vertically integrated kinetic energy, in the
geostrophic state, for different values of w while f is held constant at
f=f0. (a) w=14w0 (Ro=2),
(b) w=12w0 (Ro=1), (c) w=w0
(Ro=0.5), (d) w=2w0 (Ro=0.25), and
(e) w=4w0 (Ro=0.125).
![]()
The changes in potential and kinetic energy, compared to the
initialization, for the cases in Fig. . Panel (a)
corresponds to the change in potential energy, and panel (b) to the
changes in kinetic energy. The energy in both plots has been scaled by the
base case (f=f0, w=w0).
![]()
To compare the energy within the geostrophic state between cases, and with
the published literature, we horizontally integrate the geostrophic state
(the region shown in Fig. ) to produce a time series of both the
kinetic and potential energies. Following what was done in
, we compute the difference in these energies compared to
the initial state. The results of this are shown in Fig. . The
extent of the geostrophic state is defined as twice the distance from the
left-hand wall to the maximum in kinetic energy. Due to the nature of our
initial conditions, namely that we start with a smooth transition and still
fluid, the ratio ΔKE/ΔPE, which is used in
, is difficult to use since the change in
potential energy may be zero if it reaches its initial state during
oscillation. While it would be possible to use the reciprocal of the ratio,
we have chosen to present the differences separately (Fig. a for
ΔPE and Fig. b for ΔKE). The energy in both plots
has been scaled by the base case of f=f0 and w=w0. As in the results
presented in Fig. 8 of , for larger initial widths
(smaller Rossby numbers) there are correspondingly larger oscillations in
both potential and kinetic energy. Our simulations show that these
oscillations persist for long times, in agreement with the results of
. In a similar manner to what is seen in
, the case with Ro=1 (f0 and
12w0) appears to retain the maximum amount of potential energy
(as opposed to kinetic energy for ). However, we
have verified that the results seen in their figure can be generated by
scaling the kinetic energy by the initial energy (not shown here). We also
computed the linear kinetic energy for the geostrophic state following
's Eq. (9), using the parameter set for our base case. We
then compared this to the maximum kinetic energy in the geostrophic state. We
calculated the linear KE to be 4.06327×10-5, while our KE was
3.63771×10-5, which is roughly an 11 % difference.
The total kinetic energy located inside (blue) and outside (red) of
the geostrophic state. In this figure we have the same cases as in
Figs. and , separated into their own panels.
Panel (a) corresponds to 14w0 (Ro=2),
(b) corresponds to 12w0 (Ro=1), (c)
corresponds to w0 (Ro=0.5), (d) corresponds to 2w0
(Ro=0.25), and
(e) corresponds to 4w0 (Ro=0.125). The red line
(inner) is the energy inside the geostrophic state, while the blue line
(outer) corresponds to that outside.
![]()
We next consider how the time evolution of the total kinetic energy inside
the geostrophic state compares to that outside; this is shown in
Fig. . From Fig. a we can see that for low rotation
rates there is much less energy retained within the geostrophic state, and
that the oscillations of this remaining kinetic energy are very small. We can
compare this panel with Fig. 13 of and see
similar results, namely the dominance of the kinetic energy outside the
geostrophic state. We should also note that there is a steady decrease in
kinetic energy in the outer kinetic energy which is due to dissipation of the
waves throughout the numerical domain. As in Fig. 6 of
we see a similar separation between the energy of
the inner and outer regions for the Ro=1 case (our
Fig. b). The inner kinetic energy oscillations of
Fig. d and e do not appear to have reached a steady state by the
end of our simulation, but have reached a sufficiently close value to
interpret. In Fig. d there is an equal amount of energy in the
final inner and outer regions. These results seen in Fig. d and e
are consistent with the results from 's Fig. 14,
namely significantly more energy being retained inside the geostrophic state,
resulting in much larger amplitude oscillations.
Motivated by the results shown in Fig. , we considered the
vertical structure of the inertial oscillations. We confirmed that for all
cases the gradient Richardson number (including the v component of shear)
never dips below 0.25 in the stratified region, thereby suggesting the
inertial waves are not strong enough to induce shear instability. Indeed, the
isopycnal displacements associated with the inertial oscillations were never
larger than about 2.5 % of the total depth. For early times and low
Rossby numbers (Ro≤0.125), the spatiotemporal (in z and t)
structure of the kinetic energy field induced by the inertial waves followed
a separable structure. For the Ro=0.25 case evidence of a
nonseparable structure was clear for t>100. In comparison, for the
Ro=0.5 case a non-separable structure was evident by t=60.
However, since the inertial oscillations are smaller in this case at later
times (t>150), the signature of the inertial waves is masked by that of the
geostrophic state.
Conclusions
In this paper we have taken a systematic approach to the classical
rotation-modified stratified adjustment problem. Building on results based on
shallow-water theory presented in , we have shown that by
using the fully nonlinear incompressible Navier–Stokes equations, under the
Boussinesq approximation, the waves that are ejected from the geostrophic
state do not steepen to a shock. Once the wave front steepens sufficiently it
disperses into a primary wave packet and a tail of smaller dispersive waves.
We demonstrated that the nonlinear wave packet interpretation of the wave
train of is appropriate in some parameter
regimes, with changes in amplitude reflected in the phase of the nearly
solitary wave response. By mapping out the parameter space we have shown
that, as expected, the Rossby number is the controlling variable for the
dynamics in this problem. For Ro<1, the wave packet propagates
with a speed roughly corresponding to the linear group speed, while for
Ro>1, the packet propagates with a speed closer to the linear
phase speed. We have further characterized the nonlinear effects present in
both the wave packet and the geostrophic state. The effects of nonlinearity
were investigated by considering different initial amplitudes and changes in
polarity. As in the non-rotating case, the largest nonlinear effects occurred
as a result of changes in polarity, both in the geostrophic state and in the
wave packet ejected. Surprisingly, the high Rossby number cases yielded
nonlinear effects in both the wave train and the geostrophic state. However,
as a general rule amplitude effects were smaller than polarity effects.
A different approach to characterizing nonlinear effects is to create
different combinations of parameters that yield the same Rossby number. We
carried out this process and tracked the time dependent Froude number. While
the qualitative features of the evolution were similar in all three cases
shown, the variations in the Froude number led to significant differences in
the details of the wave train generated. The characterization of these
various nonlinear effects in a single simulation is new and significant,
providing a guideline for when linear theory can be applied and when
nonlinear effects must be considered.
Our results show that the inertial oscillations in the geostrophic state can
persist for long times, in agreement with . However, the
inertial oscillations never reach large enough amplitudes to induce shear
instability. Our results also match the work published by
, specifically matching their results for the
different Rossby number regions (Ro<1, Ro=1, and
Ro>1). By comparing the kinetic energy within the inner
geostrophic and outer non-geostrophic regions, we show that the amplitude of
the geostrophic oscillations increases quickly as the Rossby number
decreases. However, this in turn corresponds to less prominent nonlinear
effects within the geostrophic state.
Another significant finding presented is the generation,
reflection, and interaction of a
wave train propagating in the opposite direction (leftward) during the
initial generation. For any physical tank set-up, this reflected wave will
impact any measurements of the waves generated and especially any
measurements of the geostrophic state.
In addition to the work described in the previous paragraph, future work
should consider span-wise variations, especially in the case of the strong
geostrophic state for which novel instabilities may be possible (though as
noted above, on laboratory scales shear instability is not expected).
Systematic studies of the shoaling of rotation-modified solitary waves and
undular bores should also be carried out, since it is not known in what
manner these may be different from shoaling in the non-rotating case. A more
theoretical avenue could quantitatively compare weakly nonlinear and weakly
dispersive–strongly nonlinear model equations to the full stratified
equations.