The study of the adjustment to equilibrium by a stratified fluid in a rotating reference frame is a classical problem in geophysical fluid dynamics. We consider the fully nonlinear, stratified adjustment problem from a numerical point of view. We present results of smoothed dam break simulations based on experiments in the published literature, with a focus on both the wave trains that propagate away from the nascent geostrophic state and the geostrophic state itself. We demonstrate that for Rossby numbers in excess of roughly 2 the wave train cannot be interpreted in terms of linear theory. This wave train consists of a leading solitary-like packet and a trailing tail of dispersive waves. However, it is found that the leading wave packet never completely separates from the trailing tail. Somewhat surprisingly, the inertial oscillations associated with the geostrophic state exhibit evidence of nonlinearity even when the Rossby number falls below 1. We vary the width of the initial disturbance and the rotation rate so as to keep the Rossby number fixed, and find that while the qualitative response remains consistent, the Froude number varies, and these variations are manifested in the form of the emanating wave train. For wider initial disturbances we find clear evidence of a wave train that initially propagates toward the near wall, reflects, and propagates away from the geostrophic state behind the leading wave train. We compare kinetic energy inside and outside of the geostrophic state, finding that for long times a Rossby number of around one-quarter yields an equal split between the two, with lower (higher) Rossby numbers yielding more energy in the geostrophic state (wave train). Finally we compare the energetics of the geostrophic state as the Rossby number varies, finding long-lived inertial oscillations in the majority of the cases and a general agreement with the past literature that employed either hydrostatic, shallow-water equation-based theory or stratified Navier–Stokes equations with a linear stratification.

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

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

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,

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

Work has also been performed using models with higher-order nonlinearity

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

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

In the following set of experiments the dominant dimensionless number is the
Rossby number. This number is defined as

The numerical simulations presented here were performed using an
incompressible Navier–Stokes equation solver which implements a
pseudo-spectral collocation method (SPINS), presented in

We computed a series of 2-D lab-scale numerical simulations on a similar
scale to the physical experiments presented in

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.

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

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,

Rossby number of each simulation, where

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

Unless otherwise stated the following scaling is used for all figures:

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.

A space–time filled contour plot of vertically integrated kinetic
energy and density isocontours at

The result may be understood in terms of KdV theory. Using the notation of

As discussed in the introduction, a variety of model equations have been
derived that account for the effects of rotation, with

Using our definition of the Rossby number, we find that the experiments
presented in

Panels

Comparing the results seen in Fig.

Observing the structure that appears throughout the figure, we argue that
these waves closely resemble a modulated wave packet as presented in

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.

A comparison of how the horizontal velocity spectra for the

A comparison of the 1-D KE for several different cases to outline
the effects of nonlinearity. All plots are taken at

In order to investigate these effects in a more systematic manner, we started
from the case with

To investigate the nonlinear effects that arise from changes in polarity in
the geostrophic state (Fig.

Long-time simulations comparing the differences in the geostrophic
state for negative and positive initializations. For both simulations

To quantify the nonlinear behaviour of the geostrophic state and the inertial
oscillations that accompany it, Fig.

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 space–time plot of kinetic energy. The different columns
correspond to different combinations of

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.

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.

Space–time pseudocolour plots of the change in potential energy for
the

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

A major difference between the two sets of experiments is the background
stratification and density anomaly. As given explicitly in
Sect.

A comparison of the notation in our work and that of

Figure

A space–time plot of vertically integrated kinetic energy, in the
geostrophic state, for different values of

The changes in potential and kinetic energy, compared to the
initialization, for the cases in Fig.

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.

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.

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.

Motivated by the results shown in Fig.

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

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

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.

Given the length of time that this paper has been under review, the data sets are not available in a single location, and are not submitted to a repository. We are always willing to share data, and so the data for the numerical simulations, including source code, are available by email request to the corresponding author.

This research was supported by the Natural Sciences and Engineering Research Council of Canada through Discovery Grant RGPIN 3118442010. Edited by: R. Grimshaw Reviewed by: two anonymous referees