In this paper we study the spatiotemporal properties of waves in the
Lorenz-96 model and their dependence on the dimension parameter

In this paper we study the Lorenz-96 model which is defined by the equations

Recent papers with applications of the Lorenz-96 model and the values of

The Lorenz-96 model was introduced as a tool for numerical
experiments in predictability studies, rather than as a physically
realistic model. Indeed,

Hovmöller diagrams of a periodic attractor (

Power spectra of the attractors of Fig.

Table

In this paper we address the question of how the spatiotemporal
properties of waves, such as their period and wave number, in the
Lorenz-96 model depend on the dimension

In addition to

The remainder of this paper is organized as follows. In
Sect.

In this section we consider a general geophysical model in the form
of a system of ordinary differential equations:

Assume that

In this section we study waves in the Lorenz-96 model and how their
spatiotemporal characteristics depend on the parameters

Graphs of the functions

For all

The proof of Theorem 1 can be found in

From the eigenvalues that cross the imaginary axis and the
corresponding eigenvectors we can deduce the physical
characteristics of the periodic orbit that arises after a Hopf
bifurcation. When the

Note that Hopf bifurcations of an

As the equilibrium

Now assume that

Note that for even

We now consider the dimension

As Fig.

As

Bifurcation diagrams obtained by continuation of the equilibrium

Parameter values of the first Hopf bifurcation

The computations for the equilibrium

The case

The question is whether the results described above persist for
even dimensions

If

If

In spite of the aforementioned quantitative differences between the
cases

The results of Sect.

Multi-stability also occurs when

In previous work

Bifurcation diagram of the two-parameter system (Eq.

Figure

In order to explain the dynamics in a neighborhood around the
double-Hopf point, we now use Fig.

The scenario described above shows how the presence of two
subcritical Neĭmark-Sacker bifurcations emanating from
a double-Hopf bifurcation determines a region of the

Double-Hopf bifurcations are abundant in the two-parameter Lorenz-96
model of Eq. (

Figure

Continuation of periodic orbits for

As Fig.

As Fig.

Figures

The double-Hopf bifurcation has been reported in many works on
fluid dynamical models. A few examples are baroclinic flows

In this paper we have studied spatiotemporal properties of
waves in the Lorenz-96 model and their dependence on the
dimension

For

The occurrence of pitchfork bifurcations before the Hopf
bifurcation leads to multi-stability, i.e., the coexistence of
different waves for the same parameter settings. A second
scenario that leads to multi-stability is via the double-Hopf
bifurcation. For

Our results provide a coherent overview of the spatiotemporal
properties of the Lorenz-96 model for

The results presented in this paper also illustrate another
important point: both qualitative and quantitative aspects of
the dynamics of the Lorenz-96 model depend on the parity of

The scripts used for continuation with AUTO-07p are available upon request from Alef Sterk.

First note that for all

Using L'Hospital's

The number of Hopf bifurcations of the equilibrium

For the two-parameter system we can count the number of double-Hopf
bifurcations by counting the intersections of the lines in
Eq. (

DvK performed the research on traveling waves and investigated the dynamics near the double-Hopf bifurcations. AS performed the research on stationary waves and prepared the manuscript.

The authors declare that they have no conflict of interest.

The authors would like to thank the reviewers for their useful comments and suggestions that have helped to improve this paper. Edited by: Amit Apte Reviewed by: Jochen Broecker and one anonymous referee