Geostrophic Adjustment Beyond
the Traditional Approximation
Gregory M. Reznik
Introduction
In this contribution we discuss geostrophic adjustment in rotating fluid when the
angular speed of rotation Ω does not coincide in direction with the acceleration due
to gravity; the traditional and hydrostatic approximations are not used. Gyroscopic
waves (GW) are most susceptible to the “non-traditional” terms in equations of
motion and play an important role in our consideration. The GWs exist owing to
rotation (e.g., [18]); no stratification or gravity are necessary, although both of these
factors strongly affect the structure and properties of these waves. In a “pure” form
the GWs occur in a barotropic fluid layer of constant depth bounded by two rigid
lids and rotating as a whole at a constant angular speed whose direction can be
different from the gravity (see Fig. 1).
Under the traditional approximation (TA) when the horizontal component of the
angular speed Ω is neglected, the GWs in the barotropic layer are sub-inertial, i.e.
their frequencies σ do not exceed the vertical component of twice the angular speed
of rotation f = 2Ω sin φ 0 (see Fig. 1) i.e. σ ≤ f ; without the TA both sub-inertial and
super-inertial GWs with σ ≥ f are possible [4, 13]. In stably stratified fluid under the
TA the sub-inertial GWs exist together with the super-inertial internal waves only if
the minimal buoyancy frequency is N min < f [11]. In strongly stratified fluid, i.e. at
N min > f , only super-inertial internal waves are possible. However, without the TA
sub-inertial waves (so called internal inertio-gravity waves) occur even in the
strongly stratified fluid [6, 7, 13]. Like the GWs these waves cannot exist without
rotation.
The paper is organized as follows. In section “Barotropic Model” we examine
the geostrophic adjustment of a barotropic fluid layer where GWs are the only
G. M. Reznik ( ✉ )
Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russia
e-mail: greznikmd@yahoo.com
© Springer International Publishing AG, part of Springer Nature 2018
M. G. Velarde et al. (eds.), The Ocean in Motion, Springer Oceanography,
https://doi.org/10.1007/978-3-319-71934-4_20
297
the Traditional Approximation
Gregory M. Reznik
Introduction
In this contribution we discuss geostrophic adjustment in rotating fluid when the
angular speed of rotation Ω does not coincide in direction with the acceleration due
to gravity; the traditional and hydrostatic approximations are not used. Gyroscopic
waves (GW) are most susceptible to the “non-traditional” terms in equations of
motion and play an important role in our consideration. The GWs exist owing to
rotation (e.g., [18]); no stratification or gravity are necessary, although both of these
factors strongly affect the structure and properties of these waves. In a “pure” form
the GWs occur in a barotropic fluid layer of constant depth bounded by two rigid
lids and rotating as a whole at a constant angular speed whose direction can be
different from the gravity (see Fig. 1).
Under the traditional approximation (TA) when the horizontal component of the
angular speed Ω is neglected, the GWs in the barotropic layer are sub-inertial, i.e.
their frequencies σ do not exceed the vertical component of twice the angular speed
of rotation f = 2Ω sin φ 0 (see Fig. 1) i.e. σ ≤ f ; without the TA both sub-inertial and
super-inertial GWs with σ ≥ f are possible [4, 13]. In stably stratified fluid under the
TA the sub-inertial GWs exist together with the super-inertial internal waves only if
the minimal buoyancy frequency is N min < f [11]. In strongly stratified fluid, i.e. at
N min > f , only super-inertial internal waves are possible. However, without the TA
sub-inertial waves (so called internal inertio-gravity waves) occur even in the
strongly stratified fluid [6, 7, 13]. Like the GWs these waves cannot exist without
rotation.
The paper is organized as follows. In section “Barotropic Model” we examine
the geostrophic adjustment of a barotropic fluid layer where GWs are the only
G. M. Reznik ( ✉ )
Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russia
e-mail: greznikmd@yahoo.com
© Springer International Publishing AG, part of Springer Nature 2018
M. G. Velarde et al. (eds.), The Ocean in Motion, Springer Oceanography,
https://doi.org/10.1007/978-3-319-71934-4_20
297