Keita Iga

Reconsideration of Orlanski's instability theory of frontal waves

This paper complements the instability theory of frontal waves investigated by Orlanski (1968), and reinterprets the unstable modes obtained. First, the stability of a frontal model is reconsidered by using a matrix method. The major part of Orlanskis (1968) result is verified but some flaws are found in some parameter regions: unstable modes do not exist over the entire Ri-Ro region. Also, the features of the neutral waves in the one-layer subsystems are studied, in order to determine the instability of the full two-layer system. As a result, the unstable mode called the (B)-mode by Orlanski (1968) and suggested by Sakai (1989) to be Rossby-Kelvin instability caused by a resonance between a Rossby wave and a gravity wave, proves to be a geostrophic unstable mode caused by resonance between a Rossby wave and the Rossby-gravity mixed mode.

Critical layer instability as a resonance between a non-singular mode and continuous modes

The interpretation of an unstable mode in terms of resonance between neutral waves is extended to a critical layer instability. It is found that the concept of resonance can be applied to the critical layer instability, if continuous modes are taken into account: The instability occurs due to a resonance between a non-singular mode and a superposition of several continuous modes whose phase speeds are close to the velocity at the critical level. These continuous modes have pseudomomentum with its sign opposite to the gradient of the potential vorticity at the critical level, which is consistent with the interpretation of the instability in terms of resonance between neutral waves.

Various phenomena on a water vortex in a cylindrical tank over a rotating bottom

Flows in a cylindrical tank over a rotating bottom are investigated by laboratory experiments. Despite the axisymmetry of the experimental setup, various anisotropic phenomena are observed. The slow rotation of the bottom disk induces a circular flow according to the axisymmetric environment, but polygonal vortices form under faster rotation. Between these two vortex flow states, the flow undergoes a transition with clear hysteresis during which the elliptical shape assumed under faster rotation is retained when the rotation is subsequently slowed to rates that previously supported axisymmetric flow. Sloshing is also observed; here, a calm circular flow state alternates with an oscillation of the water surface along the sidewall of the container. A phase diagram showing the phenomena observed under different combinations of the initial water depth at rest and the rotation rate of the bottom disk is developed following thorough experimental testing over a wide range of parameter values. The features of the dependences of the range for each phenomenon to occur on these parameters are also elucidated.

Transition modes of rotating shallow water waves in a channel

Normal modes of shallow water waves in a channel wherein the Coriolis parameter and the depth vary in the spanwise direction are investigated based on the conservation of the number of zeros in an eigenfunction. As a result, it is generally shown that the condition for transition modes (Kelvin modes and mixed Rossby-gravity modes) to exist, besides Rossby and Poincare modes, is determined only by boundary conditions. A Kelvin mode is interpreted as a modification of a Kelvin wave or a boundary wave along a closed boundary, and a mixed Rossby-gravity mode as a modification of an inertial oscillation or a boundary wave along an open boundary. Transition modes appearing in edge and continental-shelf waves, equatorial waves and free oscillations over a sphere are systematically understood by applying the theory in this paper.

A simple criterion for the sign of the pseudomomentum of modes in shallow water systems

A simple criterion is derived for determining the sign of the pseudomomentum of neutral modes in shallow water systems. The sign of the pseudomomentum is determined by the gradient of the dispersion curve on a wavenumber vs. phase-speed plane: a mode has pseudomomentum with the opposite sign to that of the gradient of the dispersion curve. In most cases, the sign of the pseudomomentum is also determined only from the value of its phase speed: the pseudomomentum of a mode is positive if its phase speed is faster than the velocity of the basic flow at any point and vice versa, but with a few exceptions.

Transition modes in stratified compressible fluids

Normal modes which exist in stratified compressible fluids are investigated. For the analysis, the conservation of the number of zeros in an eigenfunction is used. It is generally shown that the condition for transition modes such as Lamb-wave modes to exist is determined only by boundary conditions. This mathematical result is physically explained by boundary waves, and this explanation crucially depends on which is larger, gravity acceleration g or the product of Brunt–Vaisala frequency and sound speed Ncs. This theory gives a guide to choose boundary conditions free of spurious boundary waves. It also explains why a distinct Lamb wave is not found in the ocean unlike in the atmosphere: it is simply because the ocean is not deep enough, but if the ocean were stratified a little more strongly than it is, the Lamb wave would not exist in the ocean however deep it might be.