Inertial instability

inertial instability

1. (Also called dynamic instability.) Generally, instability in which the only form of energy transferred between the steady state and the disturbance is kinetic energy.

   
   

   
   See Helmholtz instability, barotropic instability.

   
   
2. The hydrodynamic instability arising in a rotating fluid mass when the velocity distribution is such that the kinetic energy of a disturbance grows at the expense of kinetic energy of the rotation.

   
   

   For a small plane-symmetric displacement (wavenumber zero) using the parcel method, this criterion for instability is that the centrifugal force on the displaced parcels is larger than the centrifugal force acting on the environment. On the assumption that absolute angular momentum is conserved, this states that the fluid is unstable if absolute angular momentum decreases outward from the axis;

   

   where ω_a is the absolute angular speed and R the distance from the axis. If this criterion is applied to rotation of the westerlies about the earth's axis, the angular speed of the earth is so large that the inequality fails and the disturbance is stable. If applied to a system rotating about a local vertical, the criterion might be satisfied in low latitudes where the component of the earth's rotation about the local vertical is small. Inertial instability has been suggested in connection with the genesis of hurricanes.
   See rotational instability.

   
   
   Holton, J. R. 1992. An Introduction to Dynamic Meteorology. 3d edition, Academic Press, . 207–208.
   
   
   Eliassen, A., and E. Kleinschmidt 1957. Dynamic Meteorology. Handbuch der Geophysik. Vol. XLVIII, . 64– 72.
   
   

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The angular momentum as measured in an absolute coordinate system; hence, the vector product of the position vector of a particle into the absolute momentum of the particle.
In the atmosphere the absolute angular momentum M per unit mass of air is equal to the sum of the angular momentum relative to the earth and the angular momentum due to the rotation of the earth:

where r is the distance from the center of the earth to the particle, u the relative eastward component of velocity, φ the latitude, and Ω the angular rotation rate of the earth. Since the earth's atmosphere is shallow, the variable r is often replaced by the constant a, defined as the radius of the earth. The absolute angular momentum per unit mass is then approximated by

See angular-momentum balance, conservation of angular momentum.