ESCI 343 – Atmospheric Dynamics II Lesson 14 – Inertial/slantwise Instability Reference: An Introduction to Dynamic Meteorology (3rd edition), J.R. Holton Atmosphere-Ocean Dynamics, A.E. Gill Mesoscale Meteorology in Midlatitudes, P. Markowski and Y. Richardson INERTIAL INSTABILITY Imagine an air parcel in geostrophic balance at speed v on the f-plane. The diagram below shows the balance of accelerations in the lateral direction. Note that the direction of v is completely arbitrary. The coordinate r is directed to the right of the wind, while the coordinate n is in the direction of the geostrophic wind. The wind component in the transverse direction (along r) is denoted as u. The momentum equations in this coordinate system are 0Duf vDt r∂Φ= − +∂ (1) 0Dvf uDt= −. (2) Imagine that an air parcel starts in geostrophic balance. If the parcel is suddenly impelled laterally in the direction of r at a speed u, the balance of accelerations will change. Taking the time derivative of (1) gives us an equation for how the lateral acceleration changes with time, ( )0D Du D Df vDt Dt Dt r Dt ∂Φ = − +  ∂   . (3) The terms on the right-hand side of (3) are evaluated as follows: 22Du v uDt r t r r n r r∂Φ ∂Φ ∂ ∂Φ ∂ ∂Φ ∂ Φ = + + = ∂ ∂ ∂ ∂ ∂ ∂ ∂  (4) and ( ) ( )20 0 0 0 0D Dvf v f f f u f uDt Dt= = − = −, (5) so that (3) becomes fo v v r∂Φ∂r n22 2202 2D uu f uDt r∂ Φ= − −∂ (6) or 2 2202 20D uf uDt r ∂ Φ+ + = ∂  . (7) The solutions to (7) will be oscillatory provided that 22020fr∂ Φ+ >∂. (8) In this case, the parcel will oscillate around its original line of motion, and the flow is inertially stable. The angular frequency of the oscillations is 22 202frω∂ Φ= +∂. (9) If instead, 22020fr∂ Φ+ <∂, (10) then the transverse velocity will grow exponentially with time and the parcel will accelerate away from its original line of motion. PHYSICAL INTERPRETATION OF INERTIAL STABILITY The physical interpretation of inertial stability/instability is directly linked to how the pressure gradient tightens or loosens in the direction of r. The figure below shows the case of the pressure gradient becoming tighter with increasing r, which implies that 220r∂ Φ>∂. (11) The stability criteria (8) tells us that this case is inertially stable, so that a parcel displaced latitudinally will return to its base latitude. To see why this occurs, refer to the diagram below. Imagine parcel in geostrophic balance at Point 1. If the parcel is perturbed in the direction of −r, then there will also be a positive acceleration in the direction of n due to Coriolis. This will increase the v component of the wind and thus increase the3component of the Coriolis acceleration in the direction of r. Since the parcel is also moving into an area of weaker pressure gradient, there is a net acceleration on the parcel toward positive r. Thus, a lateral perturbation will result in a restoring acceleration back toward the original line of motion. For the case where the pressure gradient decreases in the r direction the physical interpretation is a little more complex. The diagram below shows this case, where 220r∂ Φ<∂. (12) As before the parcel is perturbed in the negative r direction at velocity u. There is still an acceleration due to Coriolis in the n direction, which will increase the component of the Coriolis acceleration in the r direction. However, the pressure gradient acceleration is also increasing as the parcel moves to Point 2. If the pressure gradient acceleration is larger than the Coriolis acceleration, the parcel will accelerate toward the negative r direction. If, however, the increase in Coriolis acceleration outweighs the increase in the pressure gradient acceleration, the parcel will accelerate back toward its original line of motion. Thus, a decreasing pressure gradient with increasing r is not sufficient to produce inertial instability. In order for instability to occur in this case, (8) shows us that 2202fr∂ Φ> −∂. (13) Plots of geopotential versus r for a constant pressure surface are shown in the diagrams below. For the first two diagrams the atmosphere is inertially stable, because the second derivative of Φ is either positive or zero. For the third diagram the second derivative of Φ is negative, but instability would depend on just how negative the second derivative is.4 ABSOLUTE MOMENTUM AND INERTIAL STABILITY The stability criteria (8) can be written in alternate forms as follows: ( )22 2 20 0 0 020gf f f f vr r r r∂ Φ ∂ ∂Φ ∂ + = + = + > ∂ ∂ ∂ ∂  or 00gvfr∂+ >∂. (14) For ease of notation we can rewrite (14) as ( )00gf r vr∂+ >∂, and defining a quantity called the absolute momentum1 as 0gM f r v≡ + (15) the condition for inertial stability/instability can be written as2 0 :0 :0 :MInertially stablerMInertially neutralrMInertially unstabler∂>∂∂=∂∂<∂ (16) Even though M is called absolute momentum, it is not exactly equal to the momentum as viewed from space, but is equal to it within some function of r. The reason it is defined this way is so that its r-derivative is equal to the absolute vorticity via 1 Our derivation of inertially instability and absolute momentum was done in coordinates that are completely arbitrary, with no preferred direction for the geostrophic wind. Many basic treatments of this topic do the derivation for a purely zonal geostrophic flow, and define absolute momentum as 0gM f y u= −. This form of absolute momentum always increases toward the North for purely positive zonal flow, whereas the definition (15) increases toward the South for purely positive zonal flow. 2 Since r and y are in opposite directions for positive zonal flow, derivatives with respect to r are of the opposite sign than derivatives with respect to y. With