Q-G vorticity equation Q-G thermodynamic equation We now have two equations in two unknowns,  and  We will solve these to find an equation for , the vertical motion in pressure coordinates, and for, the change of geopotential height with time. Wave-cyclones from the Q-G perspective

1 2 Derive the Q-G height tendency equation (equation for height changes of a pressure surface) 1.Assume  is constant 2.Change order of differentiation on left side of (2) 3.Multiply (2) by and (1) by f 0 4.Differentiate (2) with respect to p 4.Add to resultant equation to (1)

Both the QG omega equation and the QG height tendency equation can be derived Including the friction term and the diabatic heating term. We will not do this here, But I will simply show the result. QG OMEGA EQUATION QG HEIGHT TENDENCY EQUATION

First term is proportional to First term represents the rate at which geopotential height decreases with time

trough propagating 500 mb maps hours between panels trough deepening

First term represents the rate at which geopotential height decreases with time trough propagating 500 mb maps hours between panels trough deepening

QG HEIGHT TENDENCY EQUATION The first term on the right side of the equation describes the propagation of the height field. Propagation of the height field depends on : the advection of the relative vorticity (spin imparted by shear and curvature) and the advection of the planetary vorticity (spin imparted by the earth rotation)

A “battle” between advection of planetary vorticity and relative vorticity occurs in ridge-trough systems. High Latitudes Low Latitudes Height rises will occur due to the advection of relative vorticity Height falls will occur due to the advection of planetary vorticity Height falls will occur due to the advection of relative vorticity Height rises will occur due to the advection of planetary vorticity

Which process will win the battle? Short waves in flow Due to strong shear and sharp curvature change Relative vorticity changes substantially from ridge to trough f doesn’t change much – little deviation in latitude Advection of absolute vorticity is dominated by advection of relative vorticity: The troughs and ridges will propagate rapidly eastward

Long waves in flow f changes significantly – large deviation in latitude weak shear and wide curvature: relative vorticity changes small from ridge to trough As a result, advection of absolute vorticity is nearly equal to zero Long waves are stationary, or propagate very slowly eastward (  g > f) or westward (  g < f)

Note the speed of propagation of the waves on these maps: the shorter the wavelength becomes, the faster the wave propagates 00 Hours +12 Hours +24 Hours +36 Hours

QG HEIGHT TENDENCY EQUATION We will examine these two terms together: The vertical derivative of thickness advection (Differential thickness advection) The vertical derivative of diabatic heating (Differential diabatic heating)

ZaZa ZbZb Z A B ZaZa ZbZb Z A B ZaZa ZbZb Z A B Differential thickness (mean layer temperature) advection or Diabatic Heating Suppose we have a layer of air bounded by height Z a and Z b with level Z in the middle. Warm advection or diabatic heating in A causes layer to expand Cold advection or diabatic heating in B causes layer to contract Height Z falls to lower altitude Cold advection or diabatic cooling in A causes layer to contract Warm advection or diabatic heating in B causes layer to expand Height Z rises to higher altitude

In the real atmosphere warm and cold advection and diabatic heating and cooling decrease with height and are always strongest in the lower atmosphere 850 mb 2 Mar UTC 700 mb 2 Mar UTC 500 mb 2 Mar UTC Red circles: Strong warm advection pattern at 850, weaker at 700, very weak at 500 mb Blue circles: Strong cold advection pattern at 850, weaker at 700, very weak at 500 mb

QG HEIGHT TENDENCY EQUATION Summary Cold advection in the lower atmosphere will produce height falls amplifying trough aloft Warm advection in the lower atmosphere will produce height rises amplifying ridge aloft

QG HEIGHT TENDENCY EQUATION Surface anticyclone (term outlined > 0) Divergence, descending air and height falls Surface cyclone (term outlined < 0) Convergence, ascending air and height rises