2. Atms 4320 / 7320 lab 8 The Divergence Equation and Computing Divergence using large data sets.

3.  Divergence  a kinematic property, however, this is arguably one of the most important quantities dynamically. The Divergence Equation and Computing Divergence using large data sets.

4.  Divergence/convergence patterns result when a flow initially in geostropic balance is “forced” out-of-balance by any forcing mechanism. We don’t care what these mechanisms are, but they include forcing due to:  dynamic forcing  vorticity advection (horizontal and vertical advections)  frictional forcing  tilting The Divergence Equation and Computing Divergence using large data sets.

5.  Thermodynamic forcing:  temperature advections  diabatic heating/cooling (includes LHR, sensible heating, Radiative forcing)  adiabatic heating cooling (due to vertical motions) The Divergence Equation and Computing Divergence using large data sets.

6.  A flow that is knocked out of geostrophic balance results in ageostrophic motions, which produce divergence / convergence patterns.  These divergence/convergence patterns are then associated with vertical motions (secondary circulations), and pressure changes at a) the surface, and b) aloft. The Divergence Equation and Computing Divergence using large data sets.

7.  Thus, divergence/convergence patterns are manifestations of the fact that when a flow is knocked out of balance, the velocity field then adjusts to the mass field, as the flow attempts to establish a new geostrophic state. The Divergence Equation and Computing Divergence using large data sets.

8.  Thus, divergence is one of those quantities for which we derive a diagnostic relationship. We can use the Navier- Stokes equations. The Divergence Equation and Computing Divergence using large data sets.

9.  Take of the u equation,  and of the v equation.  Then add ‘em up. The Divergence Equation and Computing Divergence using large data sets.

10.  The resultant equation is:

11.  Where term a) is the time rate of change of the divergence.  The next are the “forcing” mechanisms that generate divergence/convergence patterns. The Divergence Equation and Computing Divergence using large data sets.

12.  term b) is the laplacian of the potential + kinetic energy terms (indirectly, this term includes thermodynamic forcing, via hydrostatic balance, the equation of state, and the first law)  term c) the vorticity flux term (transport)  term d) the vertical advection of vorticity  term e) the “tilting” term The Divergence Equation and Computing Divergence using large data sets.

13.  Many diagnostic equations (omega, Z-O, height tendency) we derive will have similar form.  We can calculate (  ) from the divergence field. The Divergence Equation and Computing Divergence using large data sets.

14.  So now, let’s calculate divergence:  Given the wind at any point, we would decompose into u and v components. This wind could be the observed wind or a geostrophic estimate. The Divergence Equation and Computing Divergence using large data sets.

15.  Use the following relationships.  However, most data sets (NCEP reanalyses, ECMWF) are gridded analyses (2.5 x 2.5 lat/lon most common format). And these data sets usually provide the u and v components for you. The Divergence Equation and Computing Divergence using large data sets.

16.  Thus, if we have a 9 - point grid, we would calculate the following way:. r,c.(3,1). (3,2). (3,3).(3,1). (3,2). (3,3). (2,1). (2,2). (2,3). (1,1). (1,2). (1,3) The Divergence Equation and Computing Divergence using large data sets.

17.  Divergence is:  In component form: The Divergence Equation and Computing Divergence using large data sets.

18.  Recall from earlier, finite difference form:  Thus, we can write our divergence relationship in finite difference form. The Divergence Equation and Computing Divergence using large data sets.

19.  And applying to our grid: The Divergence Equation and Computing Divergence using large data sets.

20.  Example (assume 500 km = dx = dy):.(-10,0). (-8.6,-5). (-8.6,-5). (-10,0). (-10,0). (-8.6,-5). (-10,0). (-10,0). (-8.6,5) The Divergence Equation and Computing Divergence using large data sets.

21.  Here’ s the calculation:  ConvDiv Conv The Divergence Equation and Computing Divergence using large data sets.

22.  Recall a few weeks ago we had used the continuity equation to calculate vertical motions, so by calculating divergence at all levels, we can estimate vertical motions.  No assignment this week since we’ve done this before, but count on there being a test question along the lines of the example! The Divergence Equation and Computing Divergence using large data sets.

23.  The End!  Any Questions? The Divergence Equation and Computing Divergence using large data sets.