Atms 4320 Lab 2

1. Atms 4320 Lab 2 Anthony R. Lupo

2. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • Recall that the total derivative can be exact  “Independent of path” • or path dependent:

3. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • Total derivative composed of the eulerian and advective derivative: • In evaluating the derivative, we estimate the partial derivative by assuming that the function is well behaved, or that the changes are “linear” from point a to point b.

4. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • For most horizontal fields, that assumption is reasonable, but there are many places where this assumption fails (i.e frontal zones). • How do we estimate the derivative given a field of regularly spaced data? • Given: the following expression:

5. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • we can “estimate” temperature change in x (foreward differencing) at point 2,2 for example: • or we can estimate using a backward difference:

6. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • Derivative estimates are most accurate when we use more points, so consider a “centered difference” • This can also be represented as the difference of two “Taylor” equations

7. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics Here they are!

8. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • With these equations we can also get an expression for the second derivative (just add (1) and (2) above):

9. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • This is called “finite differencing” since we are estimating derivatives using discrete estimates for the derivative quantities! • More precisely, what we have is second order finite differencing. We can derive higher order differences from Taylor series expansions. (ATMS 4800/7800).

10. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • Laplacian operator:

11. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • Truncation error • typically on order of the highest order term retained in estimate, thus for second order differencing, truncation error is on order of: • For 4th order differencing on order of:

12. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • Truncation Error = Difference equation - differential equation • Stability of Calculations: • Courant - Friedrichs-Levy (CFL) condition for computational stability

13. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • Typically used for evaluating schemes that estimate total derivative (e.g. leapfrog scheme)

14. Lab 2 -Methodologies for evaluating the total derviatives in the fundamental equations of hydrodynamics • Where c is phase speed of the upper air wave (propagation speed). Typically on order of 10 m s-1 • If CFL = 1 (neutral stability, no growth, but solution propagates with computational error and modes) • If CFL < 1 (stable solutions, solution propagates with computational error and modes) • If CFL > 1 (computational unstable, solution grows exponentially without bound)