Atms 4320 Lab 2 Anthony R. Lupo

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:

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.

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:

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:

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

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

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):

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).

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

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:

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

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)

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)