6.2 Slopes and moments

The tracer mass is $rm$ (kg) and the air mass is $m$ (kg) The mixing ratio is then $f=rm/m$ (kg tracer/kg air).

A tracer field is defined by 1, 4, or 10 three-dimensional arrays:

array	unit	description
rm	kg	toal tracer mass in cell
rxm, rym, rzm	kg/(half grid cel)	first order moments (slopes)
rxxm, ryym, rzzm rxym, rxzm, ryzm	kg/(half grid cel)$^2$	second order moments

The slopes and moments are properties of the concentration gradient inside the cell.

Consider a cell volumne $[0,X]\times[0,Y]\times[0,Z]$. The convention for the length scales is that a half cell has lenght $1$, thus $X=Y=Z=2$. We would like to know the concentration in a point $(x,y,z)$ in the cell. From equations (7) and (9) in [Prather, 1986] we find that the tracer mass mixing ratio (kg tracer/kg air) is given by:

$$f(x,y,z) = \left[~ {rm} \right.$$ (6.1)

$$~+~ {rm}_x\ (x - X/2) ~+~ {rm}_y\ (y - Y/2) ~+~ {rm}_z\ (z - Z/2)$$ (6.2)

$$~+~ {rm}_{xx}\ ( x^2 - X x + X^2/6 ) ~+~ {rm}_{yy}\ ( y^2 - Y y + Y^2/6 ) ~+~ {rm}_{zz}\ ( z^2 - Z z + Z^2/6 )$$ (6.3)

$$\left. ~+~ {rm}_{xy}\ (x-X/2)(y-Y/2) ~+~ {rm}_{xz}\ (x-X/2)(z-Z/2) ~+~ {rm}_{yz}\ (y-Y/2)(z-Z/2) ~\right]\ /\ m$$ (6.4)

Coding of slopes and second moments

The best way to implement slopes/second moments is to have them compiled only if certain preprocessor macro's are defined.

In cy3, at least macro 'slopes' should be defined; in addition, 'secmom' might be defined to.

      ! at least slopes should be defined:
      #ifndef slopes
        stop 'slopes not defined'
      #endif

      ! second moments as extension of slopes:
      #ifdef slopes
        rxm = 0.0
      #ifdef secmom
        rxxm = 0.0
      #endif
      #endif

      ! different codes:
      #ifdef slopes
      #ifndef secmom
        call adv_slopes
      #else
        call adv_secmom
      #endif
      #endif