4.4 Conversion to and from dry and wet mixing ratios for trace constituents in the model

There are trade offs in the various options for the representation of trace constituents $\chi$ in any general circulation model:

1. When the air mass in a model layer is defined to include the water vapor, it is frequently convenient to represent the quantity of trace constituent as a ``moist'' mixing ratio $\chi^m$, that is, the mass of tracer per mass of moist air in the layer. The advantage of the representation is that one need only multiply the moist mixing ratio by the moist air mass to determine the tracer air mass. It has the disadvantage of implicitly requiring a change in $\chi^m$ whenever the water vapor $q$ changes within the layer, even if the mass of the trace constituent does not.
2. One can also utilize a ``dry'' mixing ratio $\chi^d$ to define the amount of constituent in a volume of air. This variable does not have the implicit dependence on water vapor, but does require that the mass of water vapor be factored out of the air mass itself in order to calculate the mass of tracer in a cell.

NCAR atmospheric models have historically used a combination of dry and moist mixing ratios. Physical parameterizations (including convective transport) have utilized moist mixing ratios. The resolved scale transport performed in the Eulerian (spectral), and semi-Lagrangian dynamics use dry mixing ratios, specifically to prevent oscillations associated with variations in water vapor requiring changes in tracer mixing ratios. The finite volume dynamics module utilizes moist mixing ratios, with an attempt to maintain internal consistency between transport of water vapor and other constituents.

There is no ``right'' way to resolve the requirements associated with the simultaneous treatment of water vapor, air mass in a layer and tracer mixing ratios. But the historical treatment significantly complicates the interpretation of model simulations, and in the latest version of CAM we have also provided an ``alternate'' representation. That is, we allow the user to specify whether any given trace constituent is interpreted as a ``dry'' or ``wet'' mixing ratio through the specification of an ``attribute'' to the constituent in the physics state structure. The details of the specification are described in the users manual, but we do identify the interaction between state quantities here.

At the end of the dynamics update to the model state, the surface pressure, specific humidity, and tracer mixing ratios are returned to the model. The physics update then is allowed to update specific humidity and tracer mixing ratios through a sequence of operator splitting updates but the surface pressure is not allowed to evolve. Because there is an explicit relationship between the surface pressure and the air mass within each layer we assume that water mass can change within the layer by physical parameterizations but dry air mass cannot. We have chosen to define the dry air mass in each layer at the beginning of the physics update as

$$\delta p^d_{i,k} = (1-q^0_{i,k}) \delta^m_{i,k}$$

for column $i$, level $k$. Note that the specific humidity used is the value defined at the beginning of the physics update. We define the transformation between dry and wet mixing ratios to be

$$\chi^d_{i,k} = (\delta p^d_{i,k} / \delta p^m_{i,k}) \chi^m_{i,k}$$

We note that the various physical parameterizations that operate on tracers on the model (convection, turbulent transport, scavenging, chemistry) will require a specification of the air mass within each cell as well as the value of the mixing ratio in the cell. We have modified the model so that it will use the correct value of $\delta p$ depending on the attribute of the tracer, that is, we use couplets of $(\chi^m, \delta p^m)$ or $(\chi^d, \delta p^d)$ in order to assure that the process conserves mass appropriately.

We note further that there are a number of parameterizations (e.g. convection, vertical diffusion) that transport species using a continuity equation in a flux form that can be written generically as

$${\partial \chi \over \partial t} = {\partial F(\chi) \over \partial p}$$ (4.108)

where $F$ indicates a flux of $\chi$. For example, in convective transports $F(\chi)$ might correspond to $M_u \chi$ where $M_u$ is an updraft mass flux. In principle one should adjust $M_u$ to reflect the fact that it may be moving a mass of dry air or a mass of moist air. We assume these differences are small, and well below the errors required to produce equation 4.108 in the first place. The same is true for the diffusion coefficients involved in turbulent transport. All processes using equations of such a form still satisfy a conservation relationship

$${\partial \over \partial t} \sum_k{\chi_k \delta p_k} = F_{kbot} - F_{ktop}$$

provided the appropriate $\delta p$ is used in the summation.