This test extends the evaluation of transport schemes from
prescribed advection of inert scalars to reactive species. The test
consists of transporting two interacting chemical species in the Nair and Lauritzen 2-D idealized flow field.
The sources and sinks for these two species are given by a simple, but
non-linear, “toy” chemistry that represents combination (

Tracer transport is a basic component of any atmospheric
dynamical core. Typically, transport accuracy is evaluated in ideal tests
before being developed further or implemented in full models. Several tests
for 2-D passive and inert transport exist in the literature
(

Idealized chemical processes have readily available analytic expressions for
the forcing terms. The implementation of these processes as sub-grid-scale
forcing involves “only” solving forced continuity equations rather than the
full Navier–Stokes, primitive or shallow water equations that add extra
levels of complexity. Indeed, several simplified systems, where two species
interact non-linearly, have been developed and studied quite extensively in
the literature. For example, the Lotka and Voltera equations (also known as
predator–prey equations) are a pair of first-order differential equations
describing the dynamics of biological systems in which two species interact,
one as a predator and the other as prey. For a dynamical systems analysis of
the Lotka and Voltera equations, e.g., see Chapter 4 in

The test we develop in this paper extends the

The paper is organized as follows. In Sect.

In this section, we use the nomenclature

The non-linear toy chemistry equations for

The kinetic equations corresponding to the above system
(Eqs.

If the initial condition for

The reaction coefficient,

Contour plot of the terminator-“like” reaction coefficient

An analytic steady-state solution of the chemical concentrations for the
condition of no flow is derived in Appendix

Let

For the theoretical discussion, it is convenient to define the property
“semi-linear”: a transport operator

Since

Contour plots of the steady-state solutions, assuming no
flow, for

Several transport operators

Typically, transport operators are not applied in their unlimited versions in
full models. Shape-preserving filters are applied to ensure physically
realizable solutions such as the prevention of negative mixing ratios or
unphysical oscillations in the numerical solutions

Shape-preserving filters may render an otherwise semi-linear transport
operator non-semi-linear. Some limiters, however, are semi-linear. For
example, van Leer type 1-D limiters

We note that, instead of advecting each species separately by solving the
advection equations in Eqs. (

Coupling the chemistry parameterization with advection can be done in multiple ways. A common approach in weather/climate modeling is to update the species evolution in time incrementally by first updating the mixing ratios with respect to sub-grid-scale forcings (chemistry) and then to apply the transport operator based on the chemistry-updated state (or in reverse order). Since the computation of the sub-grid-scale tendencies in full models is computationally costly, the dynamical core (in this case, the transport scheme) is usually subcycled with respect to chemistry. For fast chemistry, this may be reversed.

A model will operate with a chemistry (physics) time step

It is, of course, up to the model developer to choose which coupling method
and time step to use. To facilitate comparison, the model developer is
encouraged to use the analytically computed forcing terms

For simplicity, the velocity field for the transport operator

It is the purpose of this section to show exploratory terminator test results. An in-depth analysis of why the limiters do not preserve linear relations (and the derivation of possible remedies) is up to the scheme developers.

Terminator test results are shown for two dynamical cores (transport schemes)
available in the CAM: CAM-FV

As discussed in detail in

For all simulations, the chemistry (physics) time step is

The sample results shown next are divided into four sections: first of all, baseline results for CAM-FV and CAM-SE using their default configurations. Next, results from experiments varying the limiter in CAM-SE are presented. Then, the consequences of using different chemistry–transport (physics–dynamics) coupling methods (in CAM-SE) are discussed. Lastly, the results are quantified.

Contour plots of

Figure

CAM-FV transport is based on the dimensionally split

CAM-SE does not preserve linear relations either and the errors in

Contour plot of

Cross sections of day 1 (left column)

To further understand this behavior, we have performed some tests (not shown)
turning the chemistry off and advecting linearly correlated cosine hills and
linearly correlated step functions. The cosine hills are

In addition to the quasi-monotone mass-conservative limiter used by default
in CAM-SE, the model has options for performing tracer advection without any
limiter and with a positive definite limiter. Results for terminator test
runs using those configurations are shown in Fig.

When using a positive definite limiter, Gibbs phenomena are eliminated near
the base of the terminator, but not near the maximum. This obviously violates
linear relations and produces large errors in

As explained in Sect.

In Fig.

In terms of the CAM-SE namelist, these configurations correspond to (a) ftype=1, nsplit=1, rsplit=6, (b) ftype=0, nsplit=2, rsplit=3, and (c) ftype=0, nsplit=6, rsplit=1.

. In all experiments, the tracer time step nd chemistry time step are held fixed:Near the western edge of the terminator (located at approximately
130

At the eastern edge of the terminator (located at approximately
30

Contour plots of

Physical parameterization packages may contain code that sets negative mixing ratios to 0. Or, similarly, there may be code that prevents tendencies from being added to the state if it is 0 or negative. The terminator test may be a useful tool to diagnose such alternations in large complicated codes.

To quantify the errors introduced in the terminator test, we suggest
computing standard error norms for

As a reference, we show the time evolution of

A simple idealized toy chemistry test case
is defined. It consists of advecting two reactive species (

The toy chemistry, by design, does not disrupt pre-existing linear relations
between the species. So, the only source of error is from the transport
scheme and/or the chemistry–transport (physics–dynamics) coupling. The
terminator test is set up so that

Time evolution of standard error norms

In addition, the terminator test assesses the accuracy of chemistry–tracer
(physics–dynamics) coupling methods in an idealized setup. Different
coupling methods (such as those available in CAM-SE) lead to different
distributions of

The terminator test is easily accessible to advection scheme developers from an implementation perspective since the software engineering associated with extensive parameterization packages is avoided. The test forces the model developer to consider how their scheme is coupled to sub-grid-scale parameterizations and, if solving the continuity equation in flux form, forces the developer to consider tracer–air mass coupling. Also, the idealized forcing proposed here has an analytic formulation, and the continuous set of forced transport equations has, contrary to the Brusselator forcing, an analytic solution for the weighted sum of the correlated species, irrespective of the flow field.

We encourage dynamical core developers to implement the toy chemistry in their test suite as it has the potential to identify tracer transport issues that standard tests (with unreactive/inert tracers) would not generate.

In the terminator test, we use the deformational flow of

To gain more insight into the toy chemistry (and to formulate “spun-up”
initial conditions), it is useful to consider the special case of no flow.
For

From the kinetic Eqs. (

For (algebraic) convenience, define the quantities

Completing the square on the right-hand side leads to the expression

The right-hand side can be factored, and the following partial
fraction expansion can be constructed:

Integration of each of these terms from time

For long times,

If the flow is slow compared to the rate at which the chemistry returns to
equilibrium, then the concentrations will stay near the steady-state solution
derived in Appendix

Since the maximum of

Relation Eq. (

When

When

Thus, for very small (negative)

Pseudo-code explaining the different levels of subcycling and chemistry–transport (physics–dynamics) coupling used in CAM-SE.

Outer loop advances solution

Compute chemistry tendencies

Update state with chemistry/physics tendencies:

subcycling of tracer advection:

The different levels of subcycling used in CAM-SE are explained via
pseudo-code in Algorithm 1 using CAM-SE namelist conventions:
nsplit and rsplit. The outer time-stepping loop starts
with a call to chemistry that computes the chemistry tendencies over the
entire chemistry time step

We distinguish between the

When
running the 3-D CAM-SE dynamical core, nsplit defines
the vertical remapping time step; if

In full model runs, if the physics time step is large, the

For the

Note however that, in CAM-FV, the tendencies are added after tracer transport and not before.

. For CAM-FV,The analytic solution of the equations leads to an explicit solution
for the change in concentrations during a time step with no flow.

In implementation,

In terms of Fortran code, the analytical forcing is given by

The tracer algorithm and dynamical core use the same time step that is
controlled by the maximum anticipated wind speed, but the dynamics uses more
stages of a second-order accurate

NCAR is sponsored by the National Science Foundation (NSF). Jean-François Lamarque, Andrew Conley and Francis Vitt were partially funded by the Department of Energy (DOE) Office of Biological & Environmental Research under grant number SC0006747. Mark Taylor was supported by the Department of Energy Office of Biological and Environmental Research, work package 12-015335, “Applying Computationally Efficient Schemes for BioGeochemical Cycles”. Thanks to Oksana Guba for discussions on the CAM-SE limiter. The authors are grateful to Michael Prather for the many discussions on simplified chemistry. We thank the reviewers for their constructive comments that greatly improved the manuscript. Edited by: F. O'Connor