From a series of zonal mean global stratospheric tracer measurements sampled in altitude vs. latitude, circulation and mixing patterns are inferred by the inverse solution of the continuity equation. As a first step, the continuity equation is written as a tendency equation, which is numerically integrated over time to predict a later atmospheric state, i.e., mixing ratio and air density. The integration is formally performed by the multiplication of the initially measured atmospheric state vector by a linear prediction operator. Further, the derivative of the predicted atmospheric state with respect to the wind vector components and mixing coefficients is used to find the most likely wind vector components and mixing coefficients which minimize the residual between the predicted atmospheric state and the later measurement of the atmospheric state. Unless multiple tracers are used, this inversion problem is under-determined, and dispersive behavior of the prediction further destabilizes the inversion. Both these problems are addressed by regularization. For this purpose, a first-order smoothness constraint has been chosen. The usefulness of this method is demonstrated by application to various tracer measurements recorded with the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS). This method aims at a diagnosis of the Brewer–Dobson circulation without involving the concept of the mean age of stratospheric air, and related problems like the stratospheric tape recorder, or intrusions of mesospheric air into the stratosphere.

In the context of climate change, possible changes in the intensity of the
Brewer–Dobson circulation have become an important research topic. Climate
models predict an intensification of the Brewer–Dobson circulation

Facing these difficulties, it is desirable to infer the atmospheric
circulation directly from tracer measurements, without going back to the age
of air concept. Multiple approaches have been developed to infer wind fields
from measured atmospheric state variables. Sequential data assimilation and,
in its optimal form, the extended Kalman filter approach

The term “so-called” is used here because it is
challenged that this method is really variational in the context of discrete
variables (

This statement refers to meteorological data assimilation. Chemical data assimilation uses chemistry transport models.

In this paper we present a method to infer two-dimensional
(latitude–altitude) circulation and mixing coefficients from subsequent
measurements of inert tracers. The application of this method, i.e., the
inference of the Brewer–Dobson circulation from global SF

Knowing the initial state of the atmosphere in terms of mixing ratio and air
density distributions, wind speed, and mixing coefficients at each grid point, a
future atmospheric state can be predicted with respect to the distribution of
any inert tracer. This procedure we call the forward problem. If no ideal
tracers are available, source and sink terms of related species have to be
included in the forward model. The goal of this work is to invert the forward
model in order to infer the circulation and mixing coefficients from tracer
measurements by the minimization of the residual between the predicted and measured
atmospheric state. This approach is complementary to free-running climate models
because it makes no assumptions about atmospheric dynamics except for the validity
of the continuity equation. It is further considered more robust than age-of-air
analysis

Our concept involves the following operations. First, a general solution of
the forward problem is formulated (Sect.

The forward model reads the measured atmospheric state at time

The local change in number density

The local change in the volume mixing ratio of gas

Since we are only interested in a two-dimensional representation of the
atmosphere in altitude and latitude coordinates, zonal advection and mixing
terms are ignored in Eqs. (

The integration of Eqs. (

For the discrete integration of the advection part of the tendencies, the

For the diffusive component we use simple Eulerian integration:

The abundance of gas

Admittedly, there exist more elaborate advection schemes than the one used
here. However, the need to provide the Jacobians needed in
Sects.

Since we do not have a closed system but have mass exchange and mixing with
the atmosphere below the lowermost model altitude and above the uppermost
altitude, the atmospheric state is not predicted for the lowermost and
uppermost altitudes. Prediction is only possible from the second altitude from the bottom up to the altitude below the uppermost one. Henceforth, we call this restricted altitude range the “nominal
altitude range”. Instead, the atmospheric state of the uppermost and
lowermost altitude is estimated by the linear interpolation of measured
values at times

We use the following convention. Atmospheric state variables are sampled on a
regular latitude–altitude grid. For some grid points, no valid measurements
may be available but we assume that, for each state variable, we have a
contiguous subset of this grid with valid measurements. For state variable

For further steps (error propagation and the solution of the inverse
problem), it is convenient to rewrite the prediction of air density and mixing ratios
in matrix notation. For this purpose, we differentiate the predicted air
densities (Eq.

For mixing ratios, the respective derivatives are

With these expressions, the prediction of air density and volume mixing ratio
can be rewritten in matrix notation for a single micro time increment. This
notation simplifies the estimation of the uncertainties of the predicted
atmospheric state and the inversion of the prediction equation.
In matrix notation, the prediction reads

Matrix structure of the right-hand part of
Eq. (

Since the source term depends on air density, the integration in matrix
notation for vmr requires simultaneous treatment of vmr and air density, and
we get the following, using notation accordant with air density:

the Jacobian

the first “row” of the Jacobian matrix includes identity

the Jacobian sub-matrices

no simple mapping mechanism between the field of atmospheric state
variables sampled at latitudes and altitudes and the vectors

Let

The measurements used are not a perfect image of the true atmospheric state
but contain some prior information. In the case of the data provided by the Institute of Meteorology and Climate Research (IMK), a priori profiles are usually set to zero, and the constraint is built
with a Tikhonov-type first-order finite differences smoothing constraint (see

In our case, the situation is different. The model is initialized with measurements of reduced altitude resolution, and the fields predicted by the model are then compared to measurements of the same altitude resolution. It is fair to assume that the model does not dramatically change the altitude resolution of the profiles, and thus comparable quantities are compared when the residuals between predicted and measured atmospheric state are evaluated.

Let the initial mixing ratio field be homogeneous except one point with
delta-type excess mixing ratio. Assume further a homogeneous velocity field
and zero mixing coefficients. If the velocity is such that the position of
the excess mixing ratio is displaced during

Again, in our case, the situation is different. The widening does not
accumulate over the

These considerations aside, there are other numerical artifacts. These are related to the numerical evaluation of partial derivatives of the state variables in our transport scheme chosen. Particularly in the case of delta functions in the state variable field, these cause side wiggles behind and smearing in front of the transported structure. To keep these artifacts small, it is necessary to set the spatial grid fine enough that every structure is represented by multiple grid points.

For convenience, we combine the variables of the initial atmospheric state and
the predicted state at the end of the macro time interval, respectively, into
the vectors

With these derivatives we linearize the prediction with respect to wind and
mixing coefficients; i.e., we linearly predict the new atmospheric state for
a given initial state

Assuming linearity and Gaussian statistics, the most likely set

The covariance matrix characterizing the uncertainty of estimated winds and
mixing coefficients is

Due to the concentration dependence of the source function and the

With

In a first step we test the predictive power of the formalism defined by
Eqs. (

In the first case,

Case studies based on real measurement data are inadequate as the sole proof
of concept because the truth is unknown and the result thus is unverifiable.
Instead we first test our scheme on the basis of simulated atmospheric states
and consider the scheme as verified if the velocities and mixing coefficients
used to simulate the atmospheric states are sufficiently well reproduced. In
the noise-free well-conditioned case, one might even expect, within the
numerical precision of the system, the exact reproduction of the reference
data; due to the – weak but non-zero – dispersivity of the numerical
transport scheme, the wiggles discussed in the previous subsection cause, at
some grid points,

During code development, a series of basic tests of increasing complexity
were performed, including a variety of mixing ratio distributions transported
with various velocity fields. The main lesson learned was that, even when the
rows of Eq. (

As an example we show the following test. An altitude-independent meridional
velocity field (in degrees per month)

Test case:

Measured distributions in September
(left column) and October (middle column) and residual distributions between
October measurements and predictions for October (right column) for
air density and mixing ratios of CFC-12, CH

The risk of case studies based on simulated data typically is that not all
difficulties encountered with real data are foreseen during theoretical
studies. In order to demonstrate applicability to real data, global monthly
latitude–altitude distributions of CFC-12, CH

For this case study, zonal monthly mean distributions of air densities and
mixing ratios of these four species from September and October 2010 were
used. Figure

Resulting circulation vectors (

The resulting circulation vectors which best explain the change in the mixing
ratio distributions from September to October 2010 are shown in the upper
left panel of Fig.

Jacobian elements with respect to

The errors in the estimated transport velocities and mixing coefficients have
been estimated according to Eq. (

Estimated uncertainties of

Larger errors above 65 km altitude and at the bins closest to the pole are
border effects, resulting from the fact that no symmetric derivatives can be
calculated there. The uncertainties in

The analysis of the age of stratospheric air can be understood as an
integrated look at the equations of motion of stratospheric air because the
total travel time of the air parcel through the stratosphere is represented.
The refinement of this method which analyzes the mean age just considers a
weighted mean of the above, but it is still an integral method. Contrary to
these integral methods, our direct inversion scheme supports a – in
approximation, due to discrete sampling in the time domain – differential
view of the same problem. The related advantages are the following:

independence of assumptions on the age spectrum because during each time step mixing is explicitly considered

insensitivity to SF

applicability to nonideal tracers in the stratosphere (since the atmospheric state is updated for each time step by measured value, depletion does not accumulate, even if no sink functions are considered) and

the circular reasoning that the lifetimes of nonideal tracers depend on their trajectories (and thus atmospheric circulation), while the determination of the circulation requires knowledge of the lifetimes, can be resolved; our scheme requires knowledge only of the local, not the global, lifetimes

the method is an empirical method which does not involve any dynamical model; i.e., the forces which cause the circulation are not required.

sensitivity of the inferred kinematic quantities to locally varying biases,

a tendency towards ill-posedness of the inversion if distributions of too few tracers with too similar a morphology are used, and

the usual artifacts arising if the numerical discretization chosen is too coarse.

We have presented a method which infers mixing coefficients and effective
velocities of a 2-D atmosphere by inversion of the continuity equation. The
main steps of this procedure are

the integration of the continuity equation over time to predict pressure and mixing ratios for given initial pressures and mixing ratios and initially guessed velocities and mixing coefficients;

the propagation of errors of initial pressures and mixing ratios onto the predicted pressures and mixing ratios, by differentiation of the predicted state with respect to the initial state and generalized Gaussian error propagation;

the estimation of the sensitivities of the predicted state with respect to the velocities and mixing coefficients; and

the minimization of a quadratic cost function involving the residual between measured and predicted state at the end of the forecasting interval by inversion of the continuity equation.

The code is still under development. The data used are compiled in
Supplement 2. The complete MIPAS data are available at

The reduction of the transport problem from three to two dimensions involves
Reynolds decomposition of the three-dimensional continuity equation and
subsequent zonal averaging and gives rise to eddy mixing and transport terms.
The inference of effective two-dimensional transport velocities and effective
mixing coefficients from measurements discussed in the main paper relies on
the fact that, within certain assumptions and approximations, all these eddy
effects can be understood as additional pseudo-advection and pseudo-mixing
terms according to the advection equation and Fick's law, with
gas-independent pseudo-velocities and pseudo-mixing coefficients. The exact
interpretation of the two-dimensional velocities and mixing coefficients
inferred from the measurements depends on the approximations made. We apply
our scheme to zonally averaged mixing ratios (no mass-weighted averaging).
Contrary to the main text, where the symbols

Assuming

that the deviations from the zonal mean are small compared to the zonal mean itself such that linearization is justifiable,

that meridional advection is negligibly small compared to zonal advection,

that the time variation of the zonal mean quantities is assumed to be much slower than the time variation of the deviations from the zonal mean, which corresponds to the assumption of a quasi-steady state,

our scheme is applied only to long-lived species, such that chemical
eddy terms can be ignored because chemical lifetimes are long compared to
transport lifetimes (see

wave disturbances are dominated by steady or periodic terms, such that
the terms with the mixed second derivative terms tend to disappear
(

We would like to emphasize that none of the approximations and assumptions discussed above are used in our proposed method to infer velocities from zonal mean mixing ratio measurements. The discussion in this Appendix only tries to relate the resulting velocities to the velocities in a 3-D world. The ambiguities in the interpretation of the inferred “effective” 2-D velocities suggests that it might be promising to switch from a theoretical to an empiricist view and to no longer conceive of the zonal mean of the 3-D velocities as the “true” 2-D velocities but as those which satisfy the 2-D continuity equation. These can be – admittedly indirectly – observed with our suggested method and can be used to validate 2-D models, including their underlying concept of solving the 2-D transport problem. With this, the adequacy of the assumptions made to approximate away the headache terms which can be expressed neither as advection nor as Fick's law terms can also be tested by means of a comparison of the measured and 2-D modeled effective, i.e., transport-relevant, velocities. This empiricist turn in argumentation might not fully solve all aspects of the problem of interpretation of the observed 2-D velocities from a 3-D perspective, but at least it moves the problem from the desk of the observation scientist onto the desk of the 2-D modeler.

The authors thank two anonymous reviewers for their thorough examination of the original manuscript and for helpful and important comments. Furthermore, T. von Clarmann wishes to thank Hendrik Elbern, Richard Menard, Peter Braesicke, Björn-Martin Sinnhuber, and Thomas Birner for drawing his attention to some important literature and for encouragement as well as Arne Babenhauserheide for helpful discussions. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: J.-U. Grooß Reviewed by: two anonymous referees