Numerical general circulation models of the atmosphere are generally required to conserve mass and energy for their application to climate studies. Here we draw attention to another conserved global integral, viz. the component of angular momentum (AM) along the Earth's axis of rotation, which tends to receive less consideration. We demonstrate the importance of global AM conservation in climate simulations with the example of the Community Atmosphere Model (CAM) with the finite-volume (FV) dynamical core, which produces a noticeable numerical sink of AM. We use a combination of mathematical analysis and numerical diagnostics to pinpoint the main source of AM non-conservation in CAM–FV. We then present a method to enforce global conservation of AM, and we discuss the results in a hierarchy of numerical simulations of the atmosphere of increasing complexity. In line with theoretical expectations, we show that even a crude, non-local enforcement of AM conservation in the simulations consistently results in the mitigation of certain persistent model biases.

The atmosphere exchanges angular momentum (AM) with the material bodies at the surface, which are, to a good approximation, in a state of motion consisting of uniform rotation about the planetary axis connecting the poles. Per unit of mass, surface AM increases in quadratic proportion to its distance from the planetary axis of rotation, from zero at the poles to a maximum at the Equator. AM is a constant of motion of the dynamical (e.g. Newton's) equations so that as air travels meridionally, it carries a specific AM that increasingly differs from that of the Earth's surface. A variety of mechanisms redistribute atmospheric AM and eventually lead to an exchange of AM between the atmosphere and the surface, mainly as a result of low-level wind shear (“surface stress”) and of small-scale wave motions over steep surface topography (“form drag”).

The importance for the atmospheric circulation of the conservation of AM in the free troposphere and of AM exchange of air with the surface was recognised long ago. Already in 1735, George Hadley, Esq, F.R.S., noted that “without the Assistance of the diurnal Motion (i.e. rotation) of the Earth, Navigation (…) would be very tedious” (Hadley, 1735) due to the absence of the trade winds. This insight still lies at the core of modern conceptual models for the atmospheric circulation (Schneider, 1977; Held and Hou, 1980; Lindzen and Hou, 1988; Pauluis, 2004; Walker and Schneider, 2006). In the upper branch of the Hadley circulation (HC), the advection of planetary angular momentum determines a sharp acceleration of the zonal wind in the mid-latitudes linked to a front-like drop in air temperatures, marking the location of the subtropical jets (STJs). Partly by baroclinic instability, the mid-latitude circulation redistributes atmospheric AM vertically and produces intense surface westerlies, whereby the air loses AM to the surface. The equatorward return flow in the surface branch of the HC in turn results in easterly “trade” winds, whereby surface stresses replenish atmospheric AM until air is lifted in cumulus convection within the inter-tropical convergence zone (ITCZ).

This circulation is the object of numerical simulations with
general circulation models (GCMs) used in meteorological
forecasting and in climate modelling. They describe the atmosphere
as a thin, density-stratified, rotating gaseous spherical shell.
These properties allow the introduction of a convenient set of
approximations in the equations of motion, which result in a system
known as the hydrostatic primitive equations (HPEs). The reader is
referred to White et al. (2005) for a detailed analysis and
discussion. Given suitable boundary conditions, the HPEs guarantee
the global conservation of three fundamental physical quantities:
mass, energy, and AM along the Earth's rotation axis. Analytic
expressions of these laws can be found e.g. in Laprise and Girard (1990). The three conservation laws determine the fundamental
character of the large-scale circulation of the atmosphere, and
virtually every climate application of GCMs is sensitive to their
enforcement when the continuum equations are discretised in space
and time. For example, the effects of changes in radiative forcing
of

CAM, the Community Atmosphere Model developed and maintained at the National Center for Atmospheric Research (NCAR) in Boulder, Colorado, is one of the atmospheric general circulations models (AGCMs) in most widespread use today. It also constitutes the core atmospheric component of NorESM, the Norwegian Earth System Model. Although it offers a choice of dynamical cores, the finite-volume (FV) dynamical core (Lin, 2004) has been, and in many instances still is, the default option. The FV dynamical core is exactly mass and vorticity conserving, and it has been employed in all model integrations submitted by NCAR and by the Norwegian Climate Centre (NCC) for the 5th phase of the Coupled Model Intercomparison Project (CMIP) contributing to the Assessment Report (AR) of the Intergovernmental Panel for Climate Change (IPCC, 2013); it is also expected to be used for phase 6 of CMIP by both institutions. Due to its high numerical efficiency, FV also continues to be the code of choice for all uses for which the overall availability of supercomputing resources is a limiting factor. This includes long historical or palaeoclimate simulations, studies with coupled chemistry and/or carbon cycle, seasonal-to-decadal coupled forecasts, academic research, and all model development efforts currently underway with NorESM.

In this paper, we employ CAM with the FV dynamical core at two
standard CESM resolutions only, a coarser one of
1.9

Figure

Numerical torque in idealised CAM simulations. The
vertically and zonally integrated apparent numerical torque is shown
as a function of latitude for CAM simulations in aquaplanet (AP;

First principles (e.g. Held and Hou, 1980; Einstein, 1926) suggest that the dissipation of AM, equivalent to a body force acting on the fluid as a sink of zonal momentum, forces a secondary circulation with the same sign as the Hadley circulation. As a result, the simulated Hadley circulation may become too vigorous. Reduced meridional advection of zonal momentum may lead to mid-latitude westerlies that are too weak or displaced poleward. The zonal momentum lost to the non-physical sink must be balanced by a matching additional eastward torque, for example in an expanded or excessively intense area of tropical easterly surface winds. Model simulations with CAM FV consistently tend to reflect such phenomenology: for example, Feldl and Bordoni (2016) and Lipat et al. (2017) show that among CMIP5 models, those based on the FV dynamical core (GFDL-x, CCSM4, and NorESM-x) simulate both relatively large overturning mass flux in the HC and a high latitude of its edge.

It is useful to illustrate these effects of AM non-conservation by
means of idealised AGCM experiments that do not include complicating
factors such as orographic form drag or parameterised bulk stresses
associated with gravity waves.
Figure

Impact of AM sink in CAM–FV integrations. Meridional
distribution of the surface stress torque (analogous to the
dashed red lines in Fig.

The other two integrations, represented by the blue and red
lines, are perturbed in an identical but opposite manner. First, the
global total numerical torque due to the FV dynamical core was
diagnosed at every time step of the reference FV simulation and
averaged in time afterwards. This was converted into a solid-body
axial rotation tendency that was applied continuously everywhere as
a constant
sink of AM
in a new integration
with the spectral dynamical core, resulting in the simulation
represented by the red curve.
Vice versa, the opposite additional solid-body rotation tendency was
applied to a new FV integration, thus compensating for its internal
numerical sink. This integration produced the physical torque
represented by the blue curve. Comparing the different curves, it may
be seen that equatorward of about 23

These results motivate us to address the issue of AM conservation in the CAM's FV dynamical core. One may speculate that systematic biases in surface stresses due to the numerical sink of AM must also impact coupled ocean–atmosphere climate simulations, with excessive Ekman and Sverdrup forcing of the subtropical gyres. The northward displacement of the mid-latitude westerlies may also result in excessive mechanical and thermal forcing of the subpolar gyres with possible implications for the Atlantic meridional overturning circulation.

In this paper, we propose ways to address the numerical dissipation of AM in CAM–FV simulations. Section 2 describes our main hypotheses as to the root cause of the error and our approaches towards rectification. Section 3 presents the result of our corrections in a set of idealised simulations. The impact on realistic simulations of the atmospheric circulation is discussed in Sect. 4. Conclusions are finally offered in Sect. 5.

The FV dynamical core (Lin, 2004) solves the HPE by updating first the advective (C grid) and then the prognostic (D grid) winds in two steps. The first step represents pure advection, i.e. the increments associated with transport, including geometric and Coriolis terms. In this step, the scheme conserves absolute vorticity exactly for the D-grid winds (Lin and Rood, 1997; hereafter LR97). The second step calculates the wind increments associated with hydrostatic pressure forces. These are computed in a special way (Lin, 1997) that differs from most Arakawa and Lamb (1981) type schemes. Violations of AM conservation may occur in either sub-step.

We first analysed the Lin (1997) treatment of the pressure-gradient
terms for conservation. A general discussion is given by Simmons and
Burridge (1981), who introduce a set of hybrid-level dimensionless
variables,

Performing Lin's (1997) path integration around the finite-volume
element on this expression yields the following form for the body
force:

In other words, Lin's (1997) expression for the pressure-gradient
term is consistent with the Simmons and Burridge (1981) prescription
for AM conservation, provided that the physical pressure
variable

Accordingly, we performed tests in which the integration variable in the relevant section of CAM–FV's dynamical core was replaced with true interface pressure. The effect was generally seen to be very small on the dynamical core's momentum conservation properties.

We note, however, that in the CAM implementation there may be
an additional problem associated with the use of the D grid.
The application of Lin's (1997) method would strictly require a
C grid, with zonal velocity points interleaving pressure (scalar)
points along the same latitude.
Thus, in CAM pressure is interpolated to the grid cell corners before
use.
While the formal expressions for the pressure forces do not change,
thus ensuring the Simmons and
Burridge (1981) total torque constraints, the inertial mass
associated
with each D-grid zonal velocity point is in fact averaged over six scalar points
surrounding it, with 1-2-1 weights along the zonal direction. This
additional zonal smoothing effectively adds
spurious terms to the zonal momentum equation of the form

AM conservation may be affected by the treatment of geometric terms in latitude–longitude coordinates, especially near the poles where such terms become large. Furthermore, convergence of the meridians forces filtering of the solution, and additional approximations need to be made. In particular, LR97 implement a flux-form semi-Lagrangian extension of Colella and Woodward's (1984) piece-wise parabolic method (PPM), which is used near the poles where Courant–Friedrichs–Lewy numbers (Courant et al., 1928) become large during the time integration. We performed several sensitivity tests on each of these aspects without being able to notice significant impacts on AM conservation.

Particularly compelling is the comparison with the performance of a
prototype implementation in CAM of the FV scheme on a cubed-sphere
grid (FV3), which lacks any poles and does not require or use
any of these special formulations (and is, in particular, run in
pure Eulerian mode, i.e. without the flux-form semi-Lagrangian
extension described in Lin and Rood, 1996).
We ran an AP simulation on the C48 grid, viz.
six pseudo-cubic faces with

In order to minimise the impact of other minor (and partly intentional) numerical sources and sinks of AM, in all idealised numerical tests presented in this paper we applied the following modifications: (1) the order of the advection scheme is kept the same (fourth) for all model layers, instead of reducing it to first in the top layer and to second up to the eighth layer; (2) an additional conservation check is applied in the vertical remapping of zonal wind , and column momentum is conserved in the moist-mass adjustment at the end of physics; (3) the surface stress residual resulting from closure of the diffusion operator (in physics) is applied in full rather than partially.

The evidence from our theoretical and diagnostic analysis points at the advective, shallow-water part of the implementation of LR97 in CAM–FV as the root of the AM conservation error. Its “vector-invariant” formulation (Arakawa and Lamb, 1981) allows for different forms of the divergence to be used in the momentum and in the mass and tracer equations, resulting in inconsistent values for the divergence of the flux of planetary AM (associated with mass divergence) and of the flux of relative AM (associated with momentum divergence). In the momentum equations, the divergence is contained in a kinetic energy (KE) gradient term, which due to the presence of a numerical symmetric instability (Hollingsworth et al., 1983) is expressed as the local gradient of a Lagrangian-average KE. Its form violates the finite-volume approximations used for other quantities, e.g. vorticity. This feature is intrinsic to the LR97 numerical discretisation scheme and cannot be eliminated.

To address the resulting violation of AM conservation,
we first note that even in AM-conserving
schemes, conservation can only be guaranteed in the zonal average
(Simmons and Burridge, 1981). We therefore do not attempt a local
correction to the scheme, which is liable to numerical instabilities
(Hollingsworth et al., 1983), and instead formulate a zonal-mean
correction as follows. We enforce the AM conservation law,

In discrete form, the last two terms must be approximated. In the
C–D grid formulation of the LR97 scheme the second one is especially
problematic. Various possibilities were explored, which resulted in
various degrees of accuracy and stability. The best compromise is to
discretise it as

Irrespective of whether the correction, as described above, is applied or not, for diagnostic purposes we calculate the apparent non-physical torque associated with the FV dynamical core advective tendencies only, i.e. excluding the increments associated with pressure gradients. These tendencies are diagnosed separately for each layer at every advective sub-step and integrated horizontally to yield the apparent numerical global total torque during the sub-step. At the same time, the layer effective moment of inertia over the sub-step is also computed.

The opposite of the ratio of these quantities gives an angular
acceleration that,
applied to the zonal wind in each layer at every advective sub-step,
enforces conservation of the AM of that layer under advection.
The application of this solid-body rotation increment at each
dynamical time step and for each layer independently is what we call
the “level” fixer.
The details of the computation are given in Appendix

Irrespective of whether they are actually applied, the fixer's
velocity increments (Eq.

A variant of the fixer was tested in CAM simulations. This variant is a “global” fixer, which still acts by applying an increment to the zonal wind at each time step. In this fixer, the apparent torque and the moment of inertia are integrated over all levels within the domain over which strict overall angular momentum conservation is desired. The zonal-wind increments are then applied as a single solid-body rotational acceleration within this domain. Experimentation showed that such acceleration should not be applied in the stratosphere, where conservation errors are small and the impact of unphysical zonal accelerations large. The necessary limitation of the domain for the global fixer, however, introduces a certain degree of arbitrariness in its application. Although sometimes used for diagnostic purposes, we do not discuss this global fixer variant any further.

Lin's (2004) FV scheme conserves mass and absolute vorticity exactly. The AM modifications described above were explicitly designed not to alter the mass flux calculations and intervene only on the rotational component of the flow in the momentum equations. Other choices, involving alterations to the calculation for the divergent flow, would have been possible. However, we judged exact mass conservation more important for climate simulations than exact vorticity conservation. The AM modifications also change the kinetic energy of the flow and thus change the total energy budget of the model. However, the unmodified FV scheme does not conserve energy. CAM–FV therefore employs an energy “fixer” (analogous to out AM fixer), described e.g. in Williamson et al. (2015). The fixer diagnoses the energy non-conservation at each time step. This allowed us to monitor the impact of the AM mods on energy non-conservation in all our experiments. We found no systematic effect, either in sign or magnitude, of the AM modifications on the energy non-conservation of the model.

Initial tests were carried out for adiabatic dynamics and flat
bottom topography from baroclinically unstable initial conditions,
as defined in Jablonowski and Williamson (2006; JW06).
Figure

AM correction and fixer in adiabatic, frictionless baroclinic
wave tests. Three sets of curves are shown for each of four
different simulations with CAM FV, indicating the time evolution of
global AM (diamond shapes) and its two components of planetary AM
(vertical crosses) and relative AM (

It may be seen that both the correction and the fixer are effective in
reducing the systematic numerical sink of AM in these integrations.
In particular, the fixer appears to remove it almost completely; in
other words, the integration with the fixer conserves global AM in the
time mean. This result is central to this paper, and it proves its two
main conclusions. The first is that the systematic non-conservation of
global AM in the FV dynamical core indeed resides in the advective wind
increments of the shallow-water part of the dynamical core. The second is
that, by virtue of its effectiveness and its formulation that is
entirely independent of the model configuration or parameterisations
(topography, physical momentum sources, etc.), the fixer is a useful
and accurate general diagnostic tool that allows us to quantify the
numerical torque in any CAM–FV integration.
By virtue of this quality, the diagnosed time-averaged fixer
tendencies were, for example, used for the perturbations in the
experiments shown in Figs.

The impact of the correction on conservation is generally smaller, and
different dynamical regimes may be seen when the size and quality of
that impact change.
In the baroclinic instability tests in Fig.

Aside from the conservation properties they are designed for, both the
correction and the fixer represent a perturbation of the numerical
solutions of the FV dynamical core. By arbitrarily modifying the
relative vorticity associated with the zonal wind, both destroy one of
the fundamental numerical properties of the LR97 formulation, viz. the
conservation of absolute vorticity under advection. (In the case of
the fixer, the vorticity input has a rigid dependency on latitude,

AM correction and fixer in an adiabatic, frictionless baroclinic
wave test. The simulations shown in Fig.

It may be noted that the
largest impact on the RMS of surface pressure arises from the
correction. Within the first 30 d this impact is formally always
well below significance (as defined in JW06; see their Fig. 10),
but it
increases in time and eventually becomes appreciable as a full global
meridional circulation is established.
Similar results hold for the vorticity field, as seen in
Fig.

Other aspects of the solution besides RMS differences also show
limited sensitivity to the application of the correction and the
fixer. Figure

AM correction and fixer in an adiabatic, frictionless baroclinic
wave test. Evolution of minimum pressure

Even if the impacts of the modifications of the FV dynamical core are relatively small on local circulations over subseasonal timescales, as shown above, the rationale for introducing them is the hope of achieving a better simulation of the state of the atmosphere in integrations under specified forcings. As explained in the Introduction, one particular expectation is that the subtropical easterlies should weaken without affecting the circulation elsewhere too heavily. In particular, the role of the correction, which alone does not ensure AM conservation, must be clarified and its eventual use justified. Here we document the results of two sets of idealised simulations that still have a simplified, equipotential lower boundary but include non-vanishing physical torques and heating tendencies.

The first set of such simulations adhere to the benchmark test of Held and Suarez (1994; HS henceforth), whereby the forcing has the form of a relaxation towards a specified three-dimensional atmospheric temperature field. Likewise, surface friction is represented by a damping of the winds within a set of levels near the bottom boundary. Apart from the small numerical diffusion, these stresses are communicated to the rest of the atmosphere by means of momentum advection in the mean circulation and of pressure fluctuation in resolved transient motions (including travelling waves). The second set of simulations follows the aquaplanet (AP) test first proposed by Neale and Hoskins (2000), wherein only a persistent field of bottom-boundary temperatures is prescribed (the QOBS profile of Neale and Hoskins, 2000), and the full set of moist atmospheric physical parameterisations of CAM6 are used to force the circulation (except for those specific to orographic processes). The bottom boundary is a notional static ocean with unlimited heat and water capacity. Surface stresses are computed by the coupler and passed to the moist atmospheric boundary-layer parameterisation, which then distributes those stresses vertically. Momentum is also transported in moist convection, where active, and further adjustments are made when the moist mass of the atmospheric column changes due to precipitation and surface evaporation processes. To simplify the analysis, the gravity-wave parameterisation of CAM6 was turned off in our AP tests. In both sets of tests, FV's advection scheme is used at PPM's standard fourth order at all levels; i.e. the numerical diffusion obtained in standard CAM–FV integrations by employing low-order calculations near the model top is avoided. For initial conditions, HS simulations are cold-started with uniform surface pressure and geopotential, as well as vanishing wind fields except for a westerly perturbation identical to that used in the dry baroclinic wave tests (necessary in order to break zonal symmetry and to allow for a non-vanishing correction). The AP simulations all take the same instantaneous atmospheric state from a previous spun-up run, even though this requires more adjustment for the corrected (fixed) simulations than for the control.

AM correction and fixer in Held–Suarez (HS) and aquaplanet
(AP) integrations. Panel

Impact of the AM correction and fixer in Held–Suarez
simulations. Time-mean latitude–pressure profiles of wind
differences between HS simulations shown in the stippled lines in
Fig.

Figure

Impact of the AM correction and fixer in aquaplanet
simulations. Same as Fig.

The effectiveness of the fixer in removing most of the AM drift
confirms that the systematic sink of AM in CAM–FV integrations arises
predominantly from the shallow-water advection calculations.
The accuracy of the correction, by contrast, depends on the features
of the circulation, with good accuracy for numerically well-resolved
features, as in the HS tests, but a poorer one when grid-scale forcing
associated with the water cycle occurs. Figure

The effect on the mean circulation of applying the correction and/or
the fixer is shown in Figs.

In AP simulations, a slowdown of the meridional circulation is still
expected and found, but the interaction between dynamical forcing by
the fixer or the correction and the physics tendencies is much more
complex and difficult to predict. The fixer now produces large
westerly differences near the Equator at all levels and a marked
weakening of the subtropical jet stream (Fig.

Impact of the AM correction and fixer in F2000
simulations. Panels

The relevance of the AM modifications to the FV dynamical core for CAM
simulations in a realistic configuration is investigated here using
F2000 cases, which are AMIP-type simulations (Gates, 1992) wherein
sea surface temperatures (SSTs) and all compositional forcings are
prescribed as a repeating annual cycle obtained from an observed
climatology of the decade spanning the turn of the century. We test at
two grid resolutions, one of 1.9

More precisely, we used a pre-release of CESM2.1.1 (no. 20, 22 March 2019). In terms of the simulations presented in this paper, the differences with the full 2.1.1 release only affect the F2000 cases at f19 resolution, for which slightly different emission datasets are used to force the simulations. The impacts of this are of negligible consequence for the results discussed in this section.

.Impact of the AM correction and fixer in F2000
simulations. Latitude–pressure maps of zonal-mean zonal-wind
climatologies for boreal winter (DJF). Panels

Impact of the AM correction and fixer in F2000 simulations.
Panels

Figure

In general, we obtain a similar conclusion as for the AP
simulations. The impact of the correction on the global
conservation of AM is modest, removing only about 15 % of the
sink at f19 resolution. However, its action is stronger on
upper-level winds (see Fig.

Figures

More in detail, it may be noted that the benefits of the AM
modifications appear more clearly for the winds in the simulation
at the lower resolution, for which the numerical sink of AM is indeed
larger. These benefits, however, are not limited to the zonal-mean
zonal winds, and they are also appreciable at the f09 resolution.
Most notable is the reduction in the strength of the Hadley
circulations (see Fig. S4
in the Supplement), which is expected from the arguments set out in the
Introduction. This has consequences for many aspects of the global
circulation.
Figure

AM conservation in CAM–FV has been substantially improved by means of
a correction that reduces the zonal-mean numerical sink of Lin and
Rood's (1997) shallow-water scheme and a fixer that ensures
the conservation of global angular momentum under advection.
The effectiveness of these modifications in terms of AM conservation in
the simulations presented here is summarised in Table

The zonal-mean correction of the shallow-water scheme is not necessary for enforcing global conservation, as this can be achieved by the fixer alone. Indeed, the correction is quite ineffective in realistic simulations of the atmosphere in terms of global conservation. Nevertheless, we find that its concomitant application with the fixer has positive impacts on the simulations. In particular, it reduces the effects of the fixer in the mid-latitudes. This can be explained with the greater effectiveness of the correction in the baroclinically unstable regions around the subtropical jet streams, where the zonal-mean numerical sink appears to be largest. Even so, because of its potentially large local effects, the utilisation of the correction under different set-ups should be tested on a case-by-case basis according to its impacts on the results.

Improving the quality of the simulation of the global distribution of surface wind stress should be expected to bring particular benefits to coupled atmosphere–ocean simulations. An adequate discussion of such a coupled simulation would exceed the scope of the present paper, which is aimed primarily at presenting the method. In particular, due to their computational expense, at the present time it is not possible to produce well spun-up coupled simulations that can provide an assessment of the impact of the AM modifications.

Simulation set-ups and the effect of AM modifications. The
percentage figures represent the numerical source (negative for
sink) of global total atmospheric AM relative to the global total
physical eastward torque acting on the atmosphere (terms

The modification to the FV dynamical core that we describe and utilise is relatively crude and causes local loss of accuracy due to violation of vorticity conservation under advection. Nevertheless, the associated detrimental impacts appear to be fairly limited, with insignificant differences under standard tests such as the Jablonowski and Williamson (2006) baroclinic wave test, which should be sensitive to local conservation. Even so, it is clear from the very same tests that simulations over weather timescales are not sensitive to AM conservation, so for such an application it is not advisable to trade enforcing such conservation for a loss of accuracy. On the longer timescales of climate simulations, by contrast, our results demonstrate the importance of the global conservation of atmospheric AM in order to obtain a realistic global circulation.

The diagnosis of the residual torque that violates AM conservation
in CAM simulations follows from the hydrostatic primitive equations
(see White et al., 2005). In our zonally and vertically integrated
diagnostics such as in Fig.

The local conservation equation for the shallow-water equations is

We note two aspects of this expression. First, there is a
significant numerical cancellation between the second and the
third lines on the right-hand side. Second, all advective terms
in the first two lines on the right-hand side can be easily
discretised according to the standard LR97 prescription and are thus
automatically defined on D-grid zonal velocity points, i.e. where required for

We therefore chose to approximate the last term as follows:

We note that setting the higher-order terms to zero implies that the
correction has no effect on a zonally symmetric flow.
If, in addition, the flow is in an exact steady state, then the
correction always vanishes identically, regardless of these
terms.
It can further be shown that, if the term in

In Eq. (

As we explain in Sect.

So, for each time step and at each level

In some regions of the model domain, it is
not desirable to apply a fixer, since dissipation is explicitly built
into in the dynamical core formulation. This is the case near the upper
boundary of CAM's domain (the lower boundary in pressure space),
where the fixer is accordingly switched off. In general, a weight

The global fixer applies the same solid-body rotation increment to
all levels within the domain where it is required. When all weights
are unity, this is simply

For diagnostic purposes, fixer increments are always calculated as in
Eq. (

The code used in the numerical simulations of this paper is available
under:

The supplement related to this article is available online at:

TT conceived the idea, proposed the work, made the calculations, implemented the code, ran the simulations, evaluated them, produced all figures, and wrote the paper. MB supported this activity through national infrastructure projects of the Norwegian Research Council. CC, BEE, and JE revised the code and included it in the official ESCOMP CESM repository. SG gave technical advice on CAM code and simulations. PHL, MB, and CJ were at hand for critical discussions of the scientific ideas and helped provide the initial impetus for this work. MB, JE, SG, and PHL also provided useful comments and suggestions on the draft paper.

The authors declare that they have no conflict of interest.

Warm thanks go to Christoph Heinze for his unbending dedication to model development in the NorESM consortium and in particular for allowing this work to go forward despite the lack of dedicated funding. We are grateful to Alok Gupta at NORCE and Cecile Hannay at NCAR for their assistance with NorESM and CESM development simulations.

This work was partially supported by the Norwegian Research Council (EVA (grant no. 229771) and INES (grant no. 270061)) as well as by the US National Science Foundation Cooperative Agreement (grant no. 1852977).

This paper was edited by Paul Ullrich and reviewed by two anonymous referees.