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- Author or Editor: Edwin K. Schneider x

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## Abstract

A simple example problem is constructed for which the continuous primitive equations are shown to have a solution. The model is then discretized in the vertical in a manner equivalent to that used in the NMC global spectral model. It is shown that, in general, the discretized version of the problem has no solution. The vertical discretization schemes used in NMC and many other atmospheric circulation models are therefore inconsistent with the continuous equations. Conditions for design of a vertical differencing scheme that does not have this limitation are suggested.

## Abstract

A simple example problem is constructed for which the continuous primitive equations are shown to have a solution. The model is then discretized in the vertical in a manner equivalent to that used in the NMC global spectral model. It is shown that, in general, the discretized version of the problem has no solution. The vertical discretization schemes used in NMC and many other atmospheric circulation models are therefore inconsistent with the continuous equations. Conditions for design of a vertical differencing scheme that does not have this limitation are suggested.

## Abstract

The maximum eddy energies reached by baroclinically unstable disturbances in some simplified numerical models (Gall, 1976b; Simmons and Hoskins, 1978) are compared to a scale-dependent measure of the energy available to the disturbances. It is found that, except for the longest wavelengths, the ratio of the maximum eddy energies to this available energy (the eddy efficiency) tends to remain constant relative to the large variations in the maximum energies. The roles of the growth rate of the initial disturbance and the β-effect are discussed with reference to results from a maximally simplified model. It is shown that dimensional arguments paralleling Charney (1971) lead to the maximum energy versus scale relationships found in the numerical results.

## Abstract

The maximum eddy energies reached by baroclinically unstable disturbances in some simplified numerical models (Gall, 1976b; Simmons and Hoskins, 1978) are compared to a scale-dependent measure of the energy available to the disturbances. It is found that, except for the longest wavelengths, the ratio of the maximum eddy energies to this available energy (the eddy efficiency) tends to remain constant relative to the large variations in the maximum energies. The roles of the growth rate of the initial disturbance and the β-effect are discussed with reference to results from a maximally simplified model. It is shown that dimensional arguments paralleling Charney (1971) lead to the maximum energy versus scale relationships found in the numerical results.

## Abstract

A steady linearized version of a general circulation model (GCM) is a potentially useful tool for diagnosis and understanding of the time-mean solutions of the GCM. A method is developed for direct solution of the linearized equations. The method is efficient in the case that the GCM employs spectral horizontal discretization and finite differences in the vertical sigma coordinate, with seven or more vertical levels.

The method uses a system of equations that is equivalent to the linearized GCM, but with an expanded number of variables. The resulting system of equations can then be solved by sparse matrix techniques. The method is applicable to general basic states.

## Abstract

A steady linearized version of a general circulation model (GCM) is a potentially useful tool for diagnosis and understanding of the time-mean solutions of the GCM. A method is developed for direct solution of the linearized equations. The method is efficient in the case that the GCM employs spectral horizontal discretization and finite differences in the vertical sigma coordinate, with seven or more vertical levels.

The method uses a system of equations that is equivalent to the linearized GCM, but with an expanded number of variables. The resulting system of equations can then be solved by sparse matrix techniques. The method is applicable to general basic states.

## Abstract

A simplified linearized two-level zonally averaged atmospheric model and the calculus of variations are used to address the following problems: Assuming fixed thermal forcing except for the longwave, radiative response to temperature, we find the equilibrium states of 1) minimum zonal kinetic energy (ZKE) and 2) minimum zonal kinetic plus zonal available potential energy (ZKE + ZAPE) for any distribution of horizontal eddy momentum fluxes.

The solution to these problems contains interesting and suggestive features, among which are:

- The states of minimum ZKE or minimum ZKE + ZAPE contain a jet stream.
- The structure of the zonal winds is independent of the thermal forcing. The strength of the zonal winds is dependent on the thermal forcing.
- The horizontal eddy momentum fluxes that minimize the ZKE or ZKF + ZAPE can be up-gradient everywhere with respect to the upper level zonal mean zonal winds, down-gradient everywhere, or some combination, depending on the thermal forcing.
- ZKE minimization leads to a jet stream at 54° latitude independent of any external parameters. Minimization of ZKE + ZAPE loads to a jet stream at 35° for a planet of the earth's radius and static stability parameter when the top of the model is at approximately 300 mb.

## Abstract

A simplified linearized two-level zonally averaged atmospheric model and the calculus of variations are used to address the following problems: Assuming fixed thermal forcing except for the longwave, radiative response to temperature, we find the equilibrium states of 1) minimum zonal kinetic energy (ZKE) and 2) minimum zonal kinetic plus zonal available potential energy (ZKE + ZAPE) for any distribution of horizontal eddy momentum fluxes.

The solution to these problems contains interesting and suggestive features, among which are:

- The states of minimum ZKE or minimum ZKE + ZAPE contain a jet stream.
- The structure of the zonal winds is independent of the thermal forcing. The strength of the zonal winds is dependent on the thermal forcing.
- The horizontal eddy momentum fluxes that minimize the ZKE or ZKF + ZAPE can be up-gradient everywhere with respect to the upper level zonal mean zonal winds, down-gradient everywhere, or some combination, depending on the thermal forcing.
- ZKE minimization leads to a jet stream at 54° latitude independent of any external parameters. Minimization of ZKE + ZAPE loads to a jet stream at 35° for a planet of the earth's radius and static stability parameter when the top of the model is at approximately 300 mb.

## Abstract

A linear stationary wave model is used to diagnose the causes of stationary waves in integrations of a general circulation model (GCM) and to indicate the sources of differences between the stationary waves of separate integrations.

The GCM generates solutions to the equations of motion. The linear model, constructed to be as similar as possible in structure to the GCM, is employed in various attempts to approximate and understand the time averages of the GCM solutions. When GCM values for internal dissipation time constants are used in the linear model, significant differences between the linear model and GCM solutions are found. These differences can be interpreted as errors due to the linear approximation. The linear simulation is improved somewhat by enhancing the scale selective horizontal diffusion.

The linear model with enhanced dissipation is used to simulate the differences between the stationary waves of two consecutive months of a GCM integration. Transient forcing turns out to be the major cause for these differences, according to the linear model.

The phase structure of the errors of the linear solutions indicates that the error source is located primarily in the tropics and subtropics. Possible explanations for the errors are inaccurate representation of the topographic forcing and reflections from critical latitudes. The latter possibility is subjected to a crude test and cannot be rejected.

## Abstract

A linear stationary wave model is used to diagnose the causes of stationary waves in integrations of a general circulation model (GCM) and to indicate the sources of differences between the stationary waves of separate integrations.

The GCM generates solutions to the equations of motion. The linear model, constructed to be as similar as possible in structure to the GCM, is employed in various attempts to approximate and understand the time averages of the GCM solutions. When GCM values for internal dissipation time constants are used in the linear model, significant differences between the linear model and GCM solutions are found. These differences can be interpreted as errors due to the linear approximation. The linear simulation is improved somewhat by enhancing the scale selective horizontal diffusion.

The linear model with enhanced dissipation is used to simulate the differences between the stationary waves of two consecutive months of a GCM integration. Transient forcing turns out to be the major cause for these differences, according to the linear model.

The phase structure of the errors of the linear solutions indicates that the error source is located primarily in the tropics and subtropics. Possible explanations for the errors are inaccurate representation of the topographic forcing and reflections from critical latitudes. The latter possibility is subjected to a crude test and cannot be rejected.

## Abstract

Numerical experiments are performed to isolate the cause of differences between the simulations of SST in the low-latitude Pacific of two coupled atmosphere–ocean general circulation models, the Center for Ocean–Land–Atmosphere (COLA) coupled model and the NCAR Climate System Model (CSM). The COLA model produces a more realistic simulation of the annual cycle of SST and interannual SST variability. The CSM has the more realistic annual mean wind stress and east–west SST gradient. The approach to finding the causes of these differences is to systematically eliminate differences in the physical parameterizations and numerics of the two models, and to examine the effects of these changes on the simulations.

The results indicate that the atmospheric models rather than the ocean models are primarily responsible for differences in the simulations. There is no dominant process in the atmospheric models that explains the differences; both physical parameterizations (convection, surface flux formulation, shortwave radiation) and numerical schemes (vertical structure, moisture advection scheme) have significant effects. The effects of the parameterization changes on the annual mean SST are linear and additive, although tuning can cause apparent nonlinearity.

In terms of the effects that directly impact the ocean, the different physics and numerics of the atmospheric models change the net heat flux into the ocean and/or the sensitivity of the wind stress to SST. These properties can be estimated by AGCM-only simulations with observed SST. Flux correction is then used to identify the process responsible for the difference between the coupled simulations. Heat flux is found to produce most of the difference, and with the sign that would be expected from the heat budget of the mixed layer. However, the larger sensitivity of the NCAR atmospheric model wind stress has a significant impact on extending the cold tongue into the western equatorial Pacific.

## Abstract

Numerical experiments are performed to isolate the cause of differences between the simulations of SST in the low-latitude Pacific of two coupled atmosphere–ocean general circulation models, the Center for Ocean–Land–Atmosphere (COLA) coupled model and the NCAR Climate System Model (CSM). The COLA model produces a more realistic simulation of the annual cycle of SST and interannual SST variability. The CSM has the more realistic annual mean wind stress and east–west SST gradient. The approach to finding the causes of these differences is to systematically eliminate differences in the physical parameterizations and numerics of the two models, and to examine the effects of these changes on the simulations.

The results indicate that the atmospheric models rather than the ocean models are primarily responsible for differences in the simulations. There is no dominant process in the atmospheric models that explains the differences; both physical parameterizations (convection, surface flux formulation, shortwave radiation) and numerical schemes (vertical structure, moisture advection scheme) have significant effects. The effects of the parameterization changes on the annual mean SST are linear and additive, although tuning can cause apparent nonlinearity.

In terms of the effects that directly impact the ocean, the different physics and numerics of the atmospheric models change the net heat flux into the ocean and/or the sensitivity of the wind stress to SST. These properties can be estimated by AGCM-only simulations with observed SST. Flux correction is then used to identify the process responsible for the difference between the coupled simulations. Heat flux is found to produce most of the difference, and with the sign that would be expected from the heat budget of the mixed layer. However, the larger sensitivity of the NCAR atmospheric model wind stress has a significant impact on extending the cold tongue into the western equatorial Pacific.

## Abstract

A model is developed to find the low-latitude upper-tropospheric atmospheric circulation which would exist in the absence of transient eddies. The forcing in the model is zonally asymmetric to take into account the effect of the distribution of oceans and continents on latent heating. Steady state solutions of the equations of motion in such a case have been termed the “modified Hadley circulation” by Lorenz. The model is the set of steady, nonlinear inviscid shallow-water equations with thermal forcing as introduced by Gill, and is solved semianalytically. The modified Hadley circulation found using forcing derived from observed OLR exhibits some similarities to observed features of the tropical and subtropical time-mean upper-tropospheric flow, particularly with regard to the velocity potential, as well as many dissimilarities.

The roles of transients and friction as indicated by the model are discussed.

## Abstract

A model is developed to find the low-latitude upper-tropospheric atmospheric circulation which would exist in the absence of transient eddies. The forcing in the model is zonally asymmetric to take into account the effect of the distribution of oceans and continents on latent heating. Steady state solutions of the equations of motion in such a case have been termed the “modified Hadley circulation” by Lorenz. The model is the set of steady, nonlinear inviscid shallow-water equations with thermal forcing as introduced by Gill, and is solved semianalytically. The modified Hadley circulation found using forcing derived from observed OLR exhibits some similarities to observed features of the tropical and subtropical time-mean upper-tropospheric flow, particularly with regard to the velocity potential, as well as many dissimilarities.

The roles of transients and friction as indicated by the model are discussed.

## Abstract

The roles of eddy and zonal mean processes in producing the earth's annual-mean zonally-averaged tropospheric wind and temperature fields are investigated using a numerical zonally-averaged atmospheric model. First a diagnostic calculation is performed to determine the beat sources and sinks and mean meridional circulation which maintain the observed annual and zonal mean temperatures and surface zonal winds, given that all zonal momentum sources except those due to the zonal-mean motions are observed or parameterized. The use of observed data and calculations of the zonally-averaged radiative heat sources and sinks, together with some simplifying assumptions, permits estimation of the surface sensible heating atmospheric latent heating, and vertical eddy sensible heat flux divergences. Then the heat and momentum sources and sinks are varied (longwave radiation is taken to depend on the zonal mean temperature), and the steady state response of the zonal winds, temperatures, and meridional circulation to the altered forcing is found. Among the more interesting results are:

- As long as there is sufficient internal friction for a steady state to exist, the Hadley circulation mass flux does not respond strongly to changes in the strength of the horizontal eddy momentum flux forcing, but does respond strongly to changes in the distribution of the tropical thermal forcing.
- The response to varying the width but not the total precipitation of the intertropical convergence zone, with fixed eddy fluxes and extratropical heat sources, bears some resemblance to the observed zonally-averaged El Niño changes.
- Given fixed latent and eddy heat sources, the horizontal eddy momentum fluxes are varied. The minimum speed of the jet stream is found in the reference state.

## Abstract

The roles of eddy and zonal mean processes in producing the earth's annual-mean zonally-averaged tropospheric wind and temperature fields are investigated using a numerical zonally-averaged atmospheric model. First a diagnostic calculation is performed to determine the beat sources and sinks and mean meridional circulation which maintain the observed annual and zonal mean temperatures and surface zonal winds, given that all zonal momentum sources except those due to the zonal-mean motions are observed or parameterized. The use of observed data and calculations of the zonally-averaged radiative heat sources and sinks, together with some simplifying assumptions, permits estimation of the surface sensible heating atmospheric latent heating, and vertical eddy sensible heat flux divergences. Then the heat and momentum sources and sinks are varied (longwave radiation is taken to depend on the zonal mean temperature), and the steady state response of the zonal winds, temperatures, and meridional circulation to the altered forcing is found. Among the more interesting results are:

- As long as there is sufficient internal friction for a steady state to exist, the Hadley circulation mass flux does not respond strongly to changes in the strength of the horizontal eddy momentum flux forcing, but does respond strongly to changes in the distribution of the tropical thermal forcing.
- The response to varying the width but not the total precipitation of the intertropical convergence zone, with fixed eddy fluxes and extratropical heat sources, bears some resemblance to the observed zonally-averaged El Niño changes.
- Given fixed latent and eddy heat sources, the horizontal eddy momentum fluxes are varied. The minimum speed of the jet stream is found in the reference state.

## Abstract

An analysis of Rossby wave dynamics in two and three dimensions is carried from the point of view of a reference frame propagating with the zonal phase speed of the wave. Since trajectories and streamlines coincide in this reference frame, the mechanism for (westward) propagation of free waves has a different interpretation than in a reference frame fixed to the ground. In the wave reference frame, propagating free-wave solutions are possible only when parcels approach from the west. When parcels approach from the east, potential vorticity cannot be conserved along trajectories.

## Abstract

An analysis of Rossby wave dynamics in two and three dimensions is carried from the point of view of a reference frame propagating with the zonal phase speed of the wave. Since trajectories and streamlines coincide in this reference frame, the mechanism for (westward) propagation of free waves has a different interpretation than in a reference frame fixed to the ground. In the wave reference frame, propagating free-wave solutions are possible only when parcels approach from the west. When parcels approach from the east, potential vorticity cannot be conserved along trajectories.

## Abstract

The linearized model presented in Part I of this study (Schneider and Lindzen, 1977) is extended to include the nonlinear advections of angular momentum by the meridional circulation. A crude cumulus heating parameterization is also introduced in certain calculations to simulate the effect of an internal hydrological cycle. The nonlinear (steady-state) boundary value problem is solved by means of an iterative scheme. The results from the nonlinear model are viewed as a reasonable basic state whose stability may be studied. It is seen that this basic state is similar in many respects to the observed zonally averaged general circulation. Surface easterlies and westerlies appear at the correct latitudes with the right magnitudes, the tropical Hadley cell has near the observed mass flux and geometry, with an ITCZ forming at the sea surface temperature maximum, mid-latitude Ferrel cells appear (although fluxes by baroclinic eddies are not modeled) with somewhat less mass flux than the observations indicate and the tropical tropospheric temperature fields are close to the observed. The main departures of the model results from the observations is in the excessive magnitude of the upper level zonal winds in middle latitudes. This difference is ascribed to the neglect of fluxes by large-scale eddies (baroclinic, barotropic and topographic) in the model. The similarity of the observed surface winds to those computed in the model is in opposition to some classical pictures [such as that of Jeffreys (1926)] in which the mid-latitude westerlies are maintained by poleward transports of zonal momentum by large-scale eddies. Results from the numerical model are used to develop arguments showing that the horizontal scale of the tropical Hadley circulation is a suitably defined global Rossby radius of deformation, and that the vertical scale cf the Hadley circulation and the tropical static stability can be adequately approximated by simple one-dimensional radiative-convective models (see the Appendix).

## Abstract

The linearized model presented in Part I of this study (Schneider and Lindzen, 1977) is extended to include the nonlinear advections of angular momentum by the meridional circulation. A crude cumulus heating parameterization is also introduced in certain calculations to simulate the effect of an internal hydrological cycle. The nonlinear (steady-state) boundary value problem is solved by means of an iterative scheme. The results from the nonlinear model are viewed as a reasonable basic state whose stability may be studied. It is seen that this basic state is similar in many respects to the observed zonally averaged general circulation. Surface easterlies and westerlies appear at the correct latitudes with the right magnitudes, the tropical Hadley cell has near the observed mass flux and geometry, with an ITCZ forming at the sea surface temperature maximum, mid-latitude Ferrel cells appear (although fluxes by baroclinic eddies are not modeled) with somewhat less mass flux than the observations indicate and the tropical tropospheric temperature fields are close to the observed. The main departures of the model results from the observations is in the excessive magnitude of the upper level zonal winds in middle latitudes. This difference is ascribed to the neglect of fluxes by large-scale eddies (baroclinic, barotropic and topographic) in the model. The similarity of the observed surface winds to those computed in the model is in opposition to some classical pictures [such as that of Jeffreys (1926)] in which the mid-latitude westerlies are maintained by poleward transports of zonal momentum by large-scale eddies. Results from the numerical model are used to develop arguments showing that the horizontal scale of the tropical Hadley circulation is a suitably defined global Rossby radius of deformation, and that the vertical scale cf the Hadley circulation and the tropical static stability can be adequately approximated by simple one-dimensional radiative-convective models (see the Appendix).