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- Author or Editor: Gerald R. North x

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## Abstract

Simple climate models employing diffusive heat transport and ice cap albedo feedback have equilibrium solutions with no stable ice cap smaller than a certain finite size. For the usual parameters used in these models the minimum cap has a radius of about 20 degrees on a great circle. Although it is traditional to remove this peculiar feature by various *ad hoc* mechanisms, it is of interest because of its relevance to ice age theories. This paper explains why the phenomenon occurs in these models by solving them in a physically appealing way. If an ice-free solution has a thermal minimum and if the minimum temperature is just above the critical value for formation of ice, then the artificial addition of a patch of ice leads to a widespread depression of the temperature below the critical freezing temperature; therefore, a second stable solution will exist whose spatial extent is determined by the range of the influence function of a point sink of heat, due to the albedo shift in the patch. The range of influence is determined by the characteristic length in the problem which in turn is determined by the distance a heat anomaly can be displaced by random walk during the characteristic time scale for radiative relaxation; this length is typically 20–30 degrees on a great circle. Mathematical detail is provided as well as a discussion of why the various mechanisms previously introduced to eliminate the phenomenon work. Finally, a discussion of the relevance of these results to nature is presented.

## Abstract

Simple climate models employing diffusive heat transport and ice cap albedo feedback have equilibrium solutions with no stable ice cap smaller than a certain finite size. For the usual parameters used in these models the minimum cap has a radius of about 20 degrees on a great circle. Although it is traditional to remove this peculiar feature by various *ad hoc* mechanisms, it is of interest because of its relevance to ice age theories. This paper explains why the phenomenon occurs in these models by solving them in a physically appealing way. If an ice-free solution has a thermal minimum and if the minimum temperature is just above the critical value for formation of ice, then the artificial addition of a patch of ice leads to a widespread depression of the temperature below the critical freezing temperature; therefore, a second stable solution will exist whose spatial extent is determined by the range of the influence function of a point sink of heat, due to the albedo shift in the patch. The range of influence is determined by the characteristic length in the problem which in turn is determined by the distance a heat anomaly can be displaced by random walk during the characteristic time scale for radiative relaxation; this length is typically 20–30 degrees on a great circle. Mathematical detail is provided as well as a discussion of why the various mechanisms previously introduced to eliminate the phenomenon work. Finally, a discussion of the relevance of these results to nature is presented.

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## Abstract

An attempt to provide physical insight into the empirical orthogonal function (EOF) representation of data fields by the study of fields generated by linear stochastic models is presented in this paper. In a large class of these models, the EOFs at individual Fourier frequencies coincide with the orthogonal mechanical modes of the system-provided they exist. The precise mathematical criteria for this coincidence are derived and a physical interpretation is provided. A scheme possibly useful in forecasting is formally constructed for representing any stochastic field by a linear Hermitian model forced by noise.

## Abstract

An attempt to provide physical insight into the empirical orthogonal function (EOF) representation of data fields by the study of fields generated by linear stochastic models is presented in this paper. In a large class of these models, the EOFs at individual Fourier frequencies coincide with the orthogonal mechanical modes of the system-provided they exist. The precise mathematical criteria for this coincidence are derived and a physical interpretation is provided. A scheme possibly useful in forecasting is formally constructed for representing any stochastic field by a linear Hermitian model forced by noise.

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## Abstract

A class of mean annual, zonally averaged energy-balance climate models of the Budyko-Sellers type are studied by a spectral (expansion in Legendre polynomials) method. Models with constant thermal diffusion coefficient can be solved exactly, The solution is approached by a rapidly converging sequence with each succeeding approximant taking into account information from ever smaller space and time scales. The first two modes represent a good approximation to the exact solution as well as to the present climate. The two-mode approximation to a number of more general models are shown to be either formally or approximately equivalent to the same truncation in the constant diffusion case. In particular, the transport parameterization used by Budyko is precisely equivalent to the two-mode truncation of thermal diffusion. Details of the dynamics do not influence the first two modes which fortunately seem adequate for the study of global climate change. Estimated ice age temperatures and ice line latitude agree well with the model if the solar constant is reduced by 1.3%.

## Abstract

A class of mean annual, zonally averaged energy-balance climate models of the Budyko-Sellers type are studied by a spectral (expansion in Legendre polynomials) method. Models with constant thermal diffusion coefficient can be solved exactly, The solution is approached by a rapidly converging sequence with each succeeding approximant taking into account information from ever smaller space and time scales. The first two modes represent a good approximation to the exact solution as well as to the present climate. The two-mode approximation to a number of more general models are shown to be either formally or approximately equivalent to the same truncation in the constant diffusion case. In particular, the transport parameterization used by Budyko is precisely equivalent to the two-mode truncation of thermal diffusion. Details of the dynamics do not influence the first two modes which fortunately seem adequate for the study of global climate change. Estimated ice age temperatures and ice line latitude agree well with the model if the solar constant is reduced by 1.3%.

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## Abstract

A simple radiative balance climate model is presented which includes the ice feedback mechanism, zonal averaging, constant homogeneous cloudiness, and ordinary diffusive thermal heat transfer. The simplest version of the model with only one free parameter is solved explicitly in terms of hypergeometric functions and is used to study ice sheet latitude as a function of solar constant. A multiple branch structure of this function is found and discussed along with comparison to earlier results. A stability analysis about the equilibrium solutions shows that the present climate as well as an ice-covered earth are stable while an intermediate solution is unstable for small perturbations away from equilibrium.

## Abstract

A simple radiative balance climate model is presented which includes the ice feedback mechanism, zonal averaging, constant homogeneous cloudiness, and ordinary diffusive thermal heat transfer. The simplest version of the model with only one free parameter is solved explicitly in terms of hypergeometric functions and is used to study ice sheet latitude as a function of solar constant. A multiple branch structure of this function is found and discussed along with comparison to earlier results. A stability analysis about the equilibrium solutions shows that the present climate as well as an ice-covered earth are stable while an intermediate solution is unstable for small perturbations away from equilibrium.

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## Abstract

We Present a simple Budyko-Sellers type climate model which is forced by a heating term whose time dependence is white noise and whose space-separated autocorrelation is independent of position and orientation on the sphere (statistical homogeneity). Such models with diffusive transport are analytically soluble by expansion into spherical harmonies. The modes are dynamically and statistically independent. Each satisfies a simple Langevin equation having a scale-dependent characteristic time. Climate anomalies in these models have an interval of predictability which can be explicitly computed. The predictability interval is independent of the wavenumber spectrum of the forcing in this class of models. We present the predictability results for all scales and discuss the implications for more realistic models.

## Abstract

We Present a simple Budyko-Sellers type climate model which is forced by a heating term whose time dependence is white noise and whose space-separated autocorrelation is independent of position and orientation on the sphere (statistical homogeneity). Such models with diffusive transport are analytically soluble by expansion into spherical harmonies. The modes are dynamically and statistically independent. Each satisfies a simple Langevin equation having a scale-dependent characteristic time. Climate anomalies in these models have an interval of predictability which can be explicitly computed. The predictability interval is independent of the wavenumber spectrum of the forcing in this class of models. We present the predictability results for all scales and discuss the implications for more realistic models.

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## Abstract

This paper treats the stability of steady-state solutions of some simple, latitude-dependent, energy-balance climate models. For north-south symmetric solutions of models with an ice-cap-type albedo feed-back, and for the sum of horizontal transport and infrared radiation given by a linear operator, it is possible to prove a “slope-stability” theorem; i.e., if the local slope of the steady-state icelinc latitude versus solar constant curve is positive (negative) the steady-state solution is stable (unstable). Certain rather weak restrictions on the albedo function and on the heat transport are required for the proof, and their physical basis is discussed in the text.

## Abstract

This paper treats the stability of steady-state solutions of some simple, latitude-dependent, energy-balance climate models. For north-south symmetric solutions of models with an ice-cap-type albedo feed-back, and for the sum of horizontal transport and infrared radiation given by a linear operator, it is possible to prove a “slope-stability” theorem; i.e., if the local slope of the steady-state icelinc latitude versus solar constant curve is positive (negative) the steady-state solution is stable (unstable). Certain rather weak restrictions on the albedo function and on the heat transport are required for the proof, and their physical basis is discussed in the text.

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## Abstract

While approximate cyclostationary processes are commonly found in climatic and geophysical studies, one great disincentive for using cyclostationary empirical orthogonal functions is their computational burden. This is especially so for the three-dimensional, space–time case. This paper discusses a simple method of computing approximate cyclostationary empirical orthogonal functions based on the theory of harmonizable cyclostationary processes. The new method is computationally much more efficient than that of Kim et al. In the new method, cyclostationary empirical orthogonal functions are easier to understand. Namely, they are naturally defined as the products of Bloch functions (inner modes) and Fourier functions (outer modes), which otherwise are the result of the factorization theorem. Bloch functions are simply the principal components (PC) of the multivariate coefficient time series, which are generally correlated. They represent the normal modes of the nested fluctuations of harmonizable cyclostationary processes. Under the assumption of independent PC time series, Bloch functions are computed independently of the outer modes, which results in a tremendous speedup in computation.

## Abstract

While approximate cyclostationary processes are commonly found in climatic and geophysical studies, one great disincentive for using cyclostationary empirical orthogonal functions is their computational burden. This is especially so for the three-dimensional, space–time case. This paper discusses a simple method of computing approximate cyclostationary empirical orthogonal functions based on the theory of harmonizable cyclostationary processes. The new method is computationally much more efficient than that of Kim et al. In the new method, cyclostationary empirical orthogonal functions are easier to understand. Namely, they are naturally defined as the products of Bloch functions (inner modes) and Fourier functions (outer modes), which otherwise are the result of the factorization theorem. Bloch functions are simply the principal components (PC) of the multivariate coefficient time series, which are generally correlated. They represent the normal modes of the nested fluctuations of harmonizable cyclostationary processes. Under the assumption of independent PC time series, Bloch functions are computed independently of the outer modes, which results in a tremendous speedup in computation.

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## Abstract

Due to the variety of periodic or quasi-periodic deterministic forcings (e.g., diurnal cycle, seasonal cycle, Milankovitch cycles, etc.), most climate fluctuations may be modeled as cyclostationary processes since their properties are modulated by these cycles. Difficulties in using conventional spectral analysis to explore the seasonal variation of climate fluctuations have indicated the need for some new statistical techniques. It is suggested here that the cyclic spectral analysis he used for interpreting such fluctuations. The technique is adapted from cyclostationarity theory in signal processing. To demonstrate the usefulness of this technique, a very simple cyclostationarity stochastic climate model is constructed. The results show that the seasonal cycle strongly modulates the amplitude of the covariance and spectrum. The seasonal variation of intraseasonal oscillations in the Tropics has also been studied on a zonally symmetric all-land planet in the absence of external forcing. The idealized planet has no ocean no topography. A 15-year length seasonal run of the atmosphere is analyzed with the NCAR Community Climate Model (CCM2, R15). Analysis of the simulation data indicates the presence of intraseaonal oscillations in the Tropics, which are also localized in the time of year.

Both examples suggest that these techniques might be useful for analysis of fluctuations that exhibit locality in both frequency and season.

## Abstract

Due to the variety of periodic or quasi-periodic deterministic forcings (e.g., diurnal cycle, seasonal cycle, Milankovitch cycles, etc.), most climate fluctuations may be modeled as cyclostationary processes since their properties are modulated by these cycles. Difficulties in using conventional spectral analysis to explore the seasonal variation of climate fluctuations have indicated the need for some new statistical techniques. It is suggested here that the cyclic spectral analysis he used for interpreting such fluctuations. The technique is adapted from cyclostationarity theory in signal processing. To demonstrate the usefulness of this technique, a very simple cyclostationarity stochastic climate model is constructed. The results show that the seasonal cycle strongly modulates the amplitude of the covariance and spectrum. The seasonal variation of intraseasonal oscillations in the Tropics has also been studied on a zonally symmetric all-land planet in the absence of external forcing. The idealized planet has no ocean no topography. A 15-year length seasonal run of the atmosphere is analyzed with the NCAR Community Climate Model (CCM2, R15). Analysis of the simulation data indicates the presence of intraseaonal oscillations in the Tropics, which are also localized in the time of year.

Both examples suggest that these techniques might be useful for analysis of fluctuations that exhibit locality in both frequency and season.

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## Abstract

Optimal space–time signal processing is used to infer the amplitude of the large-scale, near-surface temperature response to the, “11 year” solar cycle. The estimation procedure involves the following stops. 1) By correlating 14 years of monthly total solar irradiance measurements made by the *Nimbus-7* satellite and monthly Wolf sunspot numbers, a monthly solar irradiance forcing function is constructed for the years 1894–1993. 2) Using this forcing function, a space-time waveform of the climate response for the same 100 years is generated from an energy balance climate model. 3) The space-time covariance statistics in the frequency band (16.67 yr)^{−1}–(7.14 yr)^{−1} are calculated using control runs from two different coupled ocean-atmosphere global climate models. 4) Using the results from the last two stops, an optimal filter is constructed and applied to observed surface temperature data for the years 1894–1993. 5) An estimate of the ratio of the real climate response, contained in the observed data, and the model generated climate response from step 2 is given, as well as an estimate of its uncertainty. A number of consistency checks are presented, such as using data from different regions of the earth to calculate this ratio and using data lagged up to ±5 yr. Our findings allow us to reject the null hypothesis. that no response to the solar cycle is present in the data, at a confidence level of 97.4%.

## Abstract

Optimal space–time signal processing is used to infer the amplitude of the large-scale, near-surface temperature response to the, “11 year” solar cycle. The estimation procedure involves the following stops. 1) By correlating 14 years of monthly total solar irradiance measurements made by the *Nimbus-7* satellite and monthly Wolf sunspot numbers, a monthly solar irradiance forcing function is constructed for the years 1894–1993. 2) Using this forcing function, a space-time waveform of the climate response for the same 100 years is generated from an energy balance climate model. 3) The space-time covariance statistics in the frequency band (16.67 yr)^{−1}–(7.14 yr)^{−1} are calculated using control runs from two different coupled ocean-atmosphere global climate models. 4) Using the results from the last two stops, an optimal filter is constructed and applied to observed surface temperature data for the years 1894–1993. 5) An estimate of the ratio of the real climate response, contained in the observed data, and the model generated climate response from step 2 is given, as well as an estimate of its uncertainty. A number of consistency checks are presented, such as using data from different regions of the earth to calculate this ratio and using data lagged up to ±5 yr. Our findings allow us to reject the null hypothesis. that no response to the solar cycle is present in the data, at a confidence level of 97.4%.

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## Abstract

A simple Budyko-Sellers mean annual energy balance climate model with diffusive transport (North, 1975b) is extended to include a seasonal cycle. In the model the latitudinal distribution of the zonal average surface temperature is represented by a series of Legendre polynomials, while its time-dependence is represented by a Fourier sine-cosine series. The model has three parameters which are adjusted so that the observed amplitudes of the Northern Hemisphere's zonal mean surface temperature are recovered. In order to obtain the correct amplitude and phase of the surface temperature's seasonal oscillation, allowance must be made for the disparity between the thermal inertia of the atmosphere over continents and that of the ocean's mixed layer. Although the model parameters are adjusted to recover the surface temperature fields of the Northern Hemisphere, a test of the model's ability to produce the fields of the Southern Hemisphere indicates that the model responds properly to changes in boundary conditions.

The seasonal model is used to reveal how the annual mean climate and its sensitivity to changes in incident radiation differ from the predictions obtained with the corresponding mean annual model. Although the zonal temperatures obtained with the seasonal model are 1–3°C higher than those obtained with the mean annual model, the changes in the global average annual mean surface temperatures calculated with the two models are practically identical for a 1% decrease in solar constant. Furthermore, because the albedo changes in them are linked mainly to changes in surface temperature, both models respond in the same manner to changes in the incident solar radiation caused by changes in the earth's orbit. The distribution of the incident solar radiation in the models is shown to be insensitive to changes in the eccentricity and the longitude of perihelion and sensitive only to changes in the obliquity of the earth. For past orbital changes, both the seasonal and the mean annual model fail to produce glacial advances of the magnitude that are thought to have occurred.

## Abstract

A simple Budyko-Sellers mean annual energy balance climate model with diffusive transport (North, 1975b) is extended to include a seasonal cycle. In the model the latitudinal distribution of the zonal average surface temperature is represented by a series of Legendre polynomials, while its time-dependence is represented by a Fourier sine-cosine series. The model has three parameters which are adjusted so that the observed amplitudes of the Northern Hemisphere's zonal mean surface temperature are recovered. In order to obtain the correct amplitude and phase of the surface temperature's seasonal oscillation, allowance must be made for the disparity between the thermal inertia of the atmosphere over continents and that of the ocean's mixed layer. Although the model parameters are adjusted to recover the surface temperature fields of the Northern Hemisphere, a test of the model's ability to produce the fields of the Southern Hemisphere indicates that the model responds properly to changes in boundary conditions.

The seasonal model is used to reveal how the annual mean climate and its sensitivity to changes in incident radiation differ from the predictions obtained with the corresponding mean annual model. Although the zonal temperatures obtained with the seasonal model are 1–3°C higher than those obtained with the mean annual model, the changes in the global average annual mean surface temperatures calculated with the two models are practically identical for a 1% decrease in solar constant. Furthermore, because the albedo changes in them are linked mainly to changes in surface temperature, both models respond in the same manner to changes in the incident solar radiation caused by changes in the earth's orbit. The distribution of the incident solar radiation in the models is shown to be insensitive to changes in the eccentricity and the longitude of perihelion and sensitive only to changes in the obliquity of the earth. For past orbital changes, both the seasonal and the mean annual model fail to produce glacial advances of the magnitude that are thought to have occurred.