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- Author or Editor: Gerald R. North x
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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.
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.
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.
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%.
Abstract
It is natural that a book chapter honoring Joanne Simpson draw the connection between the two most important tropical meteorological observing programs in the history of meteorology: the Global Atmospheric Research Program Atlantic Tropical Experiment (GATE) and the Tropical Rainfal Measuring Mission (TRMM). Both programs were dominated by the influences of Joanne Simpson. When TRMM data are all in, these two grand experiments will have given us more information about the behavior of tropical convection and precipitation over the tropical oceans than all other tropical field campaigns combined. But some may not know how GATE data played a key role in demonstrating the feasibility of a mission like TRMM. This chapter will present a review of a number of studies that connect GATE precipitation data with TRMM, especially in the early planning stages.
Abstract
It is natural that a book chapter honoring Joanne Simpson draw the connection between the two most important tropical meteorological observing programs in the history of meteorology: the Global Atmospheric Research Program Atlantic Tropical Experiment (GATE) and the Tropical Rainfal Measuring Mission (TRMM). Both programs were dominated by the influences of Joanne Simpson. When TRMM data are all in, these two grand experiments will have given us more information about the behavior of tropical convection and precipitation over the tropical oceans than all other tropical field campaigns combined. But some may not know how GATE data played a key role in demonstrating the feasibility of a mission like TRMM. This chapter will present a review of a number of studies that connect GATE precipitation data with TRMM, especially in the early planning stages.
Abstract
Using a combination of statistical methods and monthly SST anomalies (SSTAs) from one or two ocean regions, relatively strong SSTA–precipitation relationships are found during much of the year in the United States: hindcast-bias-corrected correlation coefficients 0.2–0.4 and 0.3–0.6, on monthly and seasonal timescales, respectively. Improved rigor is central to these results: the most crucial procedure was a transform giving regression residuals meeting statistical validity requirements. Tests on 1994–99 out-of-sample data gave better results than expected: semiquantitative, mapped predictions, and quantitative, Heidke skills, are shown. Correlations are large enough to suggest that substantial skill can be obtained for one to several months' precipitation and climate forecasts using ocean circulation models, or statistical methods alone. Although this study was limited to the United States for simplicity, the methodology is intended as generally applicable. Previous work suggests that similar or better skills should be obtainable over much of earth's continental area. Ways likely to improve skills are noted.
Pacific SSTAs outside the Tropics showed substantial precipitation influence, but the main area of North Pacific variability, that along the subarctic front, did not. Instead, the east–west position of SSTAs appears important. The main variability is likely due to north–south changes in front position and will likely give PC analysis artifacts. SSTAs from some regions, the Gulf of Mexico in particular, gave very strong correlations over large U.S. areas. Tests indicated that they are likely caused by atmospheric forcing. Because unusually strong, they should be useful for testing coupled ocean–atmosphere GCMs. Investigation of differences between ENSO events noted by others showed that they are likely attributable to differing SSTA patterns.
Abstract
Using a combination of statistical methods and monthly SST anomalies (SSTAs) from one or two ocean regions, relatively strong SSTA–precipitation relationships are found during much of the year in the United States: hindcast-bias-corrected correlation coefficients 0.2–0.4 and 0.3–0.6, on monthly and seasonal timescales, respectively. Improved rigor is central to these results: the most crucial procedure was a transform giving regression residuals meeting statistical validity requirements. Tests on 1994–99 out-of-sample data gave better results than expected: semiquantitative, mapped predictions, and quantitative, Heidke skills, are shown. Correlations are large enough to suggest that substantial skill can be obtained for one to several months' precipitation and climate forecasts using ocean circulation models, or statistical methods alone. Although this study was limited to the United States for simplicity, the methodology is intended as generally applicable. Previous work suggests that similar or better skills should be obtainable over much of earth's continental area. Ways likely to improve skills are noted.
Pacific SSTAs outside the Tropics showed substantial precipitation influence, but the main area of North Pacific variability, that along the subarctic front, did not. Instead, the east–west position of SSTAs appears important. The main variability is likely due to north–south changes in front position and will likely give PC analysis artifacts. SSTAs from some regions, the Gulf of Mexico in particular, gave very strong correlations over large U.S. areas. Tests indicated that they are likely caused by atmospheric forcing. Because unusually strong, they should be useful for testing coupled ocean–atmosphere GCMs. Investigation of differences between ENSO events noted by others showed that they are likely attributable to differing SSTA patterns.
Abstract
Low earth-orbiting satellites such as the Tropical Rainfall Measuring Mission (TRMM) estimate month-long averages of precipitation (or other fields). A difficulty is that such a satellite sensor returns to the same spot on the planet at discrete intervals, about 11 or 12 h apart. This discrete sampling leads to a sampling error that is the one of the largest components of the error budget. Previous studies have examined this type of error for stationary random fields, but this paper examines the possibility that the field has a diurnally varying standard deviation, a property likely to occur in precipitation fields. This is a special case of the more general cyclostationary field.
In this paper the authors investigate the mean square error (mse) for the monthly averaging case derived from the satellites whose revisiting intervals are 12 h (sun synchronous) and off 12 h (11.75 h). In addition, the authors take the diurnal cycle of the standard deviation to be a constant plus a single sinusoid, either diurnal or semidiurnal.
The authors have derived an mse formula consisting of three parts: the errors from the stationary background, the cyclostationary part, and a cross-term between them. The separate parts of the mse allow the authors to assess the contribution of the cyclostationary error to the total mse.
The results indicate that the cyclostationary errors due to the diurnal variation appear small for both a 12-h and an off-12-h (11.75 h) revisiting satellite. In addition, the cyclostationary error amounts are similar to each other. For the semidiurnally varying field, the cyclostationary errors increase rapidly as the magnitude of the variance cycle increases for both the 12-h and off-12-h revisting satellites. However, the off-12-h sampling shows the cyclostationary error to be less than that of the exact 12-h sampling.
Furthermore, the authors have evaluated the cyclostationary error as a function of the phase of the satellite visit as it is shifted from the phase of the diurnal cycles (the sun-synchronous case or the start of the month for the off-12-h case). It is found that the cyclostationary error observed from the off-12-h satellite is much less sensitive to the phase shift than the cyclostationary error from the exact 12-h satellite.
Abstract
Low earth-orbiting satellites such as the Tropical Rainfall Measuring Mission (TRMM) estimate month-long averages of precipitation (or other fields). A difficulty is that such a satellite sensor returns to the same spot on the planet at discrete intervals, about 11 or 12 h apart. This discrete sampling leads to a sampling error that is the one of the largest components of the error budget. Previous studies have examined this type of error for stationary random fields, but this paper examines the possibility that the field has a diurnally varying standard deviation, a property likely to occur in precipitation fields. This is a special case of the more general cyclostationary field.
In this paper the authors investigate the mean square error (mse) for the monthly averaging case derived from the satellites whose revisiting intervals are 12 h (sun synchronous) and off 12 h (11.75 h). In addition, the authors take the diurnal cycle of the standard deviation to be a constant plus a single sinusoid, either diurnal or semidiurnal.
The authors have derived an mse formula consisting of three parts: the errors from the stationary background, the cyclostationary part, and a cross-term between them. The separate parts of the mse allow the authors to assess the contribution of the cyclostationary error to the total mse.
The results indicate that the cyclostationary errors due to the diurnal variation appear small for both a 12-h and an off-12-h (11.75 h) revisiting satellite. In addition, the cyclostationary error amounts are similar to each other. For the semidiurnally varying field, the cyclostationary errors increase rapidly as the magnitude of the variance cycle increases for both the 12-h and off-12-h revisting satellites. However, the off-12-h sampling shows the cyclostationary error to be less than that of the exact 12-h sampling.
Furthermore, the authors have evaluated the cyclostationary error as a function of the phase of the satellite visit as it is shifted from the phase of the diurnal cycles (the sun-synchronous case or the start of the month for the off-12-h case). It is found that the cyclostationary error observed from the off-12-h satellite is much less sensitive to the phase shift than the cyclostationary error from the exact 12-h satellite.
Abstract
In this paper point gauges are used in an analysis of hypothetical ground validation experiments for satellite-based estimates of precipitation rates. The ground and satellite measurements are fundamentally different since the gauge can sample continuously in time but at a discrete point, while the satellite samples an area average (typically 20 km across) but a snapshot in time. The design consists of comparing a sequence of pairs of measurements taken from the ground and from space. Since real rain has a large nonzero contribution at zero rain rate, the following ground truth designs are proposed: design 1 uses all pairs, design 2 uses the pairs only when the field-of-view satellite average has rain, and design 3 uses the pairs only when the gauge has rain. The error distribution of each design is derived theoretically for a Bernoulli spatial random field with different horizontal resolutions. It is found that design 3 cannot be used as a ground-truth design due to its large design bias. The mean-square error is used as an index of accuracy in estimating the ground measurement by satellite measurement. It is shown that there is a relationship between the mean-square error of design 1 and design 2 for the Bernoulli random field. Using this technique, the authors derive the number of satellite overpasses necessary to detect a satellite retrieval bias, which is as large as 10% of the natural variability.
Abstract
In this paper point gauges are used in an analysis of hypothetical ground validation experiments for satellite-based estimates of precipitation rates. The ground and satellite measurements are fundamentally different since the gauge can sample continuously in time but at a discrete point, while the satellite samples an area average (typically 20 km across) but a snapshot in time. The design consists of comparing a sequence of pairs of measurements taken from the ground and from space. Since real rain has a large nonzero contribution at zero rain rate, the following ground truth designs are proposed: design 1 uses all pairs, design 2 uses the pairs only when the field-of-view satellite average has rain, and design 3 uses the pairs only when the gauge has rain. The error distribution of each design is derived theoretically for a Bernoulli spatial random field with different horizontal resolutions. It is found that design 3 cannot be used as a ground-truth design due to its large design bias. The mean-square error is used as an index of accuracy in estimating the ground measurement by satellite measurement. It is shown that there is a relationship between the mean-square error of design 1 and design 2 for the Bernoulli random field. Using this technique, the authors derive the number of satellite overpasses necessary to detect a satellite retrieval bias, which is as large as 10% of the natural variability.
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.
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%.
