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Gerald R. North

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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Gerald R. North

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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Gerald R. North

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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Gerald R. North

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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Gerald R. North

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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.

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Gregory R. Markowski
and
Gerald R. North

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.

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Gerald R. North
and
Robert F. Cahalan

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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Gerald R. North
and
Shoichiro Nakamoto

Abstract

Space-time averages of rain rates are needed in several applications. Nevertheless, they are difficult to estimate because the methods invariably leave gaps in the measurements in space or time. A formalism is developed which makes use of the frequency-wavenumber spectrum of the rain field. The mean square error of the estimate is expressed as an integral over frequency and two-dimensional wavenumber of an integrand consisting of two factors, a design-dependent-filter multiplied by the space-time spectrum of the rain rate field. Such a formalism helps to separate the design issues from the peculiarities of rain rate random fields. Two cases are worked out in detail: a low orbiting satellite which takes cell-wide snapshots at discrete intervals and a network of raingages which are gappy in space but continuous in time.

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Mark J. Stevens
and
Gerald R. North

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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Jian-Ping Huang
and
Gerald R. North

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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