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In order to describe changes in ice volume in the cryosphere, a differential equation mathematical model is used in this paper. The dominating effects of freezing and thawing in the cryosphere enable simplification of the heat transport equations. This results in a mathematical model that can be solved exactly and is useful in investigating other climatic components, which in turn may be similarly analyzed for possible Global Circulation Model (GCM) validation. Data forms representing temperature and ice volume during the Pleistocene are available and can be directly compared with the exact solution of the simplified differential equation used in this paper. The model parameters may then be adjusted to approximate the effects of climate change; the adjusted model then run in reverse time, to develop an alternative history of ice volume of the cryosphere to be compared with the actual data interpretations previously published in the literature. In this fashion, an assessment may be made as to possible climate impacts in the cryosphere.

The use of Global Climate Models (or Global Circulation Models, GCMs) appears to be becoming a main stream course in the analysis of climate change effects in order to develop public policy, despite the mixed reactions documented in recent literature regarding the success of their use (e.g., [

The proposed conceptual model will directly relate estimates of temperature found in the literature (as derived from proxy data, such as the ratio dD; for example, see [

The heat transport effects that impact the problem domain include solar and internal heat within the planet, circulation of the atmosphere and oceans with corresponding interface and convective heat transport effects, among others. All of these components interface and integrate together and are represented by the term [Heat] . (The notation [Heat] includes all the components of heat transport that effect phase change in the cryosphere. The total heat budget can be found in [

Details regarding conceptual model parameters, data types used and corresponding proxy relationships are presented in [

Because normalized data are used, the forcing function, H(t), is linearly related to the selected temperature data proxy for the time domain of interest. That is, for some constants a_{1} and a_{2}, and assuming H(t) = a_{1} + a_{2}T(t), then when transformed into normalized N(0,1) form, both functions H(t) and T(t) plot identically. Consequently, one can work directly with the N(0,1) normalized form of a selected T(t) realization. Furthermore, the estimated temperature T(t) is typically given as a linear function of the proxy data, dD (for example, see [

Some papers suggest that ice volume estimates may be written as a linear function of d18O (e.g., [

The general solution to the conceptual model operates upon estimates of the magnitude of heat, H(t) , assumed to be a linear function of temperature. In turn, these are assumed to be a linear function of dD and produce an output of ice measure that is calibrated to estimates of ice volume. Ice volume is assumed to be a linear function of d18O (as stated previously, the above development can consider a nonlinear transformation of d18O, or other proxy data). Hence, the model solution is a mapping (i.e., a relationship between the two measured variables considered) from dD to d180.

The proxy data types considered in this paper include those listed in

In [

where

Proxy for Temperature | Proxy for Ice Volume |
---|---|

Term | Definition | Relationship to Climate Change |
---|---|---|

Ratio of the sampled hydrogen isotopic ratio compared to the V_{SMOW} standard isotopic ratio | Proxy for temperature | |

Isotopic ratio of ^{2}H versus ^{1}H in a sample | ||

Hydrogen isotopic composition of the Vienna Standard Mean Ocean Water (V_{SMOW}) | Standard as defined in 1968 by the International Atomic Energy Agency | |

Ratio of the sampled oxygen isotopic ratio compared to the V_{SMOW} standard isotopic ratio | Proxy for ice volume | |

Isotopic ratio of ^{18}O versus ^{17}O in a sample | ||

Oxygen isotopic composition of the Vienna Standard Mean Ocean Water (V_{SMOW}) | Standard as defined in 1968 by the International Atomic Energy Agency |

cryosphere; and

For details regarding the underpinnings of Equation (1), the reader is referred to [

The ratio

such as growing glaciers, or changing boundary lengths with respect to interface with ocean currents, among other effects.

It is straightforward to extend

Dividing

where the initial condition for

Multiplying by_{0} as a reference value for heat (see [

or

where G(t) is the total heat applied to the cryosphere between model times s = 0 and s = t (see [

Various fields of geoscience utilize the Cumulative Departure Model (CDM) when looking at long term trends in data where the data under examination (usually annual data) must be considered in the context of previous data influences (for example, groundwater accumulation with respect to annual rainfall). As such, the CDM method involves three steps: 1) calculate the mean for the complete data set; 2) subtract the mean from each data point to determine its relationship or departure from the mean (i.e., surplus or deficit); and 3) add the departures together to create a cumulative departure at each time step. In Equation (4), the second exponential term, G(t), indicates a long term trend of aggradation or degradation of

where

Carrying forward the formulation of Equation (5) (indicating no long term aggradation or degradation of

where

It is this formulation that is used to approximate climate change impacts upon our cryosphere based upon the above simplified mathematical model.

An opportunity is afforded, in assessing climate change impacts on the cryosphere, by using the simplified modeling equation formulation of Equations (6) through (8). Specifically, the key model D parameter of Equation (8) is a composite of five parameters which results in the variation in the diffusion parameters analogous to the variation in the components of the parameter cluster. For example, a climate change impact that has the effect of increasing the area averaged latent heat of fusion parameter would be represented by a change in the “r” parameter, which inversely impacts the D parameter. Other such influences follow accordingly.

To assess the impact from such parameter changes, the D parameter is varied by 5%, 10% and 25%. The model of (6) through (8) is then run in reverse time for 435,000 years. Plots of the revised volumetric ice contents versus time (expressed in years before present time) are shown in Figures 1-3.

From Figures 1-3, the simplified mathematical model formulation simulation results indicate that for changes in the D model parameter of 5% primarily shows more thawing of the cryosphere occurring during interglacials but

that recovery substantially occurs during the next freeze cycle. However, at a 25% variation in the D parameter, significant impacts are indicated in the cryosphere ice volume. Even a 10% variation in D indicates significant change in volumetric ice content predictions.

From Equation (8), a variation in D is achieved by variation of several heat budget parameters or a larger variation by only one. For example, changes in the climate due to increased retention of heat, such as contemplated in some climate change model scenarios by others, can be approximately modeled by a decrease in the model return heat parameter, k. From Equation (8), increasing k implies decreasing values of D. Figures 1-3 demonstrate this simplified model estimate of the effect of both increased and decreased values of the D parameter.

From the figures, even for a 5% parameter variation in D, the indicated minimum volumetric ice content is seen to be substantially lower when compared with respect to the recent experience of volumetric ice content. That is, compared to recent measures, which are occurring during a thaw cycle, the predicted ice quantities would be significantly less. However, the next freezing cycle recovers the ice volume substantially. In contrast, a 25% variation in D indicates a far more substantial impact for both the thawing and freezing cycles. It is noted that in all of these simulations, the entire freeze/thaw records of several hundred thousand years is simulated. Simulating only a portion of the record would indicate less modeling impacts.

In this paper, the simple heat budget model formulated in [

Acknowledgements are paid to the United States Military Academy, West Point, NY, Department of Mathematical Sciences, for their support to the authors during this research. Also acknowledged are the several individuals who have participated in particular tasks in developing this paper including but by no means limited to Rene Perez, Laura Hromadka, Michael Barton, T.V. Hromadka III, among others.