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The interpretation of secular Caspian Sea level records during the Holocene
Quaternary International xxx (2015) 1e5
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The interpretation of secular Caspian Sea level records during the Holocene A.V. Kislov Lomonosov Moscow State University, Faculty of Geography, Department of Meteorology and Climatology, Lenin Gory 1, 119991 Moscow, Russia
a r t i c l e i n f o
a b s t r a c t
Article history: Available online xxx
The Caspian Sea (CS) experienced significant changes during the Holocene. The standard deviation for Caspian Sea level (CSL) variations over that interval is estimated as s ¼ 1.4 m. Based on well-established views, they were climate-induced variations. There are no clear links with the calendar of climatic anomalies, and climate models do not reproduce the changes. Therefore, the question about the origin of “secular” CSL fluctuations remains open. Based on general ideas about the laws of temporal dynamics relating to massive inertial objects, the observed slow changes of the CSL under the semi-steady climate state of the Holocene can be represented as resulting from the accumulation of small anomalies in the water regime, as a kind of “self-developing” system. To test this hypothesis, the model of the water balance of the CS was used. Time scale for the sea fluctuations was estimated as ~20 years. This model is interpreted as stochastic, and from this perspective, it is a Langevin equation that incorporates the action of precipitation and evaporation as random white noise, so that the whole can be thought of as an analogue of Brownian motion. Under these conditions, the CS is represented by a system undergoing random walk. Modeling results are interpreted from the probabilistic point of view, although the model is deterministically based on the physical law of conservation of water mass. The results showed that the CSL fluctuations under steady state conditions are characterized by s ¼ 1.1 m, close to the empirical value. “Super-large” anomalies in CSL are not prohibited by the theory, but their development requires a correspondingly long time. However, during long periods of time, background conditions change, and uniformity of the Brownian process becomes disrupted. The origin of large transgressive/regressive stages can be different. For example, the low stand of the Enotaevkian Regression during the LGM was determined by a significant reduction in precipitation over the Volga River catchment and by a corresponding reduction in the volume of river runoff. Hence, based on modeling results, the possibility of “self-development” effects is not prohibited by the theory: there need not be any cause for specific level changes or shifts, merely the expected behavior of red noise processes. © 2015 Elsevier Ltd and INQUA. All rights reserved.
Keywords: Caspian Sea Regional climate change Paleodata Model of random walk
1. Introduction The Caspian Sea (CS) (36e47 N, 47e54 E) is a closed basin. Its sea level lies below the mean sea level of the ocean and has varied between 25 and 29 m in the last ~100 years of recorded history. Water level fluctuations have occurred 100 times faster in comparison to global sea level changes over the last century. The main water source is the Volga River (a total of 80% of river inflow to the CS comes from the Volga River), whose catchment area covers a large part of the Eastern European Plain (EEP). The water inflow is offset by evaporation over the CS. Other rivers, such
E-mail address: [email protected].
as the Ural, Kura, Terek, and Sefidrud, and the subsurface runoff into the sea have to be considered as well, but their contributions are significantly less, uncoordinated and irregular. The water budget of the CS and the current Caspian Sea Level (CSL) variability have been investigated in many studies, e.g., Golitsyn and Panin (1989), Rodionov (1994), Golitsyn et al. (1998), Arpe et al. (2000, 2012) and Arpe and Leroy (2007). Precise reconstructions and dating have demonstrated that during the Holocene, the CS fluctuated between regressive and transgressive stages (Rychagov, 1997). The variations were climateinduced. These significant secular-scale changes took place although only subtle changes were observed in external climate forcings (solar insolation change has been slow and gradual, and the amplitude of CO2 variation was small). Although the CS lies in
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Please cite this article in press as: Kislov, A.V., The interpretation of secular Caspian Sea level records during the Holocene, Quaternary International (2015), http://dx.doi.org/10.1016/j.quaint.2015.07.026
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A.V. Kislov / Quaternary International xxx (2015) 1e5
an area of tectonic activity (Allen et al., 2004), tectonic impacts on sea-level changes do not have to be taken into account for the postLast Glacial Maximum (LGM) timeframe. No tectonic deformations of the Holocene shorelines have been detected, and only lowdegree deformations of the Khvalynian (the end of the Late Pleistocene) shorelines have been detected (Rychagov, 1997). The question of the origin of the fluctuations remains open because there are no clear stable links of the CSL anomalies and palaeohydrological phases with the calendar of climatic anomalies. Correspondence of river runoff changes (on the bases of fluvial activity) to the Caspian Sea level changes was unstable during the Holocene: it was rather high in the second part of the Holocene and was poor before 4e5 ka BP (Panin and Matlakhova, 2015). Climate models (CMIP3 and CMIP5) do not reproduce the needed changes of precipitation, evaporation, and river runoff (Kislov et al., 2014). It is hypothesized that observed anomalies of the CSL, under assumption of a quasi-steady state Holocene climate, are derived from accumulation of small water budget anomalies of opposite signs. Their residual effect forms the CSL response, much like a random walk. Although this assumption cannot be precisely proven, it is possible to determine whether it contradicts the results of observations.
2. Caspian Sea level changes: probability distribution function and Langevin equation The Holocene (11.7 ka) was a stage of Earth history in which the global climate and the environment were characterized by a quasisteady state (compared with the post-LGM timeframe) with gradual long-term trends. Despite this relative stability, the CSL has experienced significant secular-scale and decadal-scale changes during the Holocene (Fig. 1). At each hierarchical level, the dynamic of the CS is characterized by complex oscillations, and its amplitude increases with decreasing frequency. The probability distribution function (pdf) of the CSL variations, calculated on the basis of instrumental records (~100 years), is a bimodal curve (Naidenov, 1992). However, it is reasonable that this assessment is incorrect because there is strong correlation seen in annually averaged data of the CSL. At 1 year time lag, the
Fig. 1. The Caspian Sea level changes during the Holocene (Based on: a e (Rychagov, 1997); b e (Hydrometeorology and Hydrochemistry of the Seas, 1992); c e http:// www.oceanography.ru/index.php/ru/2010-03-15-15-57-22/2010-03-15-15-59-06).
autocorrelation coefficient r1 ~0.95, and autocorrelation disappears only after ~20 years, i.e., r20 ~0 (Nikolaenko, 1997). This means that the 100-year series is equivalent to approximately five independent values (see Fortus (1998) for strong statistical conclusion). Hence, this amount of data is too small for statistical calculation (e.g., for accurate calculation of the pdf). Taking this fact into account, the last 2000-year series, consisting of a sequence of 20-year averaged quantities of reconstructed CSL data (Fig. 1b), was used to calculate the pdf. Of course, these data are not as accurate as measurements, but they are statistically independent variables. They characterize the climate of the Sub-Atlantic stage, during which the environmentaleclimatic regime did not experience significant changes. As for long-term changes taking place in the Holocene, this stage logically is named as “current climate”. It was shown (Kislov, 2011) that the pdf of the CSL can be represented by a Gaussian curve (statistical significance is 0.02 using the chi-squared test statistic); the mean CSL is 28 m, the standard deviation is 1.4 m. Using a smooth curve (Fig. 1a) for the same time interval, we can calculate (using connection between amplitude of harmonic function and its variance) that the standard deviation is 1.6 m. However, Kroonenberg et al. (2008) reconstructed the drop of CSL to 42 m a.s.l. during the Derbentian Lowstand (~1000 years ago). Use of such data leads to standard deviation increases up to 5 m. The main components of the CS water budget are the Volga River runoff and the evaporation minus precipitation (e) over the CS water surface. Analysis of time behaviour has shown that the variance of the Volga River runoff is substantially larger than the variance of e. Additionally, e fluctuates irregularly, whereas the Volga River runoff changes are characterized by long-term trends (Golitsyn and Panin, 1989). Decadal-scale Volga River runoff oscillations (reflecting the influence of precipitation and evaporation over the catchment, and soil water storage) are governed by atmospheric circulation changes. This has been demonstrated from data showing a well-established connection between the CSL changes and variations of circulation indices (dry-wet variations), as well as variations of the North Atlantic Oscillation index, the Wangenheim index, and the Southern Oscillation Index (Isaev et al., 1995; Arpe et al., 2000; Nesterov, 2001; Tugilkin et al., 2011). Explanation of the origin of the secular-scale oscillations is difficult because there is no clear link between CSL changes and the calendar of climate anomalies despite the widely accepted theory that the changes were climatically induced (Kislov, 2001; Bolichovskaya, 2011; Svitoch, 2011; Sidorchuk et al., 2012; Panin and Matlakhova, 2015). Furthermore, climate models (in the framework of CMIP3 and CMIP5) do not simulate the required secular-scale changes of precipitation and river runoff volume (Kislov et al., 2014). However, the origin of long-lasting CSL anomalies can be interpreted another way. The theory of Brownian motion argues that the multi-scale stochastic dynamic of a system is formed through interaction of its fast and slow components. Positive and negative fast anomalies do not cancel each other out and their residual effects accumulate slowly to form a large deviation from the initial state. However, negative feedbacks, which usually exist in the system, prohibit large deviations, and the steady state regime of slow chaotic oscillations is eventually realized. Hasselmann (1976) used this mechanism to describe and reproduce the stochastic behaviour of several geophysical processes. The random time evolution of a Brownian particle position (onedimensional) h ¼ h(t) approximately satisfies the Langevin equation:
dh ¼ lhðtÞ þ hðtÞ dt
(1)
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A.V. Kislov / Quaternary International xxx (2015) 1e5
Here, l1 is characteristic time of the system. The force h(t) has a Gaussian probability distribution and its autocorrelation function is approximated by:
jx zj ; 〈hðxÞhðzÞ〉 ¼ s2g exp th
(2)
where x and z are different moments and h(x)h(z) denotes ensemble average value. The form of the autocorrelation curve is assumed to be sufficiently acute to be approximated by the dfunction curve. Consistent with the property of the physical problem, th ≪ l1. The time series of h ¼ h(t) is expressed by the OrnsteineUhlenbeck process. I a variable initially is Gaussian distributed, then the solution of equation (1) (taking into account (2)) is Gaussian distributed. The solution of the task depicted by equations (1) and (2) is discussed below. To simulate a stochastic process h ¼ h(t), the parameters l1 and th should be chosen. This can be done using two methods. The first method, the standard statistical method, allows us to evaluate the parameters based on observations. In the second approach, the deterministic equations reflecting conservation laws could be used for estimation of parameters of the equivalent stochastic model. The latter method is much more reliable because it allows estimation of parameters under climate change conditions. This approach will be used for investigation of the CSL variations.
3 1 Using vþ for modern 0 ¼ 275 km /y, we can calculate l ¼ 0.05 y conditions; therefore, the characteristic time scale is l1 ~20 y. This value coincides with the assessment of the time lag required for the disappearance of autocorrelation. The forcing function h ¼ h(t) includes short-scale variations due to both river runoff changes (mainly Volga River contribution) and e(t). The one-year period of averaging is much less than l1. Hence, the CS system consists of two substantially different time scales, and behaviour of the CSL changes is governed by the interaction between fast and slow components. Their interaction can be described in terms of Brownian motion, assuming that the slow components are adapted to the total statistical stresses created by the fast component. In this approach, it is assumed that clarification of h ¼ h(t) is not required. This expression is replaced by a random function with known properties. It was shown (Frolov, 1985) that the autocorrelation of the series h ¼ h(t) is a small, but statistically significant value under the shift to one year. However, assuming that equation (4) was averaged over two years, it is plausible to express h ¼ h(t) as white noise. Therefore, let us consider when its autocorrelation function is described by Eq. (2), where th ¼ 2 years. In this case, th ≪ l1 and the mathematical theory of the Brownian motion expressed by equations (1) and (2) can be used. The solution of Eqs. (1) and (2) is a stochastic process, and its variance is given by:
s2h ¼ 3. Adaptation of Brownian motion to explain CSL changes during the Holocene The CS integrates annual variations of both river runoff and e. Accumulated anomalies form residual slow changes of CSL. Therefore, the mathematical apparatus of the theory of Brownian motion can be used for the interpretation of the CSL changes. To demonstrate the feasibility of this mechanism let us develop the stochastic model. This model should be based on the deterministic equation of the CS water budget. An inspection of the water balance equation for any lake can provide some clues:
dVðtÞ ¼ vþ ðtÞ f ðtÞeðtÞ v ðtÞ dt
(3)
This equation describes time-dependent water storage, taking into account the shape of bottom of the lake. The f is lake area. The Z hðtÞ f dz; , vþ and v are contribuvolume of water: VðtÞ ¼ V* þ h*
tions from river runoff and water outflow. The linear approximation (consider a small range of the CS volume change) is used: f ¼ a þ bh (a ¼ 366 000 km2, b ¼ 14 000 km2/m, when reference level of the CS is 28.5 m), assuming that (a þ bh)1 z 1 hb/a. For the water outflow from the CS to the Kara Bogaz Gol (a closed bay on the eastern shore of the CS) the hydraulic formula is: v ¼ v 0 þ ahðtÞ, 3 v 0 ¼ 8:8 km =y is the currently estimated time-averaged value and a ¼ 8.6 km3/y. Thus, the new form of the budget equation is (Frolov, 1985):
dh bvþ ¼ 2 hðtÞ dt a
! vþ ðtÞ v a v 0b 0 2 hðtÞ þ eðtÞ a a a
(4)
The annual averaging of each components of the equation allows us to focus on inter-annual CSL variability. It was shown (Frolov, 1985), that ðvþ hÞ vþ h. Therefore, for annually averaged values, we have an equation describing the CSL changes identical in 2 form to the Eq. (1) where l ¼ bvþ =a2 þ ða=a v 0 b=a Þ and hðtÞ ¼ vþ ðtÞ v =a eðtÞ. 0
3
th s2h 1 e2lt l
(5)
Its spectrum is given by: Sh ðuÞf1=u2 þ l2 . Hence, at high frequencies (u [ l), the spectral shape displays a red-noise continuum (Sh(u) f u2). It corresponds to situation t ≪ l1, whereby the expression (5) is reduced to:
s2h ¼ 2tr s2h t
(6) 1
On the other hand, when t [ l , the variance of CSL variations is defined as:
s2h;st ¼
th s2h l
(7)
where s2h;st is constant, characterizing the steady state conditions. The corresponding spectral curve displays a white-noise continuum (Sh(u)fl2 ¼ const). The CSL variability can be calculated. Application of formulas (5) or (6) requires information about the initial unperturbed state of the CS. However, this information is not available because it is assumed that the CS is always in a perturbed state. Therefore, it is justified to apply an expression (7) estimating the upper limit. Parameters l and th were estimated above. The variance, s2h , is determined by taking into account the variance of river runoff volume and e, which were estimated (only modern observation data can be used) as 0.026 and 0.007 (m/y)2 (Golitsyn et al., 1998), correspondingly. Due to their practical independence (Frolov, 2011), s2h ¼ 0:033 ðm=yÞ2 . Using these values, we obtain using (7) sh,st ¼ 1.15 m. It is only slightly smaller than the empirical estimation (1.4e1.6 m; less compared to Kroonenberg et al. (2008)). Taking into account the first empirical values, theoretical estimation could arise because sh,st calculated on the basis of modern short series has to be underestimated. The closeness of the calculated theoretical sh,st to the empirical value could indicate that the Brownian approach is correct. In order to satisfy the empirical estimation of Kroonenberg et al. (2008), s2h has to be 0.625 (m/y)2. This seems unlikely from the most general considerations, according to which the environmentaleclimatic regime did not experience significant
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changes during the Sub-Atlantic stage. The results of Kroonenberg et al. (2008) cannot be interpreted from the theoretical approach discussed here. The result demonstrates that the statistical properties of the observed time series of the CSL could be reproduced by Gaussian noise with the same parameters. Taking into account the properties of the Gaussian process, the probability of the time (t*) required to achieve a prescribed level (h ¼ a) is given by the formula (Feller, 1968; Crownover, 1995):
deep regression states of the Caspian and Black Seas and mature stages of the late Quaternary glacial events. An example of anomalies of the opposite sign can also be mentioned: the anomalous rise of runoff in the Volga catchment in the Late Glacial time (Sidorchuk et al., 2009), which correlates to one of the phases of the Khvalynian transgression in the Caspian Sea.
rffiffiffiffi 2 a 3=2 a2 t* pft* g ¼ exp 2 p ε 2ε t*
Large transgressive/regressive stages of the CS during the Holocene could have occurred without significant changes in external forcings. Based on general theories about the laws of temporal dynamics relating to massive inertial objects, the observed slow changes of the CSL under the quasi-steady climate state of the Holocene could result from the accumulation of small anomalies in the water regime. The model of the CSL is interpreted as stochastic; from this perspective, it is a Langevin equation that incorporates the action of precipitation and evaporation as random white noise. The process is analogous to Brownian motion. Under these conditions, the Caspian Sea is represented by a system undergoing random walk. Modelling results are interpreted from the probabilistic viewpoint, although the model is deterministically based on the physical law of conservation of water mass. This mechanism could be responsible for the appearance of decadal or secular-scale, large, irregular fluctuations of the CSL that do not correlate with known climatic events. However, it should be considered only as one of the possible mechanisms responsible for the generation of a wide range of the CSL oscillation. Sometimes a clear correlation were observed, for example during last 2 ka between long-scale CSL variations and changes of river runoff (Panin and Matlakhova, 2015) or between the CSL variations and longterm temperature anomalies over the central EEP established for the Little Ice Age and Medieval Climate Anomaly (Kislov, 2001). It is not clear whether they coincide contingently, or these a nonstochastic causal relationship to changes in regional atmospheric circulation. Thus, the increase in the CSL during the LIA was explained by increased penetration of Mediterranean cyclones on
(8)
where ε2 ≡2ts2h ¼ 0:132 m2 =y. Fig. 2 shows these histograms for several prescribed anomalies. The maximum of the curves characterizing the mostly probable transition time is distinctly expressed only in the case of small deviations, i.e., in the case when anomaly a can be achieved in a small number of time-steps. Increase of the prescribed deviation leads to transformation of the histogram: the whole curve shifts to the right along the axis (realization of large anomalies requires much more time), the peak is stretched and the one-modal curve is drastically flattened. Therefore, a large anomaly is a rare and irregular event, and the concepts of most-probable transition time or average transition time lose their meaning. Consider the possibility of the appearance of a large anomaly. According to the principle of probability theory, it is not forbidden, but it is an extremely rare event. In reality, the appearance of such large anomalies is likely influenced by other causes. During the period of time needed for the appearance of this type of anomaly, background climate conditions could change and the Brownian process of accumulation of anomalies (as a holistic process) would be destroyed. Sometimes, a reason for the appearance of large anomalies in the CS may be indicated. For example, in response to glacial conditions of the LGM, the lowering levels of the Black Sea and the Caspian Sea are simulated simultaneously (Kislov and Toropov, 2011), validating the theory of a connection between
4. Conclusion
Fig. 2. Examples of probability distribution of the time required to achieve a prescribed level (h ¼ 0, 5, 1, 3 and 4 m).
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the EEP during winter, much more intensive during cold climate anomalies, providing snow supply in the catchments of the Volga River and the Kama River and consequently CSL rise (Kislov, 2001). This appears to be a direct causal mechanism. Probably, in reality non-stochastic and red noises hydrological changes worked simultaneously (or in combination). References Allen, A., Jackson, J., Walker, R., 2004. Late Cenozoic reorganisation of the ArabiaEurasia collision and the comparison of the short-term and long-term deformation rates. Tectonics 23. TC 2008. Arpe, K., Bengtsson, L., Golitsin, G., 2000. Connection between Caspian Sea level variability and ENSO. Geophysical Research Letters 27, 2693e2696. Arpe, K., Leroy, S., 2007. The Caspian Sea Level forced by the atmospheric circulation, as observed and modeled. Quaternary International 173e174, 144e152. Arpe, K., Leroy, S., Lahijani, H., Khan, V., 2012. Impact of the European Russia drought in 2010 on the Caspian Sea level. Hydrology and Earth Systems Sciences 16, 19e27. Bolichovskaya, N.S., 2011. Time evolution of climate and landscapes of Volga River region during the Holocene. Bulletin of Moscow State University, Series 5 1, 70e77 (in Russian). Crownover, R.M., 1995. Introduction to Fractals and Chaos. Jones and Bartlett, Boston, p. 252. Feller, W., 1968. An Introduction to Probability Theory and its Applications. Wiley, Boston. Fortus, M.I., 1998. About equivalent number of independent measurements. Izvestiya, Atmospheric and Oceanic Physics 34, 591e599 (English translation). Frolov, A.V., 1985. Dynamic-stochastical Models of Lake Levels. Nauka, p. 104 (in Russian). Frolov, A.V., 2011. Discrete dynamic-stochastic model of long-term river runoff variations. Water Resources 38, 583e592. Golitsyn, G.S., Panin, G.N., 1989. The water balance and modern variations of the level of the Caspian Sea. Soviet Meteorology and Hydrology 1, 46e52 (English translation). Golitsyn, G.S., Ratkovich, D., Fortus, M., Frolov, A.V., 1998. Current rise of the Caspian sea level. Water Resources 25, 133e139 (English translation). Hasselmann, K., 1976. Stochastic climate models. Tellus 28, 473e485. Hydrometeorology and Hydrochemistry of the Seas, 1992. The Caspian Sea. V.1. Hydrometeoizdat, St. Petersburg, p. 359 (in Russian). Isaev, A.A., Klimenko, L.V., Jiltsova, O.V., 1995. Frequency of “rainy” and “dry” synoptic processes over the Volga River Catchment and the Caspian Sea water
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budget during its different stages. Bulletin of Moscow State University, Series 5 1, 70e77 (in Russian). Kislov, A., 2001. Climate in Past, Current and Future. Nauka-Interperiodika, Moscow, p. 326 c (in Russian). Kislov, A., 2011. On multiscale the Caspian Sea level variability. Bulletin of Moscow State University, Series 5 2, 49e53 (in Russian). Kislov, A., Panin, A.V., Toropov, P., 2014. Current status and palaeostages of the Caspian Sea as a potential evaluation tool for climate model simulations. Quaternary International. http://dx.doi.org/10.1016/j.quaint.2014.05.014. Kislov, A., Toropov, P., 2011. Modeling extreme Black Sea and Caspian Sea levels of the past 21,000 years with general circulation models. In: Buynevich, I.V., Yanko-Hombach, V., Gilbert, A.S., Martin, R.E. (Eds.), Geology and Geoarchaeology of the Black 15 Sea Region: beyond the Flood Hypothesis, Geological Society of America Special Paper, vol. 473, pp. 27e32. Kroonenberg, S.B., Kasimov, N.S., Lychagin, M.Yu, 2008. The Caspian Sea: a natural laboratory for sea-level change. Geography and Environmental Sustainability 1, 22e37. Naidenov, V.I., 1992. Nonlinear model of the Caspian Sea level oscillation. Mathematical Modelling 4, 50e64 (in Russian). Nesterov, E.S., 2001. Low frequency variability of the atmospheric circulation and the Caspian Sea level during second part of XX century. Russian Meteorology and Hydrology 11, 27e36 (English translation). Nikolaenko, A.V., 1997. About long-time variations of the Caspian sea level. Water Resources 24, 261e265 (English translation). Panin, A., Matlakhova, E., 2015. Fluvial chronology in the East European Plain over the last 20 ka and its palaeohydrological implications. Catena 130, 46e61. Rodionov, S.N., 1994. Global and Regional Climate Interaction: the Caspian Sea Experience. Kluwer Academic Publications, Dordrecht, p. 241. Rychagov, G.I., 1997. Holocene oscillations of the Caspian Sea, and forecasts based on palaeogeographical reconstructions. Quaternary International 41/42, 167e172. Sidorchuk, A., Panin, A., Borisova, O., 2009. Morphology of river channels and surface runoff in the Volga River basin (East European Plain) during the Late Glacial period. Geomorphology 113, 137e157. Sidorchuk, A., Panin, A.V., Borisova, O., 2012. Decreasing of volume of river runoff over the North Eurasia at Holocene Optimum. Water Resources 39, 40e53 (English translation). Svitoch, A., 2011. Holocene history of the Caspian Sea and other seas of the European Russia. Bulletin of Moscow State University, Series 5 2, 28e37 (in Russian). Tugilkin, V.S., Kosartev, A.N., Archipkin, V.S., Nikonova, R.E., 2011. Long-term variability of the Caspian Sea hydrological regime together with climate change. Bulletin of Moscow State University, Series 5 2, 62e71 (in Russian).
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Quaternary International journal homepage: www.elsevier.com/locate/quaint
The interpretation of secular Caspian Sea level records during the Holocene A.V. Kislov Lomonosov Moscow State University, Faculty of Geography, Department of Meteorology and Climatology, Lenin Gory 1, 119991 Moscow, Russia
a r t i c l e i n f o
a b s t r a c t
Article history: Available online xxx
The Caspian Sea (CS) experienced significant changes during the Holocene. The standard deviation for Caspian Sea level (CSL) variations over that interval is estimated as s ¼ 1.4 m. Based on well-established views, they were climate-induced variations. There are no clear links with the calendar of climatic anomalies, and climate models do not reproduce the changes. Therefore, the question about the origin of “secular” CSL fluctuations remains open. Based on general ideas about the laws of temporal dynamics relating to massive inertial objects, the observed slow changes of the CSL under the semi-steady climate state of the Holocene can be represented as resulting from the accumulation of small anomalies in the water regime, as a kind of “self-developing” system. To test this hypothesis, the model of the water balance of the CS was used. Time scale for the sea fluctuations was estimated as ~20 years. This model is interpreted as stochastic, and from this perspective, it is a Langevin equation that incorporates the action of precipitation and evaporation as random white noise, so that the whole can be thought of as an analogue of Brownian motion. Under these conditions, the CS is represented by a system undergoing random walk. Modeling results are interpreted from the probabilistic point of view, although the model is deterministically based on the physical law of conservation of water mass. The results showed that the CSL fluctuations under steady state conditions are characterized by s ¼ 1.1 m, close to the empirical value. “Super-large” anomalies in CSL are not prohibited by the theory, but their development requires a correspondingly long time. However, during long periods of time, background conditions change, and uniformity of the Brownian process becomes disrupted. The origin of large transgressive/regressive stages can be different. For example, the low stand of the Enotaevkian Regression during the LGM was determined by a significant reduction in precipitation over the Volga River catchment and by a corresponding reduction in the volume of river runoff. Hence, based on modeling results, the possibility of “self-development” effects is not prohibited by the theory: there need not be any cause for specific level changes or shifts, merely the expected behavior of red noise processes. © 2015 Elsevier Ltd and INQUA. All rights reserved.
Keywords: Caspian Sea Regional climate change Paleodata Model of random walk
1. Introduction The Caspian Sea (CS) (36e47 N, 47e54 E) is a closed basin. Its sea level lies below the mean sea level of the ocean and has varied between 25 and 29 m in the last ~100 years of recorded history. Water level fluctuations have occurred 100 times faster in comparison to global sea level changes over the last century. The main water source is the Volga River (a total of 80% of river inflow to the CS comes from the Volga River), whose catchment area covers a large part of the Eastern European Plain (EEP). The water inflow is offset by evaporation over the CS. Other rivers, such
E-mail address: [email protected].
as the Ural, Kura, Terek, and Sefidrud, and the subsurface runoff into the sea have to be considered as well, but their contributions are significantly less, uncoordinated and irregular. The water budget of the CS and the current Caspian Sea Level (CSL) variability have been investigated in many studies, e.g., Golitsyn and Panin (1989), Rodionov (1994), Golitsyn et al. (1998), Arpe et al. (2000, 2012) and Arpe and Leroy (2007). Precise reconstructions and dating have demonstrated that during the Holocene, the CS fluctuated between regressive and transgressive stages (Rychagov, 1997). The variations were climateinduced. These significant secular-scale changes took place although only subtle changes were observed in external climate forcings (solar insolation change has been slow and gradual, and the amplitude of CO2 variation was small). Although the CS lies in
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Please cite this article in press as: Kislov, A.V., The interpretation of secular Caspian Sea level records during the Holocene, Quaternary International (2015), http://dx.doi.org/10.1016/j.quaint.2015.07.026
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A.V. Kislov / Quaternary International xxx (2015) 1e5
an area of tectonic activity (Allen et al., 2004), tectonic impacts on sea-level changes do not have to be taken into account for the postLast Glacial Maximum (LGM) timeframe. No tectonic deformations of the Holocene shorelines have been detected, and only lowdegree deformations of the Khvalynian (the end of the Late Pleistocene) shorelines have been detected (Rychagov, 1997). The question of the origin of the fluctuations remains open because there are no clear stable links of the CSL anomalies and palaeohydrological phases with the calendar of climatic anomalies. Correspondence of river runoff changes (on the bases of fluvial activity) to the Caspian Sea level changes was unstable during the Holocene: it was rather high in the second part of the Holocene and was poor before 4e5 ka BP (Panin and Matlakhova, 2015). Climate models (CMIP3 and CMIP5) do not reproduce the needed changes of precipitation, evaporation, and river runoff (Kislov et al., 2014). It is hypothesized that observed anomalies of the CSL, under assumption of a quasi-steady state Holocene climate, are derived from accumulation of small water budget anomalies of opposite signs. Their residual effect forms the CSL response, much like a random walk. Although this assumption cannot be precisely proven, it is possible to determine whether it contradicts the results of observations.
2. Caspian Sea level changes: probability distribution function and Langevin equation The Holocene (11.7 ka) was a stage of Earth history in which the global climate and the environment were characterized by a quasisteady state (compared with the post-LGM timeframe) with gradual long-term trends. Despite this relative stability, the CSL has experienced significant secular-scale and decadal-scale changes during the Holocene (Fig. 1). At each hierarchical level, the dynamic of the CS is characterized by complex oscillations, and its amplitude increases with decreasing frequency. The probability distribution function (pdf) of the CSL variations, calculated on the basis of instrumental records (~100 years), is a bimodal curve (Naidenov, 1992). However, it is reasonable that this assessment is incorrect because there is strong correlation seen in annually averaged data of the CSL. At 1 year time lag, the
Fig. 1. The Caspian Sea level changes during the Holocene (Based on: a e (Rychagov, 1997); b e (Hydrometeorology and Hydrochemistry of the Seas, 1992); c e http:// www.oceanography.ru/index.php/ru/2010-03-15-15-57-22/2010-03-15-15-59-06).
autocorrelation coefficient r1 ~0.95, and autocorrelation disappears only after ~20 years, i.e., r20 ~0 (Nikolaenko, 1997). This means that the 100-year series is equivalent to approximately five independent values (see Fortus (1998) for strong statistical conclusion). Hence, this amount of data is too small for statistical calculation (e.g., for accurate calculation of the pdf). Taking this fact into account, the last 2000-year series, consisting of a sequence of 20-year averaged quantities of reconstructed CSL data (Fig. 1b), was used to calculate the pdf. Of course, these data are not as accurate as measurements, but they are statistically independent variables. They characterize the climate of the Sub-Atlantic stage, during which the environmentaleclimatic regime did not experience significant changes. As for long-term changes taking place in the Holocene, this stage logically is named as “current climate”. It was shown (Kislov, 2011) that the pdf of the CSL can be represented by a Gaussian curve (statistical significance is 0.02 using the chi-squared test statistic); the mean CSL is 28 m, the standard deviation is 1.4 m. Using a smooth curve (Fig. 1a) for the same time interval, we can calculate (using connection between amplitude of harmonic function and its variance) that the standard deviation is 1.6 m. However, Kroonenberg et al. (2008) reconstructed the drop of CSL to 42 m a.s.l. during the Derbentian Lowstand (~1000 years ago). Use of such data leads to standard deviation increases up to 5 m. The main components of the CS water budget are the Volga River runoff and the evaporation minus precipitation (e) over the CS water surface. Analysis of time behaviour has shown that the variance of the Volga River runoff is substantially larger than the variance of e. Additionally, e fluctuates irregularly, whereas the Volga River runoff changes are characterized by long-term trends (Golitsyn and Panin, 1989). Decadal-scale Volga River runoff oscillations (reflecting the influence of precipitation and evaporation over the catchment, and soil water storage) are governed by atmospheric circulation changes. This has been demonstrated from data showing a well-established connection between the CSL changes and variations of circulation indices (dry-wet variations), as well as variations of the North Atlantic Oscillation index, the Wangenheim index, and the Southern Oscillation Index (Isaev et al., 1995; Arpe et al., 2000; Nesterov, 2001; Tugilkin et al., 2011). Explanation of the origin of the secular-scale oscillations is difficult because there is no clear link between CSL changes and the calendar of climate anomalies despite the widely accepted theory that the changes were climatically induced (Kislov, 2001; Bolichovskaya, 2011; Svitoch, 2011; Sidorchuk et al., 2012; Panin and Matlakhova, 2015). Furthermore, climate models (in the framework of CMIP3 and CMIP5) do not simulate the required secular-scale changes of precipitation and river runoff volume (Kislov et al., 2014). However, the origin of long-lasting CSL anomalies can be interpreted another way. The theory of Brownian motion argues that the multi-scale stochastic dynamic of a system is formed through interaction of its fast and slow components. Positive and negative fast anomalies do not cancel each other out and their residual effects accumulate slowly to form a large deviation from the initial state. However, negative feedbacks, which usually exist in the system, prohibit large deviations, and the steady state regime of slow chaotic oscillations is eventually realized. Hasselmann (1976) used this mechanism to describe and reproduce the stochastic behaviour of several geophysical processes. The random time evolution of a Brownian particle position (onedimensional) h ¼ h(t) approximately satisfies the Langevin equation:
dh ¼ lhðtÞ þ hðtÞ dt
(1)
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Here, l1 is characteristic time of the system. The force h(t) has a Gaussian probability distribution and its autocorrelation function is approximated by:
jx zj ; 〈hðxÞhðzÞ〉 ¼ s2g exp th
(2)
where x and z are different moments and h(x)h(z) denotes ensemble average value. The form of the autocorrelation curve is assumed to be sufficiently acute to be approximated by the dfunction curve. Consistent with the property of the physical problem, th ≪ l1. The time series of h ¼ h(t) is expressed by the OrnsteineUhlenbeck process. I a variable initially is Gaussian distributed, then the solution of equation (1) (taking into account (2)) is Gaussian distributed. The solution of the task depicted by equations (1) and (2) is discussed below. To simulate a stochastic process h ¼ h(t), the parameters l1 and th should be chosen. This can be done using two methods. The first method, the standard statistical method, allows us to evaluate the parameters based on observations. In the second approach, the deterministic equations reflecting conservation laws could be used for estimation of parameters of the equivalent stochastic model. The latter method is much more reliable because it allows estimation of parameters under climate change conditions. This approach will be used for investigation of the CSL variations.
3 1 Using vþ for modern 0 ¼ 275 km /y, we can calculate l ¼ 0.05 y conditions; therefore, the characteristic time scale is l1 ~20 y. This value coincides with the assessment of the time lag required for the disappearance of autocorrelation. The forcing function h ¼ h(t) includes short-scale variations due to both river runoff changes (mainly Volga River contribution) and e(t). The one-year period of averaging is much less than l1. Hence, the CS system consists of two substantially different time scales, and behaviour of the CSL changes is governed by the interaction between fast and slow components. Their interaction can be described in terms of Brownian motion, assuming that the slow components are adapted to the total statistical stresses created by the fast component. In this approach, it is assumed that clarification of h ¼ h(t) is not required. This expression is replaced by a random function with known properties. It was shown (Frolov, 1985) that the autocorrelation of the series h ¼ h(t) is a small, but statistically significant value under the shift to one year. However, assuming that equation (4) was averaged over two years, it is plausible to express h ¼ h(t) as white noise. Therefore, let us consider when its autocorrelation function is described by Eq. (2), where th ¼ 2 years. In this case, th ≪ l1 and the mathematical theory of the Brownian motion expressed by equations (1) and (2) can be used. The solution of Eqs. (1) and (2) is a stochastic process, and its variance is given by:
s2h ¼ 3. Adaptation of Brownian motion to explain CSL changes during the Holocene The CS integrates annual variations of both river runoff and e. Accumulated anomalies form residual slow changes of CSL. Therefore, the mathematical apparatus of the theory of Brownian motion can be used for the interpretation of the CSL changes. To demonstrate the feasibility of this mechanism let us develop the stochastic model. This model should be based on the deterministic equation of the CS water budget. An inspection of the water balance equation for any lake can provide some clues:
dVðtÞ ¼ vþ ðtÞ f ðtÞeðtÞ v ðtÞ dt
(3)
This equation describes time-dependent water storage, taking into account the shape of bottom of the lake. The f is lake area. The Z hðtÞ f dz; , vþ and v are contribuvolume of water: VðtÞ ¼ V* þ h*
tions from river runoff and water outflow. The linear approximation (consider a small range of the CS volume change) is used: f ¼ a þ bh (a ¼ 366 000 km2, b ¼ 14 000 km2/m, when reference level of the CS is 28.5 m), assuming that (a þ bh)1 z 1 hb/a. For the water outflow from the CS to the Kara Bogaz Gol (a closed bay on the eastern shore of the CS) the hydraulic formula is: v ¼ v 0 þ ahðtÞ, 3 v 0 ¼ 8:8 km =y is the currently estimated time-averaged value and a ¼ 8.6 km3/y. Thus, the new form of the budget equation is (Frolov, 1985):
dh bvþ ¼ 2 hðtÞ dt a
! vþ ðtÞ v a v 0b 0 2 hðtÞ þ eðtÞ a a a
(4)
The annual averaging of each components of the equation allows us to focus on inter-annual CSL variability. It was shown (Frolov, 1985), that ðvþ hÞ vþ h. Therefore, for annually averaged values, we have an equation describing the CSL changes identical in 2 form to the Eq. (1) where l ¼ bvþ =a2 þ ða=a v 0 b=a Þ and hðtÞ ¼ vþ ðtÞ v =a eðtÞ. 0
3
th s2h 1 e2lt l
(5)
Its spectrum is given by: Sh ðuÞf1=u2 þ l2 . Hence, at high frequencies (u [ l), the spectral shape displays a red-noise continuum (Sh(u) f u2). It corresponds to situation t ≪ l1, whereby the expression (5) is reduced to:
s2h ¼ 2tr s2h t
(6) 1
On the other hand, when t [ l , the variance of CSL variations is defined as:
s2h;st ¼
th s2h l
(7)
where s2h;st is constant, characterizing the steady state conditions. The corresponding spectral curve displays a white-noise continuum (Sh(u)fl2 ¼ const). The CSL variability can be calculated. Application of formulas (5) or (6) requires information about the initial unperturbed state of the CS. However, this information is not available because it is assumed that the CS is always in a perturbed state. Therefore, it is justified to apply an expression (7) estimating the upper limit. Parameters l and th were estimated above. The variance, s2h , is determined by taking into account the variance of river runoff volume and e, which were estimated (only modern observation data can be used) as 0.026 and 0.007 (m/y)2 (Golitsyn et al., 1998), correspondingly. Due to their practical independence (Frolov, 2011), s2h ¼ 0:033 ðm=yÞ2 . Using these values, we obtain using (7) sh,st ¼ 1.15 m. It is only slightly smaller than the empirical estimation (1.4e1.6 m; less compared to Kroonenberg et al. (2008)). Taking into account the first empirical values, theoretical estimation could arise because sh,st calculated on the basis of modern short series has to be underestimated. The closeness of the calculated theoretical sh,st to the empirical value could indicate that the Brownian approach is correct. In order to satisfy the empirical estimation of Kroonenberg et al. (2008), s2h has to be 0.625 (m/y)2. This seems unlikely from the most general considerations, according to which the environmentaleclimatic regime did not experience significant
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changes during the Sub-Atlantic stage. The results of Kroonenberg et al. (2008) cannot be interpreted from the theoretical approach discussed here. The result demonstrates that the statistical properties of the observed time series of the CSL could be reproduced by Gaussian noise with the same parameters. Taking into account the properties of the Gaussian process, the probability of the time (t*) required to achieve a prescribed level (h ¼ a) is given by the formula (Feller, 1968; Crownover, 1995):
deep regression states of the Caspian and Black Seas and mature stages of the late Quaternary glacial events. An example of anomalies of the opposite sign can also be mentioned: the anomalous rise of runoff in the Volga catchment in the Late Glacial time (Sidorchuk et al., 2009), which correlates to one of the phases of the Khvalynian transgression in the Caspian Sea.
rffiffiffiffi 2 a 3=2 a2 t* pft* g ¼ exp 2 p ε 2ε t*
Large transgressive/regressive stages of the CS during the Holocene could have occurred without significant changes in external forcings. Based on general theories about the laws of temporal dynamics relating to massive inertial objects, the observed slow changes of the CSL under the quasi-steady climate state of the Holocene could result from the accumulation of small anomalies in the water regime. The model of the CSL is interpreted as stochastic; from this perspective, it is a Langevin equation that incorporates the action of precipitation and evaporation as random white noise. The process is analogous to Brownian motion. Under these conditions, the Caspian Sea is represented by a system undergoing random walk. Modelling results are interpreted from the probabilistic viewpoint, although the model is deterministically based on the physical law of conservation of water mass. This mechanism could be responsible for the appearance of decadal or secular-scale, large, irregular fluctuations of the CSL that do not correlate with known climatic events. However, it should be considered only as one of the possible mechanisms responsible for the generation of a wide range of the CSL oscillation. Sometimes a clear correlation were observed, for example during last 2 ka between long-scale CSL variations and changes of river runoff (Panin and Matlakhova, 2015) or between the CSL variations and longterm temperature anomalies over the central EEP established for the Little Ice Age and Medieval Climate Anomaly (Kislov, 2001). It is not clear whether they coincide contingently, or these a nonstochastic causal relationship to changes in regional atmospheric circulation. Thus, the increase in the CSL during the LIA was explained by increased penetration of Mediterranean cyclones on
(8)
where ε2 ≡2ts2h ¼ 0:132 m2 =y. Fig. 2 shows these histograms for several prescribed anomalies. The maximum of the curves characterizing the mostly probable transition time is distinctly expressed only in the case of small deviations, i.e., in the case when anomaly a can be achieved in a small number of time-steps. Increase of the prescribed deviation leads to transformation of the histogram: the whole curve shifts to the right along the axis (realization of large anomalies requires much more time), the peak is stretched and the one-modal curve is drastically flattened. Therefore, a large anomaly is a rare and irregular event, and the concepts of most-probable transition time or average transition time lose their meaning. Consider the possibility of the appearance of a large anomaly. According to the principle of probability theory, it is not forbidden, but it is an extremely rare event. In reality, the appearance of such large anomalies is likely influenced by other causes. During the period of time needed for the appearance of this type of anomaly, background climate conditions could change and the Brownian process of accumulation of anomalies (as a holistic process) would be destroyed. Sometimes, a reason for the appearance of large anomalies in the CS may be indicated. For example, in response to glacial conditions of the LGM, the lowering levels of the Black Sea and the Caspian Sea are simulated simultaneously (Kislov and Toropov, 2011), validating the theory of a connection between
4. Conclusion
Fig. 2. Examples of probability distribution of the time required to achieve a prescribed level (h ¼ 0, 5, 1, 3 and 4 m).
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Please cite this article in press as: Kislov, A.V., The interpretation of secular Caspian Sea level records during the Holocene, Quaternary International (2015), http://dx.doi.org/10.1016/j.quaint.2015.07.026