Is there a paleolimnological explanation for ‘walking on water’ in the Sea

of Galilee?

Doron Nof1,2,*, Ian McKeague3,4 and Nathan Paldor51Department of Oceanography, Florida State University, Tauahassee, FL 32303, U.S.A.; 2Geophysical FluidDynamics Institute, Florida State University, Tauahassee, FL 32303, U.S.A.; 3Department of Statistics,Florida State University; 4presently at Department of Biostatistics, Columbia University; 5Department ofAtmospheric Science, The Hebrew University of Jerusalem; *Author for correspondence (e-mail: nof@ocean.fsu.edu)

Received 13 April 2005; accepted in revised form 1 August 2005

Key words: ‘Walking on water’, Air–lake interaction, Convection, Lake freezing, Salty springs

Abstract

Lake Kinneret (the Sea of Galilee) is a small freshwater lake (148 km2 and a mean depth of 20 m)situated in northern Israel. Throughout recent history there have been no known records of a total iceformation on its top. Furthermore, given that convection requires an initial cooling of the entire lakedown to 4 �C, it is difficult to imagine how such a low-latitude lake, presently subject to two-digittemperatures during the winter, could ever freeze. Lake Kinneret is, however, unique in the sense thatthere are dense (warm and salty) springs along its western shore. The dynamics of the regions adjacent tothese springs are investigated using a one-dimensional nonlinear analytical ice model, a paleoceano-graphic record of the sea surface temperature of the Mediterranean Sea, and a statistical model. We showthat, because the water directly above the plume created by the salty springs does not convect when it iscooled down to 4 �C, freezing of the region directly above the salty springs was possible during periodswhen the climate in the region was somewhat cooler than it is today. We refer to this localized freezingsituation as ‘springs ice’.

The analytical ice-model involves a slowly varying approach where the ice is part of a thin fresh and coldlayer floating on top of the salty and warm spring water below. During the ice formation process, the ice iscooled by the atmosphere above and warmed by the spring water below. The plumes created by the springshave a length scale of 30 m, and it is argued that, during the Younger Dryas when the air temperature in theregion was probably 7 �C or more cooler than today, ‘springs ice’ (thick enough to support human weight)was formed once every 27 years or less. During the cold events 1500 and 2500 years ago (when theatmospheric temperature was 3 �C or more lower than today) springs ice occurred about once in 160 yearsor less. Since the duration of these cold events is of the same order as the springs ice recurrence time, there isa substantial chance that at least one springs ice occurred during these cooler periods. With today’s climate,the likelihood of a springs ice is virtually zero (i.e., once in more than 10,000 years).

One set of those springs associated with the freezing is situated in Tabgha, an area where manyarcheological features associated with Jesus Christ have been found. On this basis, it is proposed thatthe unusual local freezing process might have provided an origin to the story that Christ walked onwater. Since the springs ice is relatively small, a person standing or walking on it may appear to anobserver situated some distance away to be ‘walking on water’. This is particularly true if it rained afterthe ice was formed (because rain smoothes out the ice’s surface). Whether this happened or not is anissue for religion scholars, archeologists, anthropologists, and believers to decide on. As natural

Journal of Paleolimnology (2006) 35:417–439 � Springer 2006

DOI 10.1007/s10933-005-1996-1

scientists, we merely point out that unique freezing processes probably happened in that region severaltimes during the last 12,000 years.

NomenclatureSymbols:T – temperature;Cpa, Cpw – heat capacity of air, water;CD – non-dimensional drag coefficient;D – ice thickness;u – mean plume speed;U – wind speed;L – plume width;g – gravity;g¢ – ‘‘reduced gravity,’’(gDq/q);k – thermal conductivity;S – salinity;H – upper layer thickness;H – plume thickness;Q – springs volume flux;a – diffusivity coefficient;a, b – constants defined by (4.5);d – plume veering angle;qa, qw – density of air, water;u – heat flux;ki – ice heat of fusion;l – bottom slope;l0 – off-shore bottom slope

Introduction

This article contains two quite different compo-nents. The first is a rigorous theoretical analysis ofthe conditions under which local freezing couldhave happen along the shores of Lake Kinneretduring the last 12,000 years. This component isdone using conventional scientific methods. Thesecond component contains the plausible hypoth-esis that the story of Christ ‘walking on water’originated from this unusual freezing. By its verynature, this second component is less rigorousthan the first one. Nevertheless, the two compo-nents are intertwined and are presented togetherbecause the first component was developed in or-der to examine the second. While most recon-structions of lakes histories are based on cores and

sediments analysis, the reconstruction here isbased primarily on models.

The reader is warned in advance that it may bequite difficult to follow the analysis because it in-volves five very different areas of physical ocean-ography and limnology: paleoceanogarphicanalysis of the Mediterranean sea surface tem-perature with both simple and complicated climatemodels, an analysis of the springs characteristicsduring the cold events, plume dynamics, icedynamics, and a statistical analysis. Furthermore,each of these five sections is, by itself, fairlyinvolved.

The lake and the springs

Lake Kinneret is situated 45 km east of the Medi-terranean eastern shoreline at approximately32 �N. It is roughly 8 · 18 km with a maximumdepth of 44 m (Figure 1). It is stratified during thesummer and well-mixed in the winter. Although itis a fresh-water lake, its salinity is relatively high[approximately 0.5 practical salinity units (PSU);the reader who is interested in these commonsalinity units is referred to any introductory bookon physical oceanography] because of bothunderwater and above-water saltwater springs. Theorigin of the springs is still under debate. Theirlocation (on the western edge of the lake) suggeststhat they could perhaps be associated with a flowfrom the Mediterranean Sea that stands about200 m higher, providing a gradient of about 5 mper kilometer. Another possibility is that thesprings are a remnant of water that percolated intothe aquifer during the period that salty Lake Lisan(a paleo salty lake which incorporated both theDead Sea and Lake Kinneret, and stood approxi-mately 20 m higher than the current lake level) wasstill present. When the salty lake water level sub-sided, the percolated salty water started pouringback into the lake (Hurwitz et al. 2000).

Because of agricultural and urban development,the present lake is nothing like it was before sig-nificant human impacts. During the first half of the20th century, Lake Hula, a wetland (approxi-

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mately 5 · 10 km) situated north of Lake Kin-neret, was converted into agricultural fields. TheJordan River, which empties into the northernpart of Lake Kinneret, used to flow through thiswetland, which probably warmed its cold spring-fed water. During the 1960s, the salty and warmsprings at Tabgha (loose Arabic pronunciation ofHeptapegon meaning ‘seven springs’ in Latin), and

other springs located nearby, were diverted intothe downstream part of the Jordan River (i.e., thepart of the river which is south of Lake Kinneret).The lake is a major source of fresh water for theregion and it was hoped that this diversion wouldlower its relatively high salinity (0.5 PSU). It did,but, due to the submerged springs and other pro-cesses, the reduction was fairly limited.

Figure 1. A map of Lake Kinneret with the position of the Tabgha warm and salty springs. The locations of the Central Mediter-

ranean cores on which Figure 2 is based are also shown. Depth contours is given in meters.

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During the summer, the winds over LakeKinneret are primarily due to: (i) sea breezesassociated with the Mediterranean Sea, (ii) lakebreeze, and (iii) katabatic winds associated withthe surrounding mountains (see, e.g., Antenucciand Imberger 2003 and the references giventherein). During the winter, on the other hand, thewinds are primarily due to winter storms over theMediterranean Sea. As a result, the lake area issubject to stronger winds during the winter (whenthe low pressure anomaly just reaches the region)but these stronger winds occur less frequently thanthe strong summer winds. Furthermore, thesewinter winds usually die-off after the outer edgesof the low pass by.

Humans have been active along the lake formany years and, recently, archeological excava-tions have unearthed 23,000-year-old mats (Nadelet al. 2004). Evidently, Jesus Christ spent asignificant amount of time in Tabgha (see, e.g.,Pixner 1985) and the ‘walking on water’ accountmight have originated there. [Matthew 14:25: Andin the fourth watch of the night Jesus went untothem, walking on the sea. 14:26 And when thedisciples saw him walking on the sea, they weretroubled, saying, It is a spirit; and they cried outfor fear. 14:27 But straightway Jesus spake untothem, saying, Be of good cheer; it is I; be notafraid. 14:28 And Peter answered him and said,Lord, if it be thou, bid me come unto thee on thewater.]

Our analysis

We shall begin our analysis by presenting pale-oceanographic temperature data from the centralMediterranean Sea showing cooling events duringthe last 13,000 years (Section 2). These cores weretaken 2000 km away from Lake Kinneret butthey represent the closest paleoclimate recordsthat exist. We shall show that, during the aboveperiod, there have been several cooling eventswhere the sea surface temperature (SST) was 1–5 �C cooler than it is today. We shall relate theseSST anomalies to the atmospheric anomaliesabove using both a novel simple model involvingheat exchange and Ekman dynamics and the re-sults of recent general climate models (GCMs)which include much more variables than ourmodel does. The simple model shows that the

maximum ratio of the SST anomalies to theatmospheric temperature anomalies above is theair/water heat capacities ratio, which is roughly1:4. This implies that, since it is easier to cool theatmosphere than it is to cool the ocean, theatmospheric anomalies can be as high as fourtimes the oceanic anomalies. It turns out that thisis probably true for high latitudes but, for lowlatitudes, GCMs runs suggest a smaller ratio ofatmospheric to oceanic anomalies, about 1.5–2.0.We shall do our calculations for both the low andhigh values so that we will cover the entire pos-sible range.

We shall then present the analysis of stalactitesin the Soreq cave in southern Israel (approxi-mately 35 �E, 31 �N) and show that these coolingevents were associated with an increased aridity.We shall then proceed and examine the scales ofthe plumes created by the salt water springs (Sec-tion 3). It will be demonstrated that the plumemoves primarily along the coast gradually veeringto the left of the isobaths (looking downstream).Using the decreased aridity record, we shall alsoestimate the salinity of the springs during the coldevents.

We shall continue with an examination of theconditions under which freezing above theplumes (resulting from the springs) could takeplace. In this context, we shall note that a‘regular’ freezing of a fresh water lake withdimensions similar to Lake Kinneret wouldrequire a few or several months of near- orbelow-freezing temperatures because the densityof fresh water (and even brackish water) reachesa maximum around 4 �C. Consequently, con-vection occurs around this temperature implyingthat the entire lake needs to be first cooled downto 4 �C. Any further cooling produces lighterwater which stays on top and finally freeze. Thefreezing of such ‘regular’ lakes require relativelylarge amounts of cooling because, as mentioned,much cooling needs to be done before coolingprocesses can go directly to the formation of iceon top. Lake Mendota in central Wisconsin(approximately 89 �W, 43 �N) is about 25% thesize of Lake Kinneret and provides a goodexample of such a lake (see, e.g., Dutton andBryson 1962). Rough estimates suggest that,even during the Younger Dryas and the lastglacial maximum (LGM) when the estimatedatmospheric temperature reduction was relatively

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high, a complete cooling of the entire lake to4 �C is extremely unlikely. [Without a moredetailed knowledge of the evaporation rate andother variables during the Younger Dryas andthe last glacial maximum (LGM), it is impossibleto unequivocally say whether winter freezingoccurred or not.]

For the warmer Holocene, the possibility of aperiod of a few or several months of near-freezingtemperatures is even less likely. Consequently, werule this conventional general freeze out for therelatively low latitude of Lake Kinneret. Usingone-dimesional analytical models, we shall dem-onstrate, however, that local ‘springs ice’ (i.e.,freezing and ice formation in a confined regiondirectly above the plumes created by the saltysprings) requires merely a few days of below-freezing temperatures. This is because the saltysprings water creates a strong salt-inducedstratification (above the plume) that inhibitsconvection. Consequently, the cooling processgoes immediately to ice formation on top ratherthan toward convection and a total cooling of theentire lake to 4 �C. Note, however, that the re-quired cold spell of a few days must be relativelycalm, with wind speeds of less than, say, 6 m/s.Otherwise, the wind would mix the plume intothe fresh water above, destroying the stratifica-tion and preventing a no-convection coolingprocess. Such calm periods are very commonaround the lake.

After presenting the plumes’ details in Section 3,we describe the ice-model in Section 4. In thatsection, we employ the slowly varying approachand derive a single first-order nonlinear differentialequation incorporating heat exchange with boththe cold atmosphere above and the warm salinewater below. An exact solution (which is notentirely new) is found and its analysis is given inthe same section (4) where we show how long itwould take to form an ice layer of 10 cm thicknesson top of the plumes. (Such a layer is certainlysufficient to accommodate the weight of humanson top of it: on the basis of Red Cross recom-mendations, many municipalities require 15 cm ofice for safe ice skating.) Note that the use of one-dimensional models to examine freezing processesin lakes is adequate as a first approximation be-cause high-resolution three-dimensional numericalmodels typically display very similar results (Om-stedt 1999).

We then proceed, using a statistical model, tocalculate the likelihood that such a period of be-low-freezing temperatures would take place (Sec-tion 5). We do this for various mean temperaturereductions corresponding to the known coolingevents that took place during the last 13,000 years.This is followed by an analysis of the conditionsunder which there will be a few consecutive days ofno wind stronger than 6 m/s. The results aresummarized in Section 6.

The paleoceanographic temperature record

Unfortunately, there are no direct (or indirect)paleoclimatic records from the Lake Kinneret re-gion. What we can do is use paleo-records of seasurface temperature (SST) from regions in theMediterranean some distance away from the lake(approximately 2000 km) because this distance isnot any greater than the typical weather systemscale in that part of the world. This approach isjustified because the winds in the region are usuallyfrom the west so that the air above the lake usuallyoriginates in the atmospheric boundary layerabove the Mediterranean Sea. In other words, it isreasonable to take the Mediterranean SST asindicative of the Middle East climate – some claimthat even regions in the Atlantic (and the NorthAtlantic Oscillation) correlate with temperaturesin the Middle East (e.g., Eshel and Farrell 2000,Eshel et al. 2000, Cullen et al. 2002).

Sea surface temperature

Figure 2 shows the sea surface temperature (SST)record for the past 16,000 years as determinedfrom two cores, one taken near the northwestcorner of Sicily (adapted from Cacho et al. 2001)and the other in the Ionian Sea (adapted fromEmeis et al. 2000). The location of the cores isshown in Figure 1. The first core was chosen be-cause, of all the cores presented by Cacho et al.(2003), it is the most representative of the waterentering the eastern Mediterranean. As mentioned,since the synoptic atmospheric scale is of the orderof 1000 km, it is this SST which determines the airtemperature above the lake.

The Cacho’s core starts around 1000 years agoand we related it to the present-day climate with

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the aid of Reale and Dirmeyer (2000) analysis. Onthe basis of archaeological and historical data,they argue that 2000 years ago the climate was as

warm as it is today [see (their) page 165]. Hence,we take the Cacho’s temperature 2000 years ago tobe today’s temperature (see the horizontal gray

Figure 2. Upper two panels: The sea-surface temperature (SST) record during the last 16,000 years, determined from two cores, one

situated slightly to the northeast of the Strait of Messina (see Fig. 1), and the other situated in the Ionian Sea (adapted from Cacho et al.

2001, and Emeis et al. 2000). Note that, both records reflect cold events when the temperature dropped significantly below the present day

temperature (horizontal line). For convenience, they are only marked in the middle panel (see text). Lower panel: The rainfall record

during the last 8000 years as determined from stalactites in the Soreq cave in southern Israel (adapted from Bar-Matthews et al. 2003).

Note the correlation between the cold events and an increased aridity (vertical grey lines at 1500, 2500 and 5300 years ago).

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line in the upper panel of Figure 2). For consis-tency, we did the same with Emeis’ record (seethe horizontal line in the middle panel ofFigure 2). Since Emeis’ record extended to thepresent day, this consistency requirement impliedthat the first few hundred years of Emeis’ recordhad to be disregarded. This period correspondsto the first few centimeters of their core and,hence, may not be representative of the actualsituation because of disturbances that mighthave been introduced either by the coring deviceitself or by recent strong flows along the bottomthat altered the upper sediment (via scouring ordepositing).

Table 1 shows the cooling events that the twocores display. Some events occur in both cores atthe same time and others do not. We chose fiveevents (1500, 2500, 9800, 11,000 and 12,800 yearsago) that are shown with arrows near the record ofthe second core (Figure 2). We averaged the tem-peratures at these particular times even though acooling event in one record does not necessarilycorrespond to a cooling event in the other record.The mean cooling of the first three events (1500,2500 and 9800 years ago) turns out to be roughly2 �C and the last two (11,000 and 12,800 years ago)correspond to a mean cooling of about 5 �C. Notethat other choices could have been made. Forexample, we chose not to consider the cooling event5300 years ago (not numbered in the middle panel)but one could certainly consider it as well. We chosenot to, merely because it is considerably smallerthan the others (i.e., it corresponds to a mean SSTreduction of about 1 �C). It will become clear laterthat this relatively small cooling implies that therecurrence time for a springs ice is more than a1000 years – longer than the cooling event itself.

It is important to realize that the above SSTcoolings during the Younger Dryas and earlier areconsistent with ground water cooling in Oman

[situated about 2000 km to the southeast of theSea of Galilee (Weyhenmeyer et al. 2000)], withSST cooling in Barbados (Guilderson et al. 1994)and Venezuela (Lea et al. 2003), and with groundwater estimates in Brazil (Stute et al. 1995).Although there have been some questions regard-ing the ground water temperature estimates, theabove suggest that the Mediterranean cores arerepresentative of the anomalies in the region sur-rounding the lake. (As mentioned, it is unfortunatethat there are no cores closer to the lake – the onlyones that are available do not extent more than800 years before present.)

Figure 2 also shows the rainfall during the last8000 years as determined from stalactites in theSoreq cave in southern Israel (adapted from Bar-Matthews et al. 2003). We note that, from thebeginning of the Holocene to the present day,there has been a gradual decline in rainfall, inagreement with other studies. There are fourevents of reduced rainfall, three of which areclearly correlated with cooling events shown by theIonian Sea core (see the vertical gray lines and therecord of core 2 in Figure 2). This is also inagreement with other studies which showed thatcooling events are associated with an increasedaridity (e.g., De Rijk et al. 1997; Bartov et al.2003). Bryson and Bryson (1998) also show asignificantly decreased Nile fresh water flux duringroughly the same period of increased aridity.

Interestingly, Bar-Matthews et al. (2003) com-pare their rainfall data to the SST reconstructionfrom a core taken south of Cyprus that shows aninverse correlation with the cooling events, i.e., adecreased rainfall is associated with warming ra-ther than cooling [see (their) Fig.13A]. We believethat this is because this particular core is heavilyinfluenced by the Nile which empties into theMediterranean Sea a few hundred kilometers tothe south. (Note that, before the construction ofthe Aswan High Dam, the discharge of the NileRiver was much greater than it is today, about1000 m3/s.) Since river water is typically severaldegrees cooler than the ocean (because the ocean isusually warmer than the atmosphere), a reducedrainfall (and a resulting reduced river discharge)implies less cooling which, in the core record, isrepresented by an apparent warming. (Note that,even though the river water is cooler than theMediterranean Sea, it is still buoyant because it isfresh.) Hence, the Bar-Matthews et al. (2003)

Table 1. Significant cooling events during the last 13,000 years.

Event Years BP Temp.

reduction,

Core 1 (�C)

Temp.

reduction,

Core 2 (�C)

Approx.

mean

reduction

(�C)

1 1500 �2 �2 �22 2500 +1 �5 �23 9800 0 �4 �24 11,000 �1 �8 �55 12,800 �3 �7 �5

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comparison is not really a comparison of rainfallto an independent Mediterranean SST but rather acomparison of rainfall to an SST which, by itself,is influenced by the rainfall.

Relation of the SST to the atmospheric temperature

We shall now relate the SST anomalies to theatmospheric anomalies above. Before doing so, itis appropriate to point out that the only placewhere both oceanic anomalies and atmosphericanomalies have been observed at the same timeand the same location is the northern NorthAtlantic. Here, oceanographic proxy data gave theSST anomalies (for the last 100,000 years) whereasair bubbles trapped in the ice sheets over Green-land gave the corresponding atmospheric anoma-lies [see e.g., Bard (2002), the top two panels of(his) Figure 2]. For this case, the ratio of atmo-spheric temperature anomalies to SST anomalies isfairly high, approximately 3.5. Recent climatemodels runs suggest, on the other hand, that atleast in low latitudes, this ratio is perhaps as smallas 1.5 or 2.0 (e.g., Pinot et al. 1999; Shin et al.2003). In what follows we shall use a simpledynamical model to show that the above ratio is,at the most, 4.0. We shall later perform our finalcalculations for both a ratio of 1.5 and a ratio of4.0 so that we shall cover the entire possible rangeof anomalies.

We shall now present our simple novel argumentsuggesting that, during cold events, the maximumtemperature reduction in the atmosphere is relatedto the oceanic reduction via the ratio of the air/water heat capacities. Consider the situation shownin Figure 3a, b. A one-dimensional box is orientedalong the atmosphere geostrophic flow (Figure 3a).The mean Ekman flow in both the ocean andatmosphere are oriented at 45 � and 135 � relativeto the box axis. Both have components perpen-dicular to the geostrophic flow (and the box axis)and these perpendicular components are denotedby Qo and Qa. As Figure 3a illustrates, they areequal to the Ekman fluxes divided by

ffiffiffi

2p

. Note thatnone of this information is new, all of it can befound in a variety of textbooks. It is listed heremerely to illustrate how our box is oriented.

The oceanic volume flux Qw enters the boxwhere it cools from Twi to Two. (The subscripts ‘i’and ‘o’ denote ‘in’ and ‘out.’) The heat is released

to the atmosphere (which has a volume flux Qa)and, as a result, the atmosphere warms from Tai toTao. This heat exchange between the ocean and theatmosphere can be written as,

qwCpwQw Twi � Twoð Þ ¼ qaCpaQa Tao � Taið Þ; ð1Þ

where the subscripts ‘w’ and ‘a’ denote water andair, and Cpa and Cpw are the heat capacities. (Forconvenience, all variables are defined in both thetext and the Nomenclature.) Note that this equa-tion is simply a heat flux equation for a box andthat, in the regions to the right and left of the box(Figure 3b), the ocean is insulated from theatmosphere. Also, it is assumed here that there isno heat exchange between the box and the fluidabove or below.

Recalling that the atmospheric and oceanicEkman mass fluxes are the same, we immediatelyget the relationship,

qwQw ¼ qaQa: ð2Þ

Substitution of (2) into (1) gives,

Twi � Two ¼Cqa

CqwTao � Taið Þ; ð3Þ

showing that the mean reduction in the oceanictemperature is related to the mean increase of theatmospheric temperature through the ratio of theair/water heat capacities (approximately 1:4). In away, this is not very surprising – it is much easierto cool the air than it is to cool the water so thatany local climatic anomaly is expected to be morepronounced in the air than in the water below.

Suppose further that, through some processwhose details are not important for the presentdiscussion, the incoming water temperature hasbeen reduced so that the difference between theincoming and outgoing ocean water (Twi�Two)has been temporarily reduced by, say, D Tw. (Thereader who would like to envision a correspondingphysical situation can think of the incoming wateras, say, Atlantic water entering the Mediterraneanthrough the Straits of Gibraltar.) Such a reductionwould, in general, imply a change in both the in-going and outgoing water temperatures but, toobtain the maximum heat exchange between theocean and the air, we assume that it involves merelythe lowering of the incoming ocean water temper-ature Twi (i.e., we shall leave the outgoing watertemperature Two unaltered). In reality, there will besome reduction of the outgoing oceanic water too

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atmosphericsurface Ekman flow

stress on atmosphere

atmospheric Ekman flux

stress on oceansurface

atmosphericgeostrophic flow(aloft and withinthe Ekman layer)

oceanicEkman flux

oceanic Ekman fluxcomponent perpendicularto the geostrophic flow

atmospheric Ekman fluxcomponent perpendicular tothe geostrophic flow

45°

(a)

Figure 3. ((a) The orientation of the various fluxes in the ocean (blue lines) and atmosphere (red lines). The oceanic Ekman flux

is oriented at 90 � to the right of the stress exerted on the ocean by the atmosphere whereas the atmospheric Ekman flux is 90 �to the left of that stress. The atmospheric geostrophic flow above and within the atmospheric Ekman layer is directed at 45 � to

the right of the stress exerted on the ocean by the atmosphere so that, at the surface, it exactly cancels the surface atmospheric

Ekman flow which is directed at 135 � to the left of the stress on the ocean surface. (This implies that, as should be the case,

the sum of the two is zero at the surface.) The yellow and green spirals indicate the Ekman flow in the ocean and atmosphere

(respectively). None of the information shown here is new. It is displayed merely to illustrate how the box shown in Fig. 3b is

oriented (blue shading). (b) Schematic diagram of the simplified ocean–atmosphere heat exchange process. An oceanic volume

flux Qw (with an incoming temperature Twi and an outgoing temperature Two) enters a one-dimensional box where it cools and

releases its heat to the atmosphere (thick vertical arrows). The atmospheric volume flux Qa has an incoming temperature Tai

and an outgoing temperature Tao. The ocean is insulated from the atmosphere in regions to the right and left of the box. Both

the oceanic and atmospheric fluxes represent the fraction of the Ekman flux which is perpendicular to the atmospheric

geostrophic flow. (These amounts equal the total Ekman flux divided byffiffiffi

2p

.) The box is oriented along the atmospheric

geostrophic flow (directed into the page).

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so that only a fraction of the oceanic temperatureanomaly will be transmitted to the atmosphere.However, since we are merely interested here in themaximum ocean–atmosphere exchange, we takethis reduction of the outgoing water temperature tobe zero so that the entire oceanic anomaly iscommunicated to the atmosphere.

Because of the reduction in the incoming oce-anic temperature, less heat is now going to betransferred from the ocean to the atmosphere sothat the outgoing air will be cooler than before.Taking the incoming air temperature (Tai) to beunaltered (because it is insulated from the oceanoutside the box), we immediately find from (3) thatthe relationship between the reduction in temper-ature of the outgoing air and the reduction in theincoming oceanic temperature, is

DTao ¼Cqa

CqwDTwi: ð4Þ

This implies that the maximum ratio of thereduction in the temperature of the incomingocean water and the temperature of the outgoingair is the ratio of the heat capacities which is about1:4. [Note that the heat capacity of water isapproximately 4.19 kJ/kg �C whereas the heatcapacities of dry air and water vapor are 1.06 and1.84 kJ/kg �C (respectively). Since the moisturecontent of air is no more than a few percent (kgper kg), it follows that the heat capacities ratio isabout four]. In most situations, the actual ratiowill be less than the above 1:4 upper bound be-cause, as stated, there will be some alteration ofthe outgoing water temperature so that only afraction of the oceanic heat anomaly will betransferred to the air.

Consistent with this upper bound, the Green-land air/water temperature anomaly ratio is a bitsmaller than 4.0 (it is 3.5). The more involved cli-mate models mentioned earlier suggest a low lati-tude ratio that is still lower (1.5–2.0).Incorporating both ends of the estimates (i.e., a 1.5ratio and a 4.0 ratio), the atmospheric temperaturereductions for events 1 through 5 will be taken tobe roughly 3–8 �C, and 8–20 �C with the smallerestimates (1.5 ratio) corresponding to the moreconservative choice based on the climate models(which involve evaporation, humidity and manyother variables). We shall see later that a springsice cover can occur even within the framework ofthe most conservative estimates.

The dense plume

This section is divided into two parts. First, weshall discuss the salinity and temperature of thesprings during the cold events and then we shalldiscuss the resulting plume dynamics, i.e., its scalesand structure.

Springs salinity and temperature

For simplicity, we take all the closely packedTabgha springs (Figure 1) to be represented by asingle source of salty and warm water. For reasonsthat were briefly mentioned earlier and will beelaborated on shortly, we shall consider thebehavior of the plume during periods of calmwinds.

Recent measurements of both the water prop-erties and the local winds were taken by the Me-korot Water Corporation, a semi-privatecorporation that maintains the conduit divertingthe springs. There is a seasonal variability in thesalinity, temperature, and mass flux (e.g., Rimmeret al. 1999), but, for our purpose, it is sufficient toconsider the mean values which correspondapproximately to 3.7 PSU, 26 �C, and 0.7 m3/s.(Note that documents describing the limnology ofLake Kinneret typically give the salt content interms of chlorinity, which has a ratio of 1.7–2.1 tothe total salt content.)

The springs water consists of a mixture of twodistinctly separate ground water sources. The firstis a deep, relatively steady source providing waterwith a salinity similar to that of the MediterraneanSea, about 36 PSU (Rimmer et. al. 1999), and ahigh temperature of perhaps 63 �C which is thetemperature of the (unmixed) Tiberias hot springs.The second is a more variable fresh water sourceresulting from rainwater percolating down to theaquifer. The Tiberias Springs, situated severalkilometers to the south of the Tabgha, containonly the first source (i.e., no fresh water intrusion)and, hence, they are much saltier and much war-mer than the Tabgha springs. Conservation of salt[i.e., Q1S1 + Q2S2 = QS, where Q1, Q2 and S1,S2 are the volume fluxes and salinities of the twoincoming sources (so that S2 = 0) and Q and S arethe volume flux and salinity of the resulting springmixture] immediately shows that the deep source(Q1, S1) provides about 10% of the mixed water

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and that the remaining 90% comes from rain.(Note that the equivalent equation for conserva-tion of heat cannot be used because the sources ofunderground heat cannot be easily estimated.)

We shall now estimate the corresponding values(S and T) during the five cold events that tookplace during the last 13,000 years. For simplicity,we assume that, due to the increased aridity, therainfall was reduced by, say, 60% during the coldperiods (Figure 2, lower panel). Since the presentspring water is a 1: 9 mixture of deep and rainwater, this 60% reduction in rain water impliesthat, during the cold periods (when the above ratiois modified to 1:3.6), only 78% of the mixture wasfrom rain and the remaining 22% came from thedeep water source (with a 37 PSU). This meansthat the adjusted springs’ salinity that we shouldconsider is 8.14 PSU. (Note that, because thesprings’ salt source is either the Mediterranean orpreviously percolated water, the increased evapo-ration during the cold, more arid, periods hasalmost no bearing on it.)

Since both the air and the rain were at least 3 �Ccooler during the cold events than they are todayand, since most of the springs’ water is rain, wetake the cold period springs water temperature toalso be 3 �C cooler than the present springs watertemperature of 26 �C (i.e., we take it to be 23 �C).Note, however, that, even with different choicesfor the rain reduction, our final results wouldessentially be the same—they are not very sensitiveto these choices. Also, note that we shall not adjustthe mass flux Q to the increased aridity because, aswe shall shortly see, this has an effect only on theplume size, and this effect is very minor.

Our choices for the plume (T = 23 �C andS = 8.14 PSU) and winter lake variables(T = 15 �C and S = 0.5 PSU) are shown on theTS diagram Figure 4) which also displays thechange of lake water density due to cooling leadingto freezing (vertical dashed line). Figure 4 alsoshows the salinity and temperature of the waterabove the springs-induced plume both before andduring freezing (dashed-dotted lines). These werecomputed using a (steady) solution to the verticaldiffusion equation (discussed later) which gives alinear distribution. Note that cooling all the waydown to freezing does not, at any point, involveconvection because the springs-induced stratifi-cation always corresponds to water heavier thanthe cooled water above. This can be vividly seen

by noting that, as the surface water is cooled(thick dashed vertical line), a density greater thanthe water below (dashed-dotted lines) is neverreached.

Plume dynamics

Scales

The dynamics of heavy plumes have been studiedextensively and the interested reader is referred tothe first study by Smith (1975) and the more recentstudies of Shapiro and Hill (1997), Killworth(2001) and the references given therein. Since weare merely interested here in the plume’s scales,there is really no need to present any detailedsolution. It is sufficient to show the main balances.Note that none of the dynamics and balancespresented below is new – all the relationships thatwe shall employ can be found in one form oranother in the earlier articles mentioned above.

When the wind stress is relatively weak, thegeneral circulation in small lakes such as LakeKinneret is usually cyclonic (e.g., Csanady 1982)because this is the direction in which both Kelvinand topographic Rossby waves propagate (see alsoSerruya 1974). This circulation forces the plume todeflect to the right (looking off-shore). Thisdeflection is consistent with the condition that,because the offshore slope is large (1:100), theplume cannot descend offshore ‘head-on’ as itwould become unstable if it were to do so. Instead,the plume picks its own bottom slope along whichit can gradually progress without becomingunstable (Figure 5). We shall see that this pickedpath is in between the zero along-isobaths slopeand the 1:100 cross-isobaths slope. It is much closerto the isobaths than it is to the offshore direction.

We can estimate the plume’s speed, thickness,and width as well as the slope that it will follow, byconsidering the following balances. First, it is ex-pected that frictional losses along the bottom ofthe plume will be compensated for by a pressuredrop associated with the gradual decline in themean plume height, i.e.,

g0l � CDu2=H; ð5Þ

where g¢ is the known ‘reduced gravity’, (gDq/q), lthe yet unknown slope along which the plume

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progresses (as we said, not at all equal to the largecross-isobaths slope of 1:100), CD a known non-dimensional drag coefficient for water over a solidsurface (approximately 2 · 10�3), u the unknownmean speed and H the unknown thickness of theplume. Second, the flow is required to be stable sothat the Froude number is approximately unityimplying,

u2. g0H: ð6Þ

This condition (6), together with the energyconstraint above (5), immediately shows that l �CD � 2 · 10�3, implying that the plume veersslightly to the left of the isobaths (looking down-stream) at an angle d of approximately 18 �

(because the offshore slope l0 is 1:100 and thelong-shore slope is zero). The third condition is theconservation of volume,

Q � uHL; ð7Þ

where Q is the (known) constant volume fluxand L is the unknown plume width. The fourthand last condition is a geometrical constraint(involving a cross-section of the plume) implyingthat,

H � l0L ð8Þ

This condition states that, in order to have athickness ranging from zero to H, the plume’souter edge must be at least a distance L away

Figure 4. The mean density (during the cold events) of the springs water (A) and of the present-day lake water in winter (B). The

salinity for the cold periods was adjusted to reflect the increased aridity. For a comparison, we also show the highest unadjusted

variables of the most dense Tabgha spring, the Sartan Spring (D). The dashed-dotted lines represent the water above the springs before

and during the freezing process. They are based on a linear solution to the diffusion equation (for heat and salt) and on the plausible

assumption that the surface water above the springs has the same salinity as the off-shore lake water. Note that, as shown with the

vertical dashed line (BC), cooling of the lake water above the springs causes freezing (C) without convection because the density

associated with the cooling is always smaller than that of the water below.

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from the inner edge (where its thickness is zero).The four algebraic Eqs. (5–8) provide a solutionfor the four unknowns l, u,H, and L. We al-ready obtained l as 2 · 10�3 and a combinationof (2), (3), and (8) gives,

H � Q2l20=g

0� �1=5: ð9Þ

Taking Q to be 0.7 m3 s�1 and g¢ to beapproximately 3 · 10�2 ms�2 (Figure 4), we findL � 30 m, H � 30 cm and u � 8 cms�1.

The plume is expected to be present in a regionof O(1) meter deep. We shall see later that whetherit is 1, 3, or 4 m deep makes no difference to ourfinal conclusions.

Figure 5. Schematic diagram of the plume’s spreading along the bottom of the lake. Dashed lines represent the isobaths whose cross

slope l0 is about 1:100. The general cyclonic circulation of the lake forces the plume to the right (looking off-shore) and the tendency of

the plume to gradually lose height forces the plume to veer slightly to the left of the isobaths (looking downstream). The veering angle dis approximately 18 �. This gradual veering is because the offshore slope l0 is so large that a ‘head-on’ descent would make the plume

unstable. The slope that it follows is l.

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The no-mixing condition

The approximately 30 cm-thick plume which isembedded in a meter or so of lake water can, ofcourse, be mixed (with the fluid above) if the windis strong enough. The condition of no convectionrequires that such mixing will not take place dur-ing the freezing period (of a few days). Hence, werequire that the stress imposed by the wind will beso small that it will not be able to elevate theheavier plume and mix it with the lighter fluidabove. In analogy with a wind set-up situation(where the wind elevates the water surface on thedownstream side), we replace the free surface windset-up elevation with the interface displacementtimes the ratio of the density difference to thedensity. Taking into account the plume lengthscale and the strength of the wind, we find that thismixing condition is,

g0HH � qaCDU2L

qw

; ð10Þ

where qa and qw are (as before) the air and waterdensities and U is the wind speed. Relation (10)can also be interpreted in terms of energy – theright hand side represents the work done (on thewater) by the wind and the left represents the in-crease in potential energy (due to the lifting of theheavy fluid). For our earlier parameter choices, anair-on-water drag coefficient of 1/1000 and anH of1 or 2 m, this relationship shows thatU � 15 ms�1 is the necessary condition for mix-ing. This speed is relatively high because the plumeis small and the stratification is relatively high.

For the region in question, this speed is so highthat it has never been observed even once in theentire 5 years hourly record that we looked at.(The highest speed that was recorded was less than12 m/s.) To avoid this mixing situation, we requirethat the wind be less than 6 m/s (less than a half ofthe mixing speed) for the entire freezing period.This is certainly an adequate choice as the windstress is proportional to the square of the windspeed so that a quantity less than a half will pro-duce little mixing.

We shall see later that, in winter, the wind speedexceeds this value only during strong storms. Forall practical purposes, the above restriction on thewind corresponds to an almost no-wind state.Hence, it corresponds to a lake with very small

eddy diffusivity, close to the molecular diffusivity.In this context, it is of interest to note that mea-surements conducted in ice-covered lakes (whichhave no wind input but do have convection due toheat released from the sediments on the bottom)reveal a horizontal eddy diffusivity of 100 cm2/s(Bengtsson 1996). Together with the length-to-depth ratio of 100:1 in the lake, this gives ver-tical eddy diffusivity identical to the moleculardiffusivity, 0.01 cm2/s. This is consistent with theidea that, although the plume itself is turbulent,the turbulence is expected to die off at a distance ofthe order of the plume thickness (30 cm) awayfrom it.

The above decay of mixing is expected to takeplace within a few minutes after the winds diedown (because this is the time that it takes heavyparticles to sink and buoyant particles to rise). Aswe shall see, a linear profile of both temperatureand salinity will be established above the plume. Itis not easy to say how long it would take for such aprofile to be established but, if it were to start froma state of rest, then molecular diffusion would set itup within a few or several days. On the other hand,if it were to start from a completely mixed statethen it would be achieved within a few minutesafter the winds die down.

The ice model

Fresh and brackish water follow a complicatedfreezing process which involves convection andcomplete turnover. This is because the maximumdensity of fresh and brackish water is around4 �C implying that, before a fresh water lake canfreeze on top, all of its water must first be cooledto 4 �C. When this stage is reached and thecooling continues to temperatures below 4 �C, thenewly cooled water is lighter than the warmerwater underneath so that it floats and stays ontop until the freezing condition of 0 �C is reached.On the other hand, sea water (S >24 PSU)which is typically weakly stratified by the salt (i.e.,the density gradient due to salt is not muchgreater than that induced by temperature) doesnot have such a maximum. As a result, newlycooled surface sea water with a temperature be-low 4 �C is no longer lighter than the water belowso it does not stay on top. Consequently, some

430

convection (limited to the upper portion of thewater column) occurs before freezing. Namely,the freezing of fresh water involves a completeconvection and turnover whereas the freezing ofsea water normally involves a partial convection.(Note the two do not occur at the same temper-ature.)

A body of water which is strongly stratified bysalt (i.e., the salt gradient overwhelms the densitygradient) behaves in a way which is differentfrom the previous two. Here, there is no con-vection at all. This is because the cooling ofsurface water cannot overcome the stabilizingeffect of the salt. If one compares a hypotheticalstrongly stratified salty lake (i.e., a salty lakewith salt-induced density anomaly greater thanthat induced by the temperature gradients) and afresh water lake of the same size both having aninitial temperature of, say, 15 �C, then muchmore cooling is required to form ice on top ofthe fresh water lake than on top of the stronglystratified salt lake.

This is because the entire fresh water lakemust first be cooled to 4 �C whereas the salt-stratified lake can freeze on top even if most ofits water stays at 15 �C. We argue that the sit-uation above the Kinneret springs’ plume cor-responds to such a strongly stratified salty lake.This is clearly reflected in the T�S diagramshown in Figure 4. The straight, dashed-dottedlines show the density of the water above thesprings’ plume just before cooling starts andduring the freezing process. These lines wereconstructed using the linear solution to the(steady) diffusion equation, and off-springs lakewater salinity at the surface. Note that at nopoint during the cooling process (thick dashedvertical line) is the surface water heavier than thewater below.

Ice modeling background

The first ice growth model was developed by Ste-fan (1890), who correctly identified the mostimportant freezing processes – the atmospherecools the ice from above and the conversion ofwater into ice occurs along the ice-water interface(without significant exchange of heat from below).Stefan took the transfer of heat within the ice to belinear and this leads to a linear first-order

differential equation which has a straightforwardsolution. A good summary of Stefan’s work, aswell as of straightforward extensions of his work,is given in Neuman and Pierson (1966). The nextstep was made by Welander (1977), who, in effect,broadened Stefan’s work. Welander considered alaboratory situation where the ice exchanged heatwith the fluid below (which receives a constantamount of heat from a laboratory heater) and theair above. The temperature of the fluid below theice changes with time and the model equations arenonlinear. Welander obtained a numericalsolution to the problem. For more recent andmore involved numerical ice simulations, thereader is referred to Kantha and Mellor (1989),Mellor and Kantha (1989) and Hakkinen (1995).Although these recent numerical models as well asWelander’s model are of interest, they are muchmore involved and complicated than what we needhere. Since we are merely interested in a reasonableestimate, an analytical one-dimensional model issufficient.

Our adopted model differs from both Stefan’s(1890) and Welander’s (1977) models. Stefan’smodel does not include heat exchange from be-low which is important in our case because thesalty springs are warm. Welander’s model,though it included heat transport from below,allowed the temperature below to change withtime. Because the spring water temperature doesnot vary much over the time scale that the ice isformed (days) and because the ice thickness (onthe order of 10 cm) is much smaller than that ofthe relatively fresh water above the spring water(on the order of 1 m), our heat exchange withthe fluid below can be taken to be constant. Weshall see that, even though this heat flux isconstant, it still alters Stefan’s equation dra-matically. It turns out, however, that a simpletransformation allows one to obtain an exactanalytical solution. This was recognized earlierby Thorndike (1992) who considered a somewhatdifferent physical system than ours but ended upwith the same equation that we have here.Leppaeranta (1993) argues that, if one takes theair temperature to be equal to the ice surfacetemperature, then the Stephan’s model overesti-mates the ice thickness. This overestimate issupposedly particularly important for thin icesuch as ours. However, it is based on Anderson’s(1961) analysis for sea-ice which is normally

431

subject to strong winds. It is not at all obvioushow can the sea ice results be extended to inlandlakes which are usually subject to weak winds.

In view of this and the fact that we are merelyinterested in rough estimates, we shall not beconcerned with this point.

Figure 6. (a) Schematics of the initial conditions immediately prior to the initiation of ice. This situation is established within a

day or so after the winds die down, so that there is very little vertical mixing. Along the bottom, there is a layer (H�30 cm) of

warm and salty spring water. The layer on top (H�1 m) is lake water (0.5 PSU). For such a ‘steady’ situation, the diffusion

equation immediately implies that both T and S, as well as the speed u, are linearly distributed in the upper layer. The surface

temperature is taken to be zero because this is what the atmosphere imposes. Similarly, the surface salinity is taken to be

almost zero because these waters are off-shore lake water. The velocity is taken to be zero at the surface because of friction

along the ice–water interface. (b) The conditions during ice formation. The ice thickness, D (on the order of �10 cm) is taken

to be much smaller than H (on the order of 1 m) so that, below the ice, the temperature and salinity profiles are taken to be

the same as those with D = 0 (a). Note that there is no brine rejection, as the water that ultimately freezes is taken to be fresh

water (S = 0).

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Governing equations

Consider the situation shown in Figure 6a, b.Within the ice and within the light layer under-neath the ice, the temperature is governed by,

@Ti

@t¼ ai

@2Ti

@2z2(a)

@Tw

@t¼ aw

@2Tw

@2z2; (b)

ð11Þ

where Ti and Tw are the temperatures of the iceand water, and ai and aw are the diffusivity coef-ficients (a = k /q Cp where k is the thermal con-ductivity, Cp the specific heat and q, the density).An equation similar to (11a) controls the diffusionof salt above the plume. Assuming that the diffu-sion process is quasi-steady, the derivatives withrespect to time are set to zero. This immediatelyshows that the temperatures are linearly distrib-uted both within and underneath the ice. (Thesalinity is also linearly distributed above theplume.) In view of these, the heat flux within theice (ui) is given by,

ui ¼ kiTa

D; ð12Þ

where Ta is the temperature of the air above theice, D is the ice thickness, ki is thermal conduc-tivity through the ice, and the temperature of theice at the ice-water interface was taken to be zero.Similarly, the heat flux in the layer immediatelybelow the ice is

uw ¼ kw

Tw

H; ð13Þ

where H is the thickness of the layer above thespring water (1 or 2 m) and Tw is the temperatureof the salty and heavy spring water below. Notethat, for similar reasons, the velocity is also line-arly distributed above the plume.

Along the ice–water boundary the heat-transferequation can now be written as,

qiki

dD

dt¼ �ki

Ta

D

� �

� kwTw

H

� �

; ð14Þ

where qi is the ice density and ki is the ice heat offusion. The term on the left represents the changeof heat due to ice formation. The first term on theright is the heat flux through the ice (positive whenice is formed because Ta<0). The second is the

heat flux from below. When Tw fi 0 the problemreduces to the Stefan’s (1890) problem. When Tw isnot a constant but rather a function of time andheight, the problem becomes Welander’s (1977)oscillatory system. In our case, Tw is a constant of26 �C. Note that the advection of heat by thehorizontally moving water immediately below theice has been ignored because the flow there islaminar and, just like the temperature and salinity,the speed vanishes along the ice–water interface.

Setting, for convenience,

a ¼ kwTw=Hqiki; b ¼ kiTa=qiki ð15Þ

Eq. (14) becomes

DdD

dtþ aDþ b ¼ 0; ð16Þ

which is a nonlinear differential equation in D.Note that both terms defined by (15) are taken tobe constants (i.e., they do not vary with time). Wesee that, even though the term that was added toStefan’s equation is linear [second term on the leftside of (16)], it has changed the character ofthe equation from nonlinear homogeneous intoa nonhomogeneous one. As recognized byThorndike (1992), this difficulty can be overcome,however, by first re-writing (16) in the form,

dt ¼ �DdD

aDþ b: ð17Þ

Since the right-hand side is only a function of Dand the left is only a function of t, we can simplyintegrate (17),

Z

0

t

dt ¼ �Z

0

D

DdD

aDþ b; ð18Þ

where we took D = 0 at t = 0 as the initial con-dition for ice formation. The solution is,

t ¼ �D=aþ ba2

ln 1þ aDb

� �

; ð19Þ

which, for reasonable values of a, b, and D, isshown in Figure 7. Note that the second term inthe brackets is negative for ice growth because b isnegative (Ta<0). This means that a D must besmaller than |b|. When a D>|b| then more heat issupplied to the ice from below than the heat re-moved from the top, and, consequently, the icemelts. Of course, (17) describes both melting and

433

freezing, but the initial condition that we took in(18) corresponds only to freezing. (Meltingrequires that D will not be zero at t = 0.)

When the heat supplied from below is muchsmaller than the heat removed by the cold airabove (i.e.,a D<< |b|), we can expand the sec-ond term in (19) to get,

t ¼ �D2

2b1þ 2aD

3bþ a22

4b2þ � � �

� �

: ð20Þ

For a fi 0, (20) reduces to Stefan’s relationship asshould, of course, be the case. In the next section,

we shall use the results of (19), as shown in Figure7, to calculate the probability of a springs ice.

Before doing so, it is appropriate to point outthat, regardless of the plume, the shallow regionsof any high-latitude lake under moderate windconditions always freeze over first. This is becausethe wind mixes the water down as far as 20 m thusretarding the freeze in the deep parts of the lake(see e.g., Omstedt 1998, 1999). This may appear atfirst to be relevant to our process but it really hasno bearing on the problem at hand because itrequires much more initial cooling than our

Figure 7. Predicted ice thickness (D) as a function of the air temperature and time according to (19). The upper panel shows the ice

growth for a no-wind situation corresponding to a molecular diffusivity underneath the ice. Although this is the situation which we

argue is relevant to Lake Kinneret, for completeness we also show the ice growth plot for an eddy diffusivity which is five times higher

(lower panel). Point I corresponds to the cooling spell scenario which we, fairly arbitrarily, chose here. In making this choice we

assumed that, in order to support a person, the ice thickness needs to be larger than 10 cm.

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springs freeze does. As already alluded to, thisfamiliar freezing process requires getting the entirelake down to 4 �C before freezing starts along theshore. (Our freeze does not require such a severeinitial cooling.) As a result, the likelihood thatsuch a shore line freeze occurred during the past12,000 years is virtually zero. Furthermore, inthese high-latitude lakes the cooling processes lastweeks or even months so wind action is importantwhereas, in our case, the absence of wind for a fewor several days is a common situation, particularlyduring a cold spell.

Recurrence times for springs ice

In this section we shall compute the likelihood of acold spell strong enough to produce springs ice.We shall focus on the scenario shown in Figure 7.The 17-year time series data plotted in Figure 8 aremean daily air temperatures at Tabgha for Octo-ber 1986–December 2003 (as obtained from Me-korot, Israel Water Company). Our approach is tofit a seasonal time series model to these data, thento simulate output from the fitted model undervarious cooling events. Note that a ‘cooling event’is very different from what we refer to as a ‘coolingspell.’ A ‘cooling event’ corresponds to paleocli-matological cooling over a period of order of100 years or more (Figure 2), whereas a ‘coolingspell’ is a brief (on the order of few days) period of

below-freezing temperatures. The two are dis-tinctly different and it is important to keep thedistinction in mind as we shall focus on situationswhere both occur at the same time.

Simulated output from the statistical fittedmodel, over a 100,000-year time period, is used toestimate the expected number of years betweentwo freezing events (the recurrence time). Weconsider a situation where the average air tem-perature drops below �4 �C for two days. Thischoice is made on the basis of the calculationsshown in Figure 7 and corresponds to the solid dotshown there (top panel).

First, we decomposed the time series data intoseasonal and residual components using theS-PLUS function stl (Venables and Ripley, 2002,Section 14.3). The seasonal pattern was found tobe unstable over the last 8 years, but fairly stablebefore that. Thus, only the first 9 years of datawere deemed suitable for use in the subsequentmodel fitting. The residual, after removal of theseasonal component, was then modeled as a lag-3autoregressive process, i.e., AR(3). After removalof both the seasonal and AR(3) components, theresidual plots displayed some wild transitorybehavior (as often seen, for example, in financialtime series data), so we used the S+GARCHmodule to add a GARCH (1,1) component withnormal conditional distributions to further refinethe model (see Venables and Ripley, 2002, Section14.6, for further discussion and references on this

Figure 8. Tabgha daily mean air temperature (�C) vs. years, from October 1, 1986 through December 2003.

435

type of statistical time series modeling). Notethat after removal of the seasonal component andthe overall mean, the fitted AR(3) model is,

Xt ¼ 0:905Xt�1 � 0:299Xt�2 þ 0:136Xt�3 þ R3:

The fitted GARCH (1,1) model for the residualRt is,

Rt ¼ 0:042þ gt; gt ¼ rtet;

ðrtÞ2 ¼ 0:037þ 0:116ðgt�1Þ2 þ 0:876ðrt�1Þ2;

where the et are independent N(0,1) random vari-ables.

Our simulation results for the final fitted timeseries model give the expected recurrence times forsprings ice under mean air temperature cooling inthe range of 2–20 �C. As mentioned, these coldevents need to be distinguished from the cold spellsthat would cause a temporary freeze, because thecold events represent the mean cooling which tookplace over many years (as shown in Figure 2). Wefind that a mean 3 �C atmospheric cooling (i.e.,events 1, 2 and 3 subject to our lower air/watertemperature ratio of 1.5) is likely to produce (onaverage) springs ice every 119 years. For the high(upper bound) ratio of air/water temperatureanomaly (4), the atmospheric temperature anom-aly is fairly large (8 �C) and the correspondingrecurrence time is short, once in 14 years. Simi-larly, for a mean cooling of 7 �C (event 4 and 5subject to our lower ratio of 1.5), the recurrencetime is 20 years. A 20 �C air temperature reduc-tion (i.e., our high water/air temperature ratio of1:4 applied to events 4 and 5) corresponds to acontinuous production of springs ice during theentire winter. (A rough heat budget calculationshows that, with a 20 �C cooling event, freezingcovering the entire lake might have also beenpossible but our model does not apply to suchconvection and freezing process so we cannot saymuch about it.)

To arrive at our final recurrence time, we nowneed to incorporate the no-mixing conditionimplying that the cooling spells take place at timeswith no winds of more than 6 m/s. To calculatethis, we took the winter (1st of November to the1st of March) hourly wind intensity data from5 years (January 1999–December 2003) andlooked at the number of times that a period ofconsecutive 48 h elapsed without the wind inten-sity exceeding 6 m/s even once. This gives 73% as

the probability that the intensity of the winds willnot exceed 6 m/s in a 2 days period. Using thesame data set, we then looked at the correlationbetween times of strong winds and periods ofstrong cooling. Like many other parts of thenorthern hemisphere (e.g., the USA), typical win-ter cooling is associated with a low pressure systempassing over the region. First, the regionexperiences the outer parts of the low which in-volves strong winds and then, after a few daysduring which the winds died down, it experiencesthe coldest temperatures. This phase lag suggeststhat our freezing period will not be associated withstrong winds.

In view of these, we get that the recurrence timefor a freeze during events 1 and 2 and 3 is onceevery 163 years (i.e., 119/0.73) for the smallest(conservative) choice of the air temperatureanomaly (3 �C) and once in 19 years for the largest(least conservative) choice (8 �C). During events 4and 5 the recurrence time is once in 27 years forthe smallest (conservative) choice of atmosphericanomaly (7 �C). For the largest (the least conser-vative choice of 20 �C), there were continuoussprings ice forming during the entire winter. It isdifficult to tell from Figure 2 just how long coolingevents 1, 2, and 3 were. However, since 100 years iswithin the resolution of that figure, it is fair toassume that the length of these events is on theorder of 100 years, implying a large likelihood ofsprings ice occurring during events 1, 2 and 3. Thisis also the case for events 4 and 5.

Summary

We have presented two separate lines of investi-gation. The first is a rigorous theoretical andanalytical investigation of the likelihood of iceformation above the salty springs. The second is aless rigorous examination of the idea that such afreeze was the origin of the walking on water story.

With the idea that much of our cultural heritageis based on human observations of nature, wesought a natural process that could perhaps ex-plain the origin of the account that Jesus Christwalked on water. Our general approach is similarto that we adopted earlier to explain the parting ofthe Red Sea where we proposed that a strong windaction caused a wind set-down which exposed ausually submerged ridge (Nof and Paldor 1992). It

436

is also similar to Ryan and Pitman (1998), whoattempted to explain the biblical flood arguing thatit originated in the abrupt initiation of strong flowsinto the (then fresh water lake) Black Sea at thevery end of the last glaciation (about 6000 yearsago). These strong flows came about when, due tothe melting of ice sheets, the sea level rose abovethe sill depth in the Bosphoros (Ryan et al. 2003).Neither one of these theories provide an exactexplanation of the biblical stories – our own RedSea parting theory does not fit the biblical story interms of the wind direction, and Ryan and Pit-man’s explanation does not fit in terms of its lim-ited nonglobal extent. Similarly, our presentexplanation does not exactly address ‘walking onwater’ but rather provides a plausible physicalprocess that has some characteristics similar tothose described in the New Testament. Despitethese differences, and mismatches, we believe thatall of those explanations add to our understandingof our own and our ancestors’ lives.

Recognizing that a significant part of Christ’slife was spent in Tabgha next to Lake Kinneret,the thought that limited parts of the lake frozeduring some past cooling spells came to mind.

Initial calculations showed that, due to convection,which is a common pre-freezing condition infreshwater lakes, a total freeze of the entire lake oreven isolated parts next to the shore was notpossible because there was not sufficient cooling tobring the lake down to 4 �C. Noting that the dense(warm and salty) Tabgha springs (Figure 1), whichuntil the 1960s emptied into the lake, can provide alocalized shield from convection, we focused onthe dynamics of the region next to the springs.

Since the springs water is heavier than the lakewater, they set up a stratified region above theirassociated plumes (Figure 6a). As the T�S dia-gram (Figure 4) shows, this stratification is abarrier to convection (see also Figure 6b). In theseregions above the plumes, freezing during events 1,2 and 3 appears to be possible because only the topwater had to be cooled down to the freezing point,i.e., the entire lake stayed much warmer than 4 �C.Using the nonlinear one-dimensional ice-waterequations, we derive a solution to the ice growthproblem [Figure 7 and (19)]. It shows, for exam-ple, that, although the top is warmed from belowby the spring water, atmospheric cooling down to�4 �C for merely 2 days is sufficient to produce an

Figure 9. Typical recurrence time of springs ice in Lake Kinneret as a function of mean air temperature cooling. Note that these are

cooling spells and not cooling events. The wild fluctuations in the plot represent Monte Carlo error which decreases with cooling. The

upper line represents the case where the lake (above the plume) is cooled to �2 �C for 9 days whereas the lower represents the case

where the lake is cooled to �4 �C for 4 days. These are hypothetical cases and are shown here merely for clarity. The value that we

actually chose for our detailed calculation (�4 �C cooling for 2 days) is discussed at length in the text and is not shown here.

437

ice cap which is 10 cm thick, i.e., thick enough tosupport human weight.

It is, of course, desirable to extend these resultsof our analytical (nonlinear) one-dimensionalmodel to three dimensions (particularly to examinethe question of whether a significant amount ofnew lake water can be advected from the sides intothe region above the plume while the freezing istaking place). This can be done numerically but isleft as a subject for future investigation becauseprevious three-dimensional high-resolutionnumerical studies of ice formation have shownthat the difference between their predictions andthose of the analytical one-dimensional models isfairly limited (e.g., Omstedt, 1999).

We also performed a statistical analysis basedon both air temperature records and wind records,and concluded that the above cooling spells had arecurrence time of once every 163 years during thefirst (1500 years ago), second (2500 years ago) andthird (9800 years ago) cooling events (Figures 2, 8and 9). Since these cooling events appeared to havelasted longer than our most conservative estimateof a 163-year period, we concluded that the like-lihood of a springs ice occurring during the lasttwo cooling events is substantial. During theYounger Dryas, springs ice formation was morecommon.

Because the size of the springs ice (which can, bythe way, be partially anchored to the shoreline) isrelatively small (30 m), a person standing or walk-ing on it would appear to a distant observer to be‘walking on water.’ This is especially true for thosewhowere not used to see ice on the lake, particularlyif it rained after the ice was formed. In a way, this issimilar to what is frequently observed today in highlatitude lakes – when it rains, the ice surface be-comes so smooth that it appears to a distant ob-server as if skaters are ‘skating on water’.

We hesitate to draw any conclusion regardingthe implications of this study to the actual eventsthat took place at Tabgha during the last few (orseveral) thousand years. Our springs ice calcula-tion may or may not be related to the origin of theaccount of Christ walking on water. The wholestory may have originated in local ancient folklorewhich happened to be told best in the ChristianBible. It is hoped, however, that archeologists,religion scholars, anthropologists and believerswill examine such implications in detail.

Acknowledgements

This work could not have been completed withoutthe help of many individuals. Steve VanGorderhelped with the calculations leading to Figure 7and critically read the manuscript more than once.Through Benjamin TalTesch, Mekorot, IsraelWater Company, provided us with both the airtemperature record and the wind record, as well asinformation regarding the temperature, salinity,and mass flux of the springs. Communicationswith Matthew Huber, Alon Rimmer, Ayal Anis,Lakshmi Kantha, Kevin Speer, David Lea, JeffChanton, Georges Weatherly, Lou St. Laurent,John A. Young and Dale Haidvogel were all veryhelpful. (Note, however, that they never reallyknew exactly what we were working on so theircommunications with us do not necessarily reflecttheir support of our views.) Important commentson an earlier draft were made by Hezi Gildor;Barbara French provided comments on the socialaspects of the work. Although none of our grantswere specifically awarded with the purpose ofsupporting this particular study, our recent re-search has been supported by the National ScienceFoundation Grants OCE- 9911324 and OCE-0241036, the National Aeronautics and SpaceAdministration Grants NAG5-7630 and NGT5-30164, and the Binational Science Foundationgrant BSF 96-105. Both Florida State University(through the Oceanography Department, theGeophysical Fluid Dynamics Institute and theStatistics Department), and the Hebrew University(HU) provided support through our regular fac-ulty appointments, and the Ring Foundation atthe HU supported DN’s recent visit to Jerusalemin 2003.

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