This paper uses derived distribution analysis to explore the process controls of the onset of and recovery from hypoxic conditions in Tokyo Bay, Japan. A conceptual, lumped model of dissolved oxygen (DO) dynamics in Tokyo Bay is proposed and, through comparison with a three-dimensional simulation model, is verified to have sufficient accuracy for the prediction of the onset of and recovery from hypoxia. This conceptual DO model was implemented in continuous simulation mode, with 14 years of wind data and data on streamflow entering the Tokyo Bay, and was used to identify and quantify the various process controls of the onset of and recovery from hypoxia. The underlying process controls were identified to be streamflow discharge, as well as duration and strength of both northeast “positive” winds and southwest “negative” winds. The analysis helped to isolate, in particular, the potential for rapid and strong recovery from hypoxia due to “strong negative winds”, (i.e., negative winds that exceed a wind speed threshold of 10 m s−1) and the critical roles of the duration of these strong winds and the antecedent DO concentration on the strength of DO recovery. Motivated by these results, derived distribution analysis is adopted to predict the strength of DO recovery during periods of strong winds, using a simplified model of DO recovery, focused on isolated strong winds, that explicitly captures the effects of both wind duration and antecedent DO concentration.
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 The calmness of waters associated with enclosed bays contributes to their development as focal points for marine transportation, leading to increasing human populations centered on these locations and to the development of large cities. One of the common consequences of increased human activity resulting from such development is excessive nutrient inputs into enclosed bays through, for example, riverine inflows and effluent discharges. Although the open ocean by itself is generally nutrient poor, the suppression of seawater exchange between the inside and outside of an enclosed bay leads to nutrient accumulation, contributing to the general eutrophication of enclosed bays [Wolanski, 2006; Okada and Nakayama, 2007]. In such eutrophied systems, algae blooms are a common occurrence during summer due to strong sunshine, which causes the settlement of detritus and the accumulation of sludge and polluted sediments at the seafloor [Okada et al., 2009].
 The accumulation of sediments and the accompanying nutrients lead to consumption of dissolved oxygen (DO) from the water body adjacent to the seafloor [Aure et al., 1996]. Thus, in an enclosed bay, DO concentrations decrease with increasing stratification, with the result that hypoxia develops in the lower layer adjacent to the seafloor, leading to harmful impacts on both aquatic plants and animals. Under these circumstances, upwelling induced by wind stress can sometimes bring this hypoxic water to the surface, where it is then transported outward and onward due to horizontal circulations within the bay, causing the spread of hypoxic water at the surface over large areas, which is a major public health concern [Stevens and Imberger, 1996; Okely and Imberger, 2007; Maruya et al., 2010]. Therefore, it is highly important that the mechanisms leading to the occurrence and transportation of hypoxic waters are clarified, the role of external forces such as wind stress and riverine freshwater discharges quantified, and the episodic occurrence of and recovery from hypoxic conditions predicted. This provides the motivation for the work presented here.
1.1. Environmental Problems in Tokyo Bay
 Tokyo Bay is an excellent example of an enclosed bay facing the kind of environmental problems outlined above (Figure 1). The ecological health of Tokyo Bay has been greatly impacted by the rapid development of the Tokyo Metropolitan Region. Increased nutrient loading has led to the accumulation of organic matter on the seafloor around the head and center of the bay [Wolanski, 2006; Furukawa and Okada, 2006; Nakayama, 2006a] and, as a result, Tokyo Bay can be considered as a eutrophied enclosed bay. Hypoxia often appears in the lower layer of the bay head due to the consumption of oxygen at the seafloor, with adverse consequences for water quality and ecological health in general. The cutoff value of DO concentration for the occurrence of hypoxia is generally taken as 2 mg L−1 based on the tolerance of the benthic macrofauna in Japan [Diaz and Rosenberg, 1995; Furota, 2003]. However, the cutoff value can vary depending upon the species of interest, such as bivalves for which the value could increase to about 3 mg L−1; for this reason the more stringent 3 mg L−1 cutoff is used in this paper as the criterion for hypoxia [Furota, 2003].
 Previous studies have demonstrated that there are three important factors that influence the fluctuations of DO concentrations within Tokyo Bay: wind stress, riverine discharge, and intrusion from the ocean. Sato et al.  highlighted the different roles of southwest winds (hereinafter called negative winds), northeast winds (hereinafter called positive winds), riverine freshwater discharges, and their combined effects on the onset of and recovery from hypoxia in Tokyo Bay. Fujiwara and Yamada  demonstrated that intrusion from the ocean could also potentially contribute to subsurface hypoxia.
 Although intrusions from the ocean can have a high impact on hypoxia, they do not occur too frequently. As mentioned above, riverine discharge can also be a significant factor, but previous observational and modeling studies indicate that wind stress remains the biggest contributor to both the onset of hypoxia and recovery from it [Sato et al., 2007]. Indeed, Sato et al.  presented a highly complex picture of the effects of wind stress (both positive and negative) and steam flow on the onset of and recovery from hypoxia. In particular, they highlighted the role of strong negative winds above a threshold of 10 m s−1 in bringing about rapid and strong recovery from hypoxia. The normal tendency is for positive winds to enhance exchanges with the deep ocean and thus help maintain high DO concentrations, whereas negative winds tend to suppress exchanges with the deep ocean and thus contribute to the onset of hypoxia. However, the same southwest winds can actually play a completely opposite role once wind speed exceeds the threshold value of 10 m s−1, causing the overturning of stratification and the triggering of strong exchange with the open ocean, and resulting in rapid recovery from the hypoxic conditions that the bay had been experiencing during the preceding period of sustained (regular) negative wind. Even though sudden overturning of stratification and rapid recovery are rare events, their impact on overall DO concentrations in Tokyo Bay outside of these times, and on ecosystem health in general, can be quite strong. Therefore, in order to predict the onset of and recovery from hypoxia, it is necessary to develop a predictive understanding of the critical role of strong southwest winds, i.e., winds with speeds in excess of the 10 m s−1, and the factors that control the frequency and strength of the recovery that results from these “strong wind” events. This is the particular focus of this paper.
1.2. Outline of the Paper
 As outlined above, the onset of and recovery from hypoxia are clearly and strongly governed by the stochastic variability of wind speed and direction: Northeast (positive) winds, southwest (negative) winds, and strong negative winds over a threshold of 10 m s−1. In particular, the occurrence, duration, and intensity of strong southwest winds have been shown to cause strong recovery from hypoxia, through a mechanism that is fundamentally different to the mechanisms that contribute to the onset of and recovery from hypoxia during otherwise normal wind events [Sato et al., 2007]. Since the jump in DO concentrations due to strong southwest winds appears suddenly and episodically, there is a need to fully characterize the stochastic nature of strong southwest winds in Tokyo Bay, and to explore their role in the recovery from hypoxia.
 We approach the problem through a derived distribution approach, widely used in flood frequency analysis research [Eagleson, 1972; Sivapalan et al., 2005], through a combination of a deterministic model of system dynamics, with the main external driver of the system (i.e., wind speed) being treated as a stochastic input. In section 2 we present a simple conceptual model of DO dynamics in Tokyo Bay, which will serve as the basis of the stochastic simulations, and verify the adequacy of this model for DO simulations by comparing its predictions against predictions of a more complete three-dimensional (3-D) coupled hydrodynamic and water quality model of Tokyo Bay. The first stage of the derived distribution analysis involves continuous simulations of DO dynamics in Tokyo Bay using the above conceptual model implemented with 14 years of observed wind data, the results of which are presented in section 3. The analysis is aimed at characterizing the episodes of the onset of and recovery of hypoxia in considerable detail, their association with periods of positive winds, negative winds, riverine discharge, and in particular, episodes of strong winds that have a possible impact on strong recovery. On the basis of this analysis, and with a focus on the recovery due to strong winds, the derived distribution analysis will then follow an analytical approach, using a simplified version of the conceptual model of DO recovery applicable to periods of strong winds only, as the means to isolate the critical controls of strong recovery, namely the duration of strong wind and the initial DO concentration. Details of this analysis and the associated results are presented in section 4. Finally, section 5 presents a synthesis of the results presented in this paper and the general conclusions that can be drawn from this study.
2. Mechanisms of Estuarine Circulation and Hypoxia in Tokyo Bay: Conceptual Model of Dissolved Oxygen in Tokyo Bay
 In Tokyo Bay, DO concentrations in the lower layer decrease during summer because of the consumption of oxygen at the seafloor due to high amounts of particulate organic matter in the sediments [Nakayama et al., 2005; Laval et al., 2003]. From spring to autumn, stratification is likely to progressively strengthen because of strong sunshine, which inhibits the supply of oxygen from the upper to the lower layers of the bay. The introduction of freshwater, in the form of streamflow from land, is likely to contribute to the transport of the upper layer water toward the open ocean, with the result that deep oceanic water then intrudes into the lower layer to compensate for the withdrawal of the upper layer water, leading to the onset of estuarine circulation (Figure 2a) [Guo and Yanagi, 1998; Nakayama et al., 2005]. Since DO concentrations in the open ocean are saturated due to vertical convection, deepwater intrusion through the above estuarine circulation supplies oxygen to the lower layer of the Tokyo Bay head [Nakayama et al., 2005; Guo and Yanagi, 1998] (Figure 2a).
Unoki and Kishino  estimated the residence time of seawater, due to the above mentioned seawater exchange between the inside and outside of Tokyo Bay, to vary from about 30 days in summer to 60 days in winter. In Tokyo, it rains predominantly in summer, with winters being relatively dry. Since estuarine circulation is a result of longitudinal density gradients, the larger the river discharge the stronger the estuarine circulation (Figure 2b, upper image) [Okada et al., 2010]. The shorter residence time in summer is therefore a direct result of the enhanced circulation that results from the freshwater discharge generated during summer rain events.
 Estuarine circulation due to baroclinic effects triggered by freshwater river inflows may be seen as the default or baseline phenomenon in enclosed bays. Wind stress can additionally influence the strength of this estuarine circulation, since average water depth in enclosed bays is small, for example, only 30 m in Tokyo Bay. Depending on the direction of the wind, wind stress may either enhance or inhibit estuarine circulation, and thereby increase or decrease, respectively, DO concentrations in the lower layer of the bay head [Sato et al., 2007]. Northeast (positive) winds, which happen to be in the same direction as freshwater inflows, thus enhance estuarine circulation, which in turn supplies more oxygenated water from the ocean (Figure 2b lower image). In contrast, southwest (negative) winds, which are in the opposite direction to freshwater discharges, under normal circumstances inhibit estuarine circulation and reduce the supply of oxygen from the ocean (Figure 2c).
 However, the advent of negative winds over a threshold of 10 m s−1, termed here “strong winds”, gives rise to a whole new sequence of steps that eventually result in a sudden increase in DO concentration, and hence recovery from the hypoxia that may have previously developed [Sato et al., 2007]. This is in sharp contrast to regular negative winds of less than 10 m s−1, which, as we have seen before, inhibit oxygen supply from the ocean (Figure 2c). This is a classic threshold phenomenon, where the functioning of the system in question undergoes a sharp regime change once the driver or state variable exceeds a threshold value [Zehe and Sivapalan, 2009]. When strong wind events occur, less dense water is transported into the bay head and the stratification that had previously built up, breaks down. To compensate for the accumulation of the less dense water in the bay head, the water of the lower layer, including hypoxic water, moves toward the ocean (Figure 3). This may cause complete mixing of hypoxic water with oceanic water, inducing sudden recovery of DO concentrations and the consequent alleviation of hypoxia (Figure 3). In other words, strong wind events induce a rapid overturning of the stratification that had been previously built up and trigger an antiestuarine circulation, which drives sudden increases in DO concentrations through mixing of oxygen-rich ocean waters and hypoxic waters in the bay head. At the end of strong wind events, stratification and normal regime of estuarine circulation resumes, as shown in Figure 2a, minus the hypoxic conditions that had prevailed previously.
 It should be noted that the impact of strong wind events on DO concentrations is different from the exchange of bottom water demonstrated in some other previous studies [Torgersen et al., 1997; Hofer et al., 2002; Rao et al., 2008]. Torgersen et al.  demonstrated that the combination of horizontal and vertical transport of oxygen is a significant factor controlling the onset of hypoxia. Deepwater intrusion is also demonstrated as the key factor controlling the exchange of water in the bottom, which may control DO concentrations [Hohmann et al., 1997; Aeschbach-Hertig et al., 1996]. With regard to other factors controlling DO concentrations at the bottom, Boegman et al.  demonstrated the importance of the mixing effect due to the breaking of internal solitary wave trains, which are deformed from low frequency internal waves, such as internal seiches. Nakayama and Imberger  also revealed the importance of mass transport from the littoral to the pelagic zones in the lower layer due to the breaking of internal waves, which may lead to the increase in DO concentration in the lower layer. The sudden increase in DO concentrations studied in this paper is driven by the transport of hypoxic waters from the inside of the bay to the ocean, which causes the mixing of hypoxic water with the high DO concentration oceanic water. Therefore, the sudden increase in DO concentrations arising from strong wind events is a unique mechanism, different from the renewal of bottom water highlighted in previous studies.
 As mentioned above, riverine freshwater discharge, positive winds, negative winds, and strong negative winds over a threshold of 10 m s−1, are the predominant controls on the nature of estuarine circulation in Tokyo Bay, and hence control the dynamics of DO concentrations in the lower layer of the bay head. Together these components determine the onset of and recovery from hypoxia and the frequency and strength of these fluctuations. Our objective in this paper is a stochastic characterization of the onset of and recovery of hypoxia in Tokyo Bay, and the elucidation of the underlying process controls. There are several 3-D hydrodynamically based numerical models of estuarine circulation that can be used to simulate DO dynamics in Tokyo Bay, including a model that includes dynamic pressure effects [Nakayama, 2006b; Kakinuma, 2008] and a high performance hydrostatic model [Hodges, 2000; Hodges et al., 2000; Simanjuntak et al., 2009]. However, based on their high computational costs and input data requirements, these models are not amenable for the kind of stochastic analyses that are envisaged here nor for the kind of inferences we would like to make. A simpler model that focuses directly on DO dynamics would be more preferable.
 For this reason a simple conceptual model of DO dynamics that is capable of incorporating the important factors governing DO concentrations in the lower layer of the head of Tokyo Bay is proposed (see Figure 4 for a schematic describing the conceptual foundations of this model). Note that in Figure 4, σt (kg m−3) is the density of the waters within the bay minus 1000 kg m−3 as predicted by a 3-D model of Tokyo Bay [Sato et al., 2007]. Therefore σt can be used as a measure of salinity, and is used as a tracer to highlight the interactions between freshwater from rivers and deep ocean water. This model conceptualizes Tokyo Bay, as far as DO dynamics is concerned, as a well-mixed, linear reservoir and captures the dynamic balance between consumption of oxygen at the seabed (a sink term) and the various mechanisms by which oxygen is exchanged between the enclosed bay and the open ocean [Sato et al., 2007]. This dynamic balance is expressed in terms of the following ordinary differential equation,
where DO is the average DO concentration in the lower layer of the bay head (mg L−1), t is time (hr), wind is the wind speed along the longitudinal direction of Tokyo Bay (m s−1), and q is the river discharge (mm hr−1). The lower layer of the bay head is defined as the area enclosed by the thick, black line in Figure 4c. The term ZDO on the right-hand side (RHS) of equation (1) represents consumption of DO at the seafloor, while the term (ADO + XDOwind + CDOq + YDO)(9 - DO) on the RHS of equation (1) combines various components associated with the exchange of DO between the inside and the outside (open ocean) of the bay. The open ocean is assumed to be oxygen saturated, with a DO concentration of 9 mg L−1, and the term (9 – DO) represents the difference in concentration between the inside and outside of the bay. The term ADO is the exchange coefficient associated with the baseline or default estuarine circulation corresponding to a residence time of 30 days (Figure 2a), that is 0.00139 hr−1, excluding the effects of wind-stress and river discharge. XDO is the exchange coefficient associated with estuarine circulation that is induced by (regular) wind stress alone, 0.0008 s m−1 hr−1 (Figures 2b and 2c). CDO is the exchange coefficient associated with estuarine circulation that is induced by river discharge alone, 0.0045 hr−1 (Figure 2b). YDO is the exchange coefficient associated with the recovery of DO concentration due to “strong winds” (Figure 3a), and YDO is set to 0.0417 hr−1 when southwest winds exceed 10 m s−1; it is set to zero otherwise. ZDO is the exchange coefficient associated with the consumption of oxygen by seabed sediment and was measured in a laboratory experiment by measuring the change in DO over sampled sediment at the seafloor. Estimated consumption of oxygen at the seabed is 2 g m−2 d−1, which corresponds to ZDO = 0.012 mg L−1 hr−1 with the average depth of 7 m in the lower layer of the bay head. The simplicity of the model is that all of these exchanges are expressed in terms of constant exchange coefficients, which is clearly an approximation to the overall dynamics that can be captured by the more complete three-dimensional Estuary, Lake, and Coastal Ocean Model and the Computational Aquatic Ecosystem Dynamics Model (ELCOM-CAEDYM) [Sato et al., 2007]. Table 1 presents the coefficients that have been developed for Tokyo Bay on intuitive grounds or based on matching against 3-D model predictions [Okada et al., 2010]. In general, the wind effect, XDO, is a dominant factor in bottom DO concentrations. However, when floods occur or “strong winds” dominate, the contributions rate of CDO and YDO becomes larger than the wind effect, XDO.
= 1/(24 × 30); derived from resident time from April to October, 30 days. ADO is set to the higher value of 0.0047 hr−1 from November to March
XDO = 0.0008 s m−1 hr−1
Estimated by trial and error
CDO = 0.0045 mm−1
= (watershed flowing into Tokyo Bay)/(volume of the lower layer of Tokyo Bay)
YDO = 0.0417 hr−1
Estimated by trial and error; YDO is set to 0.0417 hr−1 when southwest winds exceed 10 m s−1 and is set to zero otherwise
ZDO = 0.012 mg L−1 hr−1
Obtained from field measurement
 In order to ensure the representativeness of the conceptual model, predictions of the average DO concentration in the lower layer of the bay head by the conceptual model were calibrated against predictions by a 3-D model developed and tested previously [Sato et al., 2007]. The features of the 3-D model are: (1) It is a hydrostatic three-dimensional coupled hydrodynamic and ecological model; (2) it uses a mixed-layer turbulence closure scheme and a z coordinate system to inhibit numerical diffusion [Adcroft et al, 1996; Simanjuntak et al, 2009]; and (3) it uses standard meteorological data, tidal data, and fresh water discharges for the specification of boundary conditions. In the ecological part of the model, phytoplankton, ammonium, nitrate, filterable reactive phosphorus, particulate organic carbon, particulate organic nitrogen, particulate organic phosphorus, dissolved organic carbon, dissolved organic nitrogen, dissolved organic phosphorus, silica and dissolved oxygen data are included.
 The 3-D simulations were conducted over the spring to autumn period of 2003 (Figure 5), and showed very good agreement between the vertical profile of density and field observations at station A (shown in Figure 1). In fact, vertical profiles of density were measured at 27 stations in Tokyo Bay once or twice a month from May to August 2003; all field observations were compared to the 3-D model predictions and were found to agree very well. Also, we made a comparison of DO concentration at the bottom of station B (shown in Figure 1). That comparison also showed good agreement between field observations and the 3-D model predictions (Figure 6).
 It should be noted that the 3-D numerical computations can also be used to reveal the details of disappearance of hypoxia in the lower layer of the bay head. Figure 7 presents the dynamics of estuarine circulation within Tokyo Bay during and after the strong negative wind event of 18–22 June 2003, and highlights the space-time dynamics of salinity and DO concentrations. Before the strong wind event, hypoxia was found to exist due to consumption of DO at the seafloor (Figures 7a and 7b). A strong wind event brings less dense water into the bay head, which induces compensating flow from the bay head toward the ocean at the sea-bottom (Figures 7c and 7d). The hypoxic water body was moved toward the ocean by the compensating flow, which enhances mixing with the DO saturated sea water. Following completion of the strong wind event, the now-mixed hypoxia returned toward the bay head, and contributed to the recovery from hypoxia that occurred on 22 June 2003. These results tend to confirm the validity of the assumptions behind the conceptual model proposed here.
 The results of model implementation during the spring to autumn period of 2003 are presented in Figure 8a, which includes a comparison of the predictions by the 3-D model with those of the conceptual model. Apart from the exchange coefficients, which are prescribed, the main inputs are the observed time series of freshwater discharge and wind speed, which were presented in Figures 3b and 3c, respectively. The initial value of DO concentrations in the bay as of 1 April is assumed to be the saturation value of 9 mg L−1.
 First of all, it is quite remarkable that, in spite of the gross simplifications made about the DO dynamics, the conceptual DO model is able to match the predictions of the 3-D model very well, giving us confidence about its ability to reproduce the statistics of the onset of and recovery from hypoxia. A number of distinct features can also be recognized in the results presented in Figure 8. Wind speed profiles shown in Figure 8c show alternative fluctuations between positive and negative winds, but they show three episodes of strong negative winds (>10 m s−1) on or about 10 May, 20 June, and 10 August. Two of these episodes, 20 June and 10 August, happened when DO concentrations had dropped to hypoxic values (i.e., 3 mg L−1), whereas DO values were much higher on 10 May. Figure 8a shows that there was a rapid recovery of DO values to almost saturated conditions on 20 June and 10 August, whereas the strong winds on 10 May did not cause a major jump in DO values (since they were high already).
 Apart from the recovery during strong wind episodes, the DO dynamics show alternative periods of DO decline and mild recovery as well as periods of high DO concentrations. These are illustrated in Figure 8a using indicator bars that link these periods to alternative periods of negative and positive winds, including a combination of period of positive winds and high freshwater discharge. DO concentrations show a general decline from 1 April to 20 June (apart from a brief period of recovery due to positive winds and freshwater discharge around 15–20 May), and generally high DO concentrations during the entire month of July (more than 5 mg L−1) due to the combination of high positive winds and persistent freshwater discharge (area enclosed by squares in Figures 8b and 3c). In spite of high freshwater discharges, hypoxia developed quickly from the beginning of August, due to persistent negative winds, culminating in a period of strong negative winds around 10 August that caused a rapid recovery from hypoxia. These results indicate that in spite of the importance of freshwater discharge, wind stress remains the biggest control on the onset of and recovery from hypoxia: High DO concentrations resulting from persistent positive and negative winds contribute to the onset of hypoxic conditions, confirming the results of previous studies based on 3-D simulations [Sato et al., 2007].
3. Stochastic Analysis of Onset of and Recovery From Hypoxia: Continuous Simulations Using Conceptual DO Model
 Having established the ability of the conceptual model to reproduce the DO dynamics generated by the 3-D model, and having understood the relative dominance of the underlying process controls (i.e., freshwater discharge, positive and negative winds, and strong negative winds), we are now in a position to undertake a continuous simulation of DO dynamics using observed wind speeds and predicted streamflows over the 14 year period, 1991 to 2004. In this case, the Automated Meteorological Data Acquisition System (AMeDAS) and meteorological data collected by the Ministry of Land, Infrastructure, Transport and Tourism of Japan were used to compute wind speed over Tokyo Bay and rainfall intensity over the adjoining catchment area that feeds the streams flowing to the bay. Spatial distributions of wind vectors were computed by interpolating wind vectors from 9 stations, which were then used to obtain the average wind speed along the longitudinal direction over Tokyo Bay. River discharge was estimated using a lumped conceptual model of the catchment area (details in Sato et al. ).
 The results of these continuous simulations, along with the corresponding wind speed profiles are presented in Figure 9. Note that in these simulations, the value of ADO was set to the higher value of 0.00417 hr−1 from November to March of the subsequent year, which has the net effect of mixing the upper and lower layers significantly and generating the saturated conditions expected during these winter months (Figure 9a). The saturated condition was confirmed to occur from field observations. Over the spring–summer period, however, the DO concentrations exhibited strong within season fluctuations, no doubt affected by wind speed and freshwater discharges. They also exhibited considerable interannual variability. In 4 of the 14 years (i.e., 1994, 1995, 2000, and 2004) DO concentrations dropped to almost zero, and in two consecutive years (i.e., 1994 and 1995) the hypoxic conditions persisted for many days at a time. On the other hand, episodes of hypoxia were infrequent in 1991–1992, 1997–1998, and 2001–2002. In order to see if this variability is in any way connected to interannual climate variability, such as the El Niño–Southern Oscillation (ENSO) cycle, we have highlighted periods of El Niño and La Nina in Figure 9a. Coincidentally, it appears that the periods of El Niño events are also 1991–1992, 1997–1998, and 2002–2003, pointing to the influence of interannual climate variability on the occurrence of hypoxic periods; no doubt through their impact on wind speeds [Ishii et al., 2005] (Figure 9). This question will be addressed in detail in section 4.2.
 To further understand the statistical characteristics of the onset of and recovery from hypoxic events, we set an hypoxia threshold of 3 mg L−1, which helped to partition the entire study period of 14 years into periods of hypoxia (DO < 3 mg L−1) and recovery (DO > 3 mg L−1), and analyzed the statistical properties of the duration of the hypoxic periods, tDOw, and the recovery periods, tDOb (Figure 10d). The numbers of hypoxic and recovery periods were 28 and 17, respectively (note that these analyses do not include winter periods, which explains the lower number of recovery periods). The mean and standard deviation of the hypoxic periods, tDOw, were computed as 141 hr and 199 hr, respectively, whereas the mean and standard deviation of the recovery periods, tDOb, were 262 hr and 445 hr.
 To find any correlation between the onset of hypoxic events and the underlying causal factors, the duration of the hypoxic periods was plotted against that of the recovery periods that immediately followed. The results do not indicate any correlation (Figure 10a), and suggest that other factors may be controlling the onset of hypoxia. As described in section 3, wind stress largely controls estuarine circulation and thus influences DO dynamics. Since positive wind enhances estuarine circulation, resulting in the supply of more oxygen into the lower layer from the ocean, it is expected that the greater the wind speed the longer the tDOb. The results presented in Figure 10b show that small tDob values are associated with small (negative) average winds, while large tDob values are associated with larger (positive) average winds, and barring a few exceptions, the magnitude of tDOb tended to increase with increasing average wind. Strong wind events were previously shown to cause rapid recovery from hypoxic conditions. In order to gauge the influence of strong winds on the duration of recovery periods, the number of strong wind events was plotted against tDob (Figure 10c). In this case, the relationship was much stronger, that is, the larger the number of strong wind events the longer the tDOb, with the exception of one anomalous result. It so happens that on this occasion there was larger mean wind, which may have enhanced estuarine circulation and resulted in high DO concentrations, overwhelming the effects of the few strong wind events.
 To clarify how often recovery from hypoxia occurs, all 28 recovery events were investigated in detail (Figures 6 and 7). Among these, 15 recoveries were driven by positive wind and freshwater discharge effects (Figure 11). Note that the recovery induced by positive wind and discharge was defined to start and end when DO concentrations were at a minimum (DOinitial) and when DO concentration reached the cutoff of 3 mg L−1, respectively. Further analysis showed that the mean DOinitial was 2.49 mg L−1 for these events, and the rate of recovery of DO concentrations was 0.02 mg L−1 hr−1, which is larger than the consumption rate of DO at the seabed of 0.012 mg L−1 hr−1. Figure 11b presents an excellent and almost linear relationship between the duration of positive wind and ΔDO, the recovery of DO concentrations.
 On the other hand, the number of recoveries due to strong wind events was 13 out of the total. Note that this number is just 3% of the total number of strong wind events (470), excluding those in the winter period (Figure 12). In this case, the mean DOinitial was computed as 1.73 mg L−1 and the rate of recovery of DO concentrations was 0.15 mg L−1 hr−1, which is 13 times the consumption rate of DO at the seabed, indicating that strong wind events had a greater role in the recovery from hypoxia than the combined effects of positive wind and freshwater discharge. In contrast to the positive wind effect, mean DOinitial was found to be smaller prior to strong wind events. The estimated relationship between DOinitial and ΔDO is presented in Figure 13, which reveals that the lower the DOinitial the larger the ΔDO. Therefore, a lower DOinitial is the necessary condition for a recovery from hypoxia, and this is the reason why only 3% of strong winds lead to a recovery from hypoxic conditions.
4. Stochastic Analysis of Recovery Due to Strong Wind Events
 The simulations presented above highlighted the importance of strong wind events to the recovery from hypoxic conditions, which motivated our further stochastic analysis. In order to understand the effects of strong winds on the recovery from hypoxia in more detail, 14 years of hourly wind data between 1991 and 2004 collected by the Ministry of Land, Infrastructure, Transport and Tourism of Japan was analyzed using AMeDAS. The threshold for the strong wind events was chosen to be 10 m s−1, and the threshold for the associated kinetic energy, defined as KEw= wind2, is then 100 m2 s−2, where wind is the wind speed along the longitudinal axis of Tokyo Bay (Figure 14a). Our analyses revealed that strong wind events occurred 813 times during the entire study period (Figure 14a).
4.1. Stochastic Analysis of Strong Wind Events
 The nature of temporal variability of strong winds is similar in at least one aspect to the temporal variability of rainfall, namely that of intermittency. The main difference is that in the case of the precipitation, the rainy or wet periods are separated by zero precipitation, whereas in the case of strong winds the periods of strong winds are separated by periods of not zero, but more moderate, positive or negative winds. There is a rich literature in hydrology, as exemplified by Robinson and Sivapalan  and Sivapalan et al. , that enables the characterization of precipitation variability through identification of discrete events in the rainfall record, and then the quantification of rainfall variability through the use of probability distributions of event durations, interstorm period and mean intensity, and the characterization of within event variability through the use of, for example, random cascade models.
 In this paper, following Robinson and Sivapalan  and Sivapalan et al. , we follow a similar approach and analyze observed wind data to identify episodic strong wind events exceeding a wind speed threshold of 10 m s−1, and then characterize the variability of these strong wind events in terms of the probability density functions of event duration, tw, interevent period, tb, and event averaged kinetic energy, KEw (Figure 14b).
 In order to estimate these probability density functions (pdfs), we analyzed the entire observational record (over 14 years), sampled the 813 strong wind events, estimated the event characteristics, and converted these into histograms (Figure 15). The number of bins for the histogram was chosen using the formula of Sturges ,
where N is the total number of strong wind events, and n the needed number of levels or bins in the histogram, which is 18 in this case.
 In contrast, the gamma distribution was chosen for characterizing the interevent interval (i.e., the time interval between strong wind events) instead of the exponential distribution, guided by the means and standard deviations, which were computed to be 140 hr and 218 hr (Figure 15b and equation (4)), respectively,
where κ is 0.4, ν is 0.00286 hr−1, and fb(tb) is the pdf (hr−1).
 In the case of the kinetic energy of strong wind events, the mean and standard deviation were computed as 143.9 m2 s−2 and 47.5 m2 s−2, respectively, which suggested the suitability of using a truncated form of the exponential distribution, with a threshold 100 m2 s−2 (Figure 15c and equation (5)),
where λKE is 47.5 m2 s−2 and fKE(KEm) is the pdf (m−2 s2).
 Correlation coefficients between the duration tw, interevent interval tb, and kinetic energy KEw of strong wind events were also computed. Correlation coefficients for tw and tb, tb and KEm, and tw and KEm were 0.043, 0.47, and 0.11, respectively. The highest correlation coefficient is 0.47 for tb and KEm. Given that the duration of strong wind, tw, is the quantity that is most influential on the recovery of DO concentration (more than the other components) and since the correlation coefficient between event duration tw and the other two components, tb and KEm, are very small, we can disregard the other components, especially tb and KEm, during any further analysis of DO recovery during strong wind events (Figure 12).
4.2. Stochastic Analysis of DO Recovery Due to Strong Winds: Derived Distribution Analysis
 In Section 3, the role of strong wind in the recovery from hypoxia was demonstrated through continuous simulations using a conceptual model of DO dynamics, suggesting the need to clarify the stochastic characteristics of sudden recovery events. The total number of events of quick recovery from hypoxia due to strong winds from 1991 to 2004 was just 13, which does not permit a complete characterization of the pdf of sudden recovery events. Therefore, in this section we present a derived distribution approach to deriving the pdf of DO recovery analytically. The derived distribution approach builds on straightforward statistical methods available in mathematical statistics for deriving the pdf of y from the pdf of x, where y = g(x), where g(.) is any deterministic function [Mood et al., 1974].
 The derived distribution analysis will use an analytical integration of equation (1) over the duration of strong events only, that is, integration from time zero (the start time of strong wind event) to time tw (end of the event), assuming all other components to be constant. The recovery, ΔDO, of DO concentration from the initial condition, DOinitial is derived as follows:
where A = ADO + XDOwind + CDOq. Equation (6) represents a deterministic, analytical relationship between the DO recovery during strong wind events and duration tw and DOinitial, with all associated coefficients kept at constant values. We denote this function as gΔDO(tw∣DOinitial).
 We have shown that the recovery of DO concentrations from hypoxia due to strong wind is much larger than due to other factors such as normal wind and freshwater discharge. Thus, A can be approximated by ADO by turning wind and q to zero. To test the validity of this approximation, we compare predictions of the simple analytical model (equation (6)) with the assumption of A=ADO, against predictions of ΔDO by the conceptual model (equation (1)) run in simulation mode over the 1991–2004 period, as presented in Figure 12b. The predictions by the simple analytical model (i.e., equation (6)) are presented in Figure 16 for DOinitial values ranging from 0 mg L−1 to 3 mg L−1. Previous estimates by the complete conceptual model are included in Figure 16 in the form of circles. Generally good agreement is found between predictions of the simple model and those of the complete conceptual model. In particular, the average DOinitial of 1.73 mg L−1, highlighted in Figure 12a, is shown to capture the mean relationship between ΔDO and tw.
 To implement derived distribution analysis, we need to invert equation (6) to estimate the duration of strong wind that will be needed to bring about a recovery of ΔDO, for a specified known value of DOinitial. This is given by
which can be denoted as gΔDO−1(ΔDO∣DOinitial). Given the pdf of tw, and the above deterministic relationship(s) between tw and ΔDO, the derived distribution theory [Mood et al., 1974; Eagleson, 1972; Sivapalan et al., 2005] can be invoked to derive the pdf of ΔDO, as follows:
Figure 17 presents the pdfs of ΔDO computed using equation (9) in combination with the pdf of tw given by equation (3) (λw = mean of tw = 7.93 hr), and repeated for DOinitial values ranging from 0 mg L−1 to the cutoff value for hypoxia of 3 mg L−1. Firstly, the results show that ΔDO values of more than 6 mg L−1 during strong winds are rare, which is confirmed by the fact that a sudden recovery of more than 6 mg L−1 occurred only once in the 14 years between 1991 and 2004 (Figures 7b and 12a). Secondly, small DOinitial values tend to favor the occurrence of larger ΔDO values, whereas larger DOinitial values tend to favor the occurrence of smaller ΔDO values (Figure 17a).
 To mimic the effects of long term climate variability and climate change, through their potential impacts on the duration of strong winds, the cumulative distribution function (cdf) of ΔDO was computed, for a fixed DOinitial value of 1.73 mg L−1, for strong wind duration tw. The results are presented in Figure 17b. The mean tw during El Niño years is larger than the overall mean over the 1991–2004 period. The results presented in Figure 17b indicate that strong wind events during El Niño years are likely to result in larger magnitude recovery of DO concentrations than during an average year. The maximum and minimum of the mean tw values over the 1991–2004 period occurred in 1997 and 1994, respectively, and the results of Figure 17b show that these years would correspond to the years with, respectively, the smallest and biggest recoveries from hypoxia due to strong winds.
5. Discussion and Conclusions
 In this paper we have utilized a series of models of DO dynamics in Tokyo Bay to elucidate the controls on the onset of and recovery from hypoxia. The first step in this direction was the development of a conceptual model of DO dynamics in Tokyo Bay, and its validation through comparison against a more detailed 3-D model of DO dynamics. This was followed by the continuous simulation of DO dynamics in Tokyo Bay with the use of the simple conceptual model, driven by observed wind speed profiles over a 14 year period, which helped to understand the process controls on the fluctuations of DO.
 The key controls on fluctuations of DO concentrations and alternative periods of hypoxia and recovery were: (1) Freshwater discharge; (2) northeast (positive) winds; (3) southwest (negative) winds; and, in particular, (4) strong negative winds. Persistent negative winds tend to suppress estuarine circulation, contributing to the onset of hypoxia. Recovery from hypoxia can come about through persistent positive winds, assisted by freshwater discharges into the bay, which together enhance estuarine circulation and help keep the DO high. Continuous simulations with the conceptual model over a 14 year period highlighted the crucial role played by strong winds, that is, southwest winds whose speeds exceed 10 m s−1. Recovery during strong wind events is brought about by complete overturning of the stratification that prevails in enclosed bays and the consequent reversal of the normal regime of estuarine circulation, which thus represents a threshold phenomenon of the functional kind [Zehe and Sivapalan, 2009] that occur when a system undergoes a sharp regime change once the driver or state variable exceeds a threshold value.
 The final step was the development and use of a simple, analytical model that focused explicitly and exclusively on the recovery from hypoxic conditions during strong winds, and the factors that contribute to the recovery. Derived distribution analysis was carried out using the pdf of the duration of strong winds estimated from observed wind data. This theoretical analysis helped to isolate the critical role played by the duration of strong wind events and antecedent DO concentrations: the longer the duration of strong winds, the stronger is the recovery of DO concentrations. Also, the analyses demonstrated that antecedent DO concentration was the most important component influencing the rapid recovery from hypoxia during strong wind events: the lower the initial DO concentration, the higher is the jump in concentration as a result of strong (negative) winds.
 The duration of strong wind events is the most significant component for characterizing the current condition of water quality in Tokyo Bay. If the duration of strong wind events is shorter than usual, one can expect that hypoxia will tend to occur more often than usual. However, when the total duration of hypoxia is the same as in previous years although the duration of strong winds is shorter than in previous years, the results obtained here suggest that the consumption of DO concentration at the seabed has decreased, which demonstrates that water quality in Tokyo Bay may have been improved. In this way the analytical results can assist toward making policy by taking into account the current condition of water quality. The results presented here are also valuable for the long-term management of environmental problems, including through anticipation of potential effects of long-term climate variability (e.g., ENSO events) and climate change that may give rise to major changes in the frequency and strength of strong winds in the future, with implications for the onset of and recovery from hypoxia.
 In this paper, we showed how a series of analysis steps systematically helped to illuminate the factors that control the onset of and recovery from hypoxic conditions in Tokyo Bay. These analysis steps involved a progressive telescoping of important factors describing oxygen dynamics in Tokyo Bay, leading to the use of increasingly simpler models and analysis approaches, but with a sharper focus on dominant process controls: From a 3-D model of hydrodynamics and water quality, to a lumped conceptual DO model, and finally to a simple analytical model that is applicable during strong wind events only. This sharpening of focus helped to increase our process understanding and also to improve the fidelity of the predictions.
 We wish to thank C. Dallimore for the ELCOM-CAEDYM computation. This work has been supported by the Japan Society for the Promotion of Science.