{"id":49,"date":"2022-12-30T18:15:03","date_gmt":"2022-12-30T18:15:03","guid":{"rendered":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/stable-and-unstable-density-configurations\/"},"modified":"2023-01-09T05:30:05","modified_gmt":"2023-01-09T05:30:05","slug":"stable-and-unstable-density-configurations","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/stable-and-unstable-density-configurations\/","title":{"raw":"4.2 Stable and Unstable Density Configurations","rendered":"4.2 Stable and Unstable Density Configurations"},"content":{"raw":"
\r\n

The Simmons and others (2002b) experiment provides an example of a potentially unstable density layering where higher density water overlies lower density water. The reason why the term \u201cpotentially unstable\u201d is used (as opposed to unconditionally unstable)\u00a0is that there are other factors, such as the permeability of the sediment and the background flow field, that determine whether gravitational instabilities as shown in Figure\u00a08b<\/a> will develop and amplify. These additional factors are considered in Section\u00a05<\/a>.<\/p>\r\n

On the opposite side of the spectrum are stable density configurations, where the lower density fluid rests on top of higher density fluid. The difference between the two regimes is illustrated in this video<\/span><\/a>, which is an animated version of the snapshot at 10,000 years shown in Figure\u00a09. The temperature is color-coded. The left and right images show the temperature distribution in a 500 \u00d7\u00a0500\u00a0m cross-section through a homogeneous aquifer with no external hydraulic head gradients to drive forced convection. In the unstable\u00a0regime of Figure\u00a09b, the temperature is fixed to 1.5\u00a0\u00b0C along the top and the system is heated from below by a heat flux of 100\u00a0mW\u00a0m\u22122<\/sup>. Conditions like these occur, for example, below the bottom of the world\u2019s oceans underlain by basalt aquifers (Fisher and Geoffrey, 2010). In the stable regime shown in Figure\u00a09a, the upper and lower boundary conditions are reversed, so the temperature is fixed along the bottom and the heat flux is applied from above. Because the density of water decreases with increasing temperature (for pure water this is true above 4\u00a0\u00b0C but for seawater, this is true for all temperatures above freezing point) the warmer water over cooler water in the system on the left represents the stable regime. Because the groundwater is stagnant in Figure\u00a09a, locations throughout the flow system where flow vectors would be plotted appear as dots (i.e., the groundwater has zero velocity, thus the flow vector has no magnitude nor direction). In the absence of flow, the rock and the groundwater convey heat only by conduction.<\/a><\/p>\r\n

\"Figure<\/p>\r\n

Figure\u00a0<\/strong>9<\/strong>\u00a0-\u00a0Stable and unstable temperature fields 10,000 years after heat was applied to the boundary. a) Heat conduction in a stable state where heat is applied to the top so water density is lower at the top; and, b) free convection cells in an unstable state where heat is applied to the bottom so water density is lower at the bottom. The Rayleigh number (discussed in Section\u00a05.2<\/a>) is on the order of Ra<\/em>\u00a0=\u00a03\u00a0\u00d7\u00a010<\/span>2<\/sup>. Video at this link<\/span><\/a> provides the animated version of this figure.<\/p>\r\n

In the case of the unstable regime shown in (Figure\u00a09b), the situation is entirely different. The video at this link<\/span><\/a> shows that the system is initially stable as the aquifer is heated from below. Then, after approximately 2250\u00a0years, the horizontal temperature stratification is disrupted and the system transitions into a new regime in which there is vigorous, circulatory groundwater flow. Cold water is transported downward in the center of the aquifer and warm groundwater rises along the left and right boundaries. The video shows that the temperature distribution does not change after approximately 5000\u00a0years, nevertheless, the convective flow pattern persists as evidenced by the circulating particles. In Figure\u00a09b, this flow pattern is shown by flow vectors, which in contrast to the stable case of Figure\u00a09a, are arrows with magnitude and direction indicating the groundwater is in motion.<\/p>\r\n

The counter-rotating cells, shown in Figure\u00a09b, are called free convection cells.<\/em> Cooler water from the top of the aquifer descends to replace the warmer fluid that rises. Something similar happens when you heat water to cook or make a cup of tea. Just before the water starts to boil, you may recognize the water movement that is similar to the pattern displayed in Figure\u00a09b, but because all the water has the same color, the convection cells can be hard to see. A clearer and more entertaining household example is the recipe for a marble cake. When alternate layers of vanilla and chocolate cake mix are carefully put more or less horizontally in a baking tray, the result after baking is a pattern where counter-rotating cells are formed in the baking process (Figure\u00a010). It is noted that this example is not entirely representative of groundwater because of the phase change that occurs (from liquid to solid when baking a cake).<\/a><\/p>\r\n

\"Photograph<\/p>\r\n

Figure\u00a0<\/strong>10<\/strong>\u00a0-\u00a0Free convection cake. <\/a>Box 1<\/span><\/a> provides the recipe.<\/p>\r\n

A classic example of a stable density configuration is encountered in coastal aquifers. In this case, dense seawater water sits under less dense fresh groundwater, which creates a seawater wedge that can penetrate inland from the coast for some distance. Coastal aquifer hydrogeology is extensively treated in books (Bear et al., 1999; Cheng and Ouazar, 2004; Jiao and Post, 2019) as well as in literature reviews on this topic (Reilly and Goodman, 1985; Werner et al., 2017, 2013). The quintessential conceptual model of the salinity distribution and flow pattern in groundwater in an aquifer near the sea is depicted in Figure\u00a011. It shows the results of a model simulation in which there is lateral inflow of fresh water across the left boundary, as well as across the top boundary of the landward part of the aquifer. On the seaward part of the aquifer, the hydraulic head along the top boundary is equal to sea level. The seawater wedge protrudes inland from the coast. As presented earlier in Figure\u00a01<\/a>, the higher pressure of saline compared with fresh groundwater at the same depth is responsible for the formation of the wedge. The rotational flow that was illustrated in Figure\u00a05<\/a> can also be seen in Figure\u00a011. It exists because the groundwater that originates inland, mixes with the intruded seawater along the interface and carries with it some of the dissolved salt, which is lost from the system when the groundwater mixture discharges. To compensate for this loss of solutes, there must be a landward flow of seawater below the seafloor. This mechanism maintains the rotational flow cell that exists seaward of the interface.<\/p>\r\n

\"Figure<\/p>\r\n

Figure\u00a0<\/strong>11<\/strong>\u00a0-\u00a0Salinity distribution and flow pattern in a coastal aquifer. The upper image shows the salinity distribution and is depicted at the true aspect ratio. The red color indicates seawater salinity, while the blue color depicts fresh water and varying salinity level between the two end members are shown buy transitional colors in the mixing zone. The white arrows in the lower image represent the magnitude and direction of specific discharge vectors. An animated version of this figure is available at this link<\/span><\/a>. In the animated version, particles are periodically added to the left and top boundaries to reveal the flow pattern illustrated by the arrows in the still image.<\/p>\r\n\r\n<\/div>","rendered":"

\n

The Simmons and others (2002b) experiment provides an example of a potentially unstable density layering where higher density water overlies lower density water. The reason why the term \u201cpotentially unstable\u201d is used (as opposed to unconditionally unstable)\u00a0is that there are other factors, such as the permeability of the sediment and the background flow field, that determine whether gravitational instabilities as shown in Figure\u00a08b<\/a> will develop and amplify. These additional factors are considered in Section\u00a05<\/a>.<\/p>\n

On the opposite side of the spectrum are stable density configurations, where the lower density fluid rests on top of higher density fluid. The difference between the two regimes is illustrated in this video<\/span><\/a>, which is an animated version of the snapshot at 10,000 years shown in Figure\u00a09. The temperature is color-coded. The left and right images show the temperature distribution in a 500 \u00d7\u00a0500\u00a0m cross-section through a homogeneous aquifer with no external hydraulic head gradients to drive forced convection. In the unstable\u00a0regime of Figure\u00a09b, the temperature is fixed to 1.5\u00a0\u00b0C along the top and the system is heated from below by a heat flux of 100\u00a0mW\u00a0m\u22122<\/sup>. Conditions like these occur, for example, below the bottom of the world\u2019s oceans underlain by basalt aquifers (Fisher and Geoffrey, 2010). In the stable regime shown in Figure\u00a09a, the upper and lower boundary conditions are reversed, so the temperature is fixed along the bottom and the heat flux is applied from above. Because the density of water decreases with increasing temperature (for pure water this is true above 4\u00a0\u00b0C but for seawater, this is true for all temperatures above freezing point) the warmer water over cooler water in the system on the left represents the stable regime. Because the groundwater is stagnant in Figure\u00a09a, locations throughout the flow system where flow vectors would be plotted appear as dots (i.e., the groundwater has zero velocity, thus the flow vector has no magnitude nor direction). In the absence of flow, the rock and the groundwater convey heat only by conduction.<\/a><\/p>\n

\"Figure<\/p>\n

Figure\u00a0<\/strong>9<\/strong>\u00a0–\u00a0Stable and unstable temperature fields 10,000 years after heat was applied to the boundary. a) Heat conduction in a stable state where heat is applied to the top so water density is lower at the top; and, b) free convection cells in an unstable state where heat is applied to the bottom so water density is lower at the bottom. The Rayleigh number (discussed in Section\u00a05.2<\/a>) is on the order of Ra<\/em>\u00a0=\u00a03\u00a0\u00d7\u00a010<\/span>2<\/sup>. Video at this link<\/span><\/a> provides the animated version of this figure.<\/p>\n

In the case of the unstable regime shown in (Figure\u00a09b), the situation is entirely different. The video at this link<\/span><\/a> shows that the system is initially stable as the aquifer is heated from below. Then, after approximately 2250\u00a0years, the horizontal temperature stratification is disrupted and the system transitions into a new regime in which there is vigorous, circulatory groundwater flow. Cold water is transported downward in the center of the aquifer and warm groundwater rises along the left and right boundaries. The video shows that the temperature distribution does not change after approximately 5000\u00a0years, nevertheless, the convective flow pattern persists as evidenced by the circulating particles. In Figure\u00a09b, this flow pattern is shown by flow vectors, which in contrast to the stable case of Figure\u00a09a, are arrows with magnitude and direction indicating the groundwater is in motion.<\/p>\n

The counter-rotating cells, shown in Figure\u00a09b, are called free convection cells.<\/em> Cooler water from the top of the aquifer descends to replace the warmer fluid that rises. Something similar happens when you heat water to cook or make a cup of tea. Just before the water starts to boil, you may recognize the water movement that is similar to the pattern displayed in Figure\u00a09b, but because all the water has the same color, the convection cells can be hard to see. A clearer and more entertaining household example is the recipe for a marble cake. When alternate layers of vanilla and chocolate cake mix are carefully put more or less horizontally in a baking tray, the result after baking is a pattern where counter-rotating cells are formed in the baking process (Figure\u00a010). It is noted that this example is not entirely representative of groundwater because of the phase change that occurs (from liquid to solid when baking a cake).<\/a><\/p>\n

\"Photograph<\/p>\n

Figure\u00a0<\/strong>10<\/strong>\u00a0–\u00a0Free convection cake. <\/a>Box 1<\/span><\/a> provides the recipe.<\/p>\n

A classic example of a stable density configuration is encountered in coastal aquifers. In this case, dense seawater water sits under less dense fresh groundwater, which creates a seawater wedge that can penetrate inland from the coast for some distance. Coastal aquifer hydrogeology is extensively treated in books (Bear et al., 1999; Cheng and Ouazar, 2004; Jiao and Post, 2019) as well as in literature reviews on this topic (Reilly and Goodman, 1985; Werner et al., 2017, 2013). The quintessential conceptual model of the salinity distribution and flow pattern in groundwater in an aquifer near the sea is depicted in Figure\u00a011. It shows the results of a model simulation in which there is lateral inflow of fresh water across the left boundary, as well as across the top boundary of the landward part of the aquifer. On the seaward part of the aquifer, the hydraulic head along the top boundary is equal to sea level. The seawater wedge protrudes inland from the coast. As presented earlier in Figure\u00a01<\/a>, the higher pressure of saline compared with fresh groundwater at the same depth is responsible for the formation of the wedge. The rotational flow that was illustrated in Figure\u00a05<\/a> can also be seen in Figure\u00a011. It exists because the groundwater that originates inland, mixes with the intruded seawater along the interface and carries with it some of the dissolved salt, which is lost from the system when the groundwater mixture discharges. To compensate for this loss of solutes, there must be a landward flow of seawater below the seafloor. This mechanism maintains the rotational flow cell that exists seaward of the interface.<\/p>\n

\"Figure<\/p>\n

Figure\u00a0<\/strong>11<\/strong>\u00a0–\u00a0Salinity distribution and flow pattern in a coastal aquifer. The upper image shows the salinity distribution and is depicted at the true aspect ratio. The red color indicates seawater salinity, while the blue color depicts fresh water and varying salinity level between the two end members are shown buy transitional colors in the mixing zone. The white arrows in the lower image represent the magnitude and direction of specific discharge vectors. An animated version of this figure is available at this link<\/span><\/a>. In the animated version, particles are periodically added to the left and top boundaries to reveal the flow pattern illustrated by the arrows in the still image.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-49","chapter","type-chapter","status-publish","hentry"],"part":127,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/49","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":13,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/49\/revisions"}],"predecessor-version":[{"id":368,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/49\/revisions\/368"}],"part":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/parts\/127"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/49\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/media?parent=49"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapter-type?post=49"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/contributor?post=49"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/license?post=49"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}