Gravity currents in the cabbeling regime

In this study, we present a series of simulations of gravity currents where ambient and intruding temperatures are on opposite sides of the temperature of maximum density. We use these simulations to describe how cabbeling (mixing of parcels of fluid that leads to a parcel of fluid that is denser) affects the evolution of gravity currents. We show that initially buoyant currents (called hypopycnal currents) undergo mixing in the body and tail region of the gravity current, which generates dense water from cabbeling. We investigate the maximum distance that the initial current progresses as a function of a parameter controlling the nonlinearity of the equation of state (referred to as $\mathrm{\Theta }$). We find that the maximum distance that the current progresses is a nonlinear function of $\mathrm{\Theta }$. For low $\mathrm{\Theta }$ hypopycnal currents reverse direction after they begin flowing, and cabbeling occurs over a limited spatial extent. As $\mathrm{\Theta }$ is increased, currents achieve a larger maximum distance of propagation, allowing the dense water formed by cabbeling to sink and form a secondary current along the bottom of the domain (called a hyperpycnal current). As we increase the parameter controlling the nonlinearity of the equation of state, the hyperpycnal current becomes much larger in scale. We discuss some general characteristics of the hyperpycnal current and highlight that once it forms, larger values of $\mathrm{\Theta }$ lead to a larger spatial extent of the current but narrower distributions of density and temperature.

Figure 1

A schematic of the initial condition demonstrating the initial locations of the intrusion and ambient, which are defined to have temperatures of one and zero respectively. In all cases, the intrusion is initially located adjacent to the top left corner of the domain. The initial hypopycnal current depth is varied across cases. The lock length is given by $x_{\ell }$, the tank length given by $l$, and the tank depth given by $h$. $x$ is the horizontal coordinate, and $z$ is the vertical coordinate, and gravity points in the negative $z$ direction. Lengths are nondimensionalized by $z_0$, so the initial intrusion height serves as the reference length.

Figure 2

A schematic representation of different values and intervals of $\mathrm{\Theta }$ and what they imply about the expected behavior of the resulting gravity current. Each set of axes shows the quadratic equation of state and the markers represent the temperatures of the intruding fluid (triangles) and the ambient fluid (squares), as well as the temperature of maximum density (stars). It is assumed that $\mathrm{\Delta }{\tilde{T}}_1>0$. For cases where $\mathrm{\Delta }{\tilde{T}}_1<0$ is of equal magnitude and opposite sign, the location of the temperature markers are flipped across the line $\tilde{T}={\tilde{T}}_{md}$. Dotted vertical lines represent the average of the intruding and ambient temperatures and are included only where necessary.

Figure 3

Temperature fields for T3h2, given by (2). The nondimensional temperature of maximum density for T3h2 ($\frac{1}{\mathrm{\Theta }}$) is marked on the colorbar. Times are located on each panel. Panels (a)–(c) show the general evolution of the hypopycnal (near-surface) current, panel (d) shows a transition time, and panels (e) and (f) show the hyperpycnal (near-bottom) current. Notice the increase in head height in panel (f) relative to panel (a). Notice also that the hyperpycnal current quickly outruns the hypopycnal current.

Figure 4

Horizontal component of nondimensional flow velocity for T3h2. The times of each plot are the same as in Fig. 3. Notice the decrease in the velocity of the current in panel (f) relative to panel (a).

Figure 5

Time series of the total kinetic energy for all cases with $h=2$. Panel (a) shows the total kinetic energy for cases with $2.2\le \mathrm{\Theta }\le 3$. Panel (c) shows the same quantity for cases with $3\le \mathrm{\Theta }\le 8$ (i.e., T3h2 is included on all four plots). Panels (b) and (d) are detailed views of the early evolution.

Figure 6

Time series of the total kinetic energy comparing between cases with varying $h$ ($\mathrm{\Theta }$ held constant in each panel). Panel (a) shows the cases with $\mathrm{\Theta }=3$, and panel (b) shows the cases with $\mathrm{\Theta }=6$.

Figure 7

Hovmoller plots of the temperature at a height of $0.95h$ for the cases with varying $\mathrm{\Theta }$ and $h$. $\mathrm{\Theta }$ is constant along each row of panels, and $h$ is constant along each column. Arrows are included in panels (a), (b), and (c) to highlight the flow reversal.

Figure 8

Panel (a) shows the variation of $\mathcal{L}$ for the large intrusion cases ($⧫$), the medium intrusion cases ($▴$), and the small intrusion cases ($\blacksquare$). VT5h4 and VT6h4 are also included and denoted by $•$. Panel (b) shows the area of the hyperpycnal current normalized by the area of the entrainment zone $\mathcal{A}$ after the cabbeling is quenched.

Figure 9

Panels (a)–(d) show the rescaled density field for several cases. Case names and times are located on the plots. A density of 0 corresponds to the ambient density while a density of 1 corresponds to the maximum density. Panel (e) shows box and whisker plots of the temperature distribution within the current. Panels (f) and (g) show the same data on rescaled axes where 1 corresponds to a temperature $T=\frac{1}{\mathrm{\Theta }}$ (the nondimensional temperature of maximum density) and 0 is the ambient temperature. The interquartile range is denoted by the blue boxes and the median temperature is denoted by the red line. The mean temperature of the current is indicated by $⧫$ and $\frac{1}{\mathrm{\Theta }}$ is indicated by $•$.