Two-dimensional simulations of flow in ice-covered lakes with horizontal variations in surface albedo

We present numerical simulations of radiatively driven convection at temperatures below the temperature of maximum density, as observed in ice-covered lakes in early spring. The purpose of these simulations is to isolate the phenomenon of lateral circulation driven by horizontal variations in surface albedo (e.g., due to partial snow cover) in an idealized and simplified system. The system we consider is one with uniform solar radiation except in a small “shadowed” region at the center of the domain which has damped radiation intensity. By comparing cases with and without a shadowed region, we identify gravity currents at the surface flowing away from the shadowed region. Not only do these gravity currents represent a mechanism for lateral transport at the surface below ice cover, but they also act as a catalyst for inducing earlier vertical mixing that develops at a faster rate than the Rayleigh-Taylor-like instabilities which drive vertical convection away from the shadow. To the authors' knowledge, only bathymetry and wind forcing at the surface have been presented as major mechanisms for lateral circulation in ice-covered lakes, and hence these simulations may provide a hitherto unreported mechanism for inducing lateral circulation.

Figure 1

Schematic of the system used for the simulations in this paper. The region between the vertical dotted lines, centered at $x_s$ with width $2w_d$, is the shadowed region. The regions to the left and right of these dotted lines are referred to as the unshadowed regions. The dashed lines in each region denote the vertical shape of the solar forcing term in Eq. (1c) in each region. $\lambda$ is fixed between regions, but the magnitude of the forcing at the surface is reduced by a factor of $(1-A_l)$ in the shadowed region. See Tables 1 and 2 for parameter definitions.

Figure 2

Initial conditions of the two passive tracers are presented here. (a) Tracer $C_H$ initialized as a column at the center of the domain with a maximum concentration of 1 at the center. (b) Tracer $C_V$ initialized in a region 1 m in height at the bottom of the domain with a concentration of 1 at the bottom.

Figure 3

The coefficient of $\mathrm{\Delta }\rho$ [Eq. (11)] for each shadowed case (see Table 2 for full list). This coefficient is the rate of change of the density difference across the shadow boundary due to the background temperature alone.

Figure 4

Snapshots of the temperature field for T0B (a), (c), (e), (g) and T02 (b), (d), (f), (h) at (a), (b) 600 s, (c), (d) 1200 s, (e), (f) 1800 s, (g), (h) 2400 s. Both simulations use $T_0=2.5^{\circ }\mathrm{C}$, $\lambda =0.4$ m. T02 uses $w_d=1$ m and $A_l=0.9$. T0B is entirely determined by the development of RT instabilities. T02 additionally has the shadowed region to generate systematic lateral motion.

Figure 5

(a) Evolution of the horizontally averaged temperature field for T0B. Vertical dashed line is superimposed to indicate the approximate transition time between a solar-radiation-dominated phase and a convective-dominated phase (defined in the text). (b) Deviation of the horizontally averaged temperature field from the background $T_b$ [Eq. (10)]. The solid black line retains the same meaning as in panel (a). Positive values indicate that the horizontally averaged temperature is warmer than just the background, negative values are colder than the background.

Figure 6

Horizontally averaged, horizontal temperature flux where $T_0=2.5^{\circ }\mathrm{C}$ for (a) T0B (unshadowed) and (b) T02 (shadowed). T02 in panel (b) is averaged over the left half of the horizontal domain to avoid cancellation of the lateral intrusion [see Figs. 4 and 4 for signs of symmetry which would cancel in an average]. Vertical dashed lines refer to the times corresponding to each row of Fig. 4, in order of increasing value.

Figure 7

Vertically averaged, vertical temperature flux where $T_0=2.5^{\circ }\mathrm{C}$ for (a) T0B (unshadowed) and (b) T02 (shadowed). Vertical dashed lines indicate the edge of the shadowed region. Vertical dot-dashed lines refer to the approximate edge of the shadow boundary. Horizontal dashed lines refer to the times corresponding to each row of Fig. 4, in order of increasing value.

Figure 8

The horizontal and vertical mean of passive tracers highlighting enhanced transport due to the lateral motion generated by the shadowed region. Initial conditions given in Fig. 2. (a), (b) Tracer arranged in a column centered at the center of the domain with width $2w_d$. Unshadowed case in panel (a) and shadowed case in panel (b). (c), (d) Tracer aligned in a row at the bottom of the domain with a width of 1 m. Unshadowed case in panel (c) and shadowed case in panel (d).

Figure 9

(a) Average kinetic energy per unit mass as a function of time for the three values of $T_0$ 1.5, 2.5, and $3.5^{\circ }\mathrm{C}$. The average kinetic energy per unit mass is given for both the shadowed (solid) and unshadowed (dashed) case, for each $T_0$. Horizontal and vertical axes on logarithmic scales. (b) Percent difference between the shadowed and unshadowed ${\mathrm{KE}}_{\mathrm{avg}}$. (c) Fraction of horizontal KE in the shadowed (solid) and unshadowed (dashed) cases. All axes in each panel are dimensionless.

Figure 10

The rate of change of the average kinetic energy per unit mass with respect to time is a function of (a) the average buoyancy flux per unit mass, ${\mathrm{\Phi }}_{\mathrm{avg}}$ and (b) the average dissipation per unit mass, ${\varphi }_{\mathrm{avg}}$. In both panels the quantity for both the shadowed and unshadowed equivalent are given. The legend in panel (b) also applies to panel (a). All axes in both panels are dimensionless.

Figure 11

Position of intrusion front as a function of time at 60 s intervals for each case listed in Table 2 in both (a) dimensional and (b) dimensionless variables. The front position is tracked using the tracer $C_H$ defined in Sec. 2b1 with initial conditions defined in Fig. 2. There is the main system T02, presented in Sec. 3a, and two variations from this system for each parameter varied in Sec. 3b: $T_0$, $A_l$, and $\lambda$.

Figure 12

Averaged and full temperature fields for T01, T02, and T03. (a), (c), (e) Vertically averaged temperature field vs time. Vertical average evaluated from $z=L_z\to L_z-\lambda$. (b), (d), (f) Snapshots of the full temperature field at $t=1500$ s for each value of $T_0$. Corresponds to the horizontal dashed line in (a), (c), (e). (a), (b) $T_0=1.5^{\circ }\mathrm{C}$, (c), (d) $T_0=2.5^{\circ }\mathrm{C}$, (e), (f) $T_0=3.5^{\circ }\mathrm{C}$. Solid horizontal line indicates the approximate transition time between the solar-radiation-dominated phase and the convection-dominated phase of RT instability development. Solid arrows are a guide to the eye which indicate the bounds of the character V shape discussed further in the text.

Figure 13

Intrusion front positions of T01 and T03 as a function of the intrusion front position for the main shadowed system (T02). Symbols are front positions extrapolated from SPINS output separated by a time interval of 60 s, connecting lines are a guide to the eye. If a curve is above the diagonal, dashed line, then it has propagated further than T02 up to that time. If it is below the diagonal line T02 has propagated further over the same time interval. Curves parallel to the diagonal are propagating at the same speed.

Figure 14

Averaged and full temperature fields for T02, Al6 and Al3 (a), (c), (e) Vertically averaged temperature field vs time. Vertical average evaluated from $z=L_z\to L_z-\lambda$. (b), (d), (f) Snapshots of the full temperature field at $t=1500$ s for each value of $T_0$. Corresponds to the horizontal dashed line in (a), (c), (e). (a), (b) $A_l=0.9$ (T02), (c), (d) $A_l=0.6$ (Al6), (e), (f) $A_l=0.3$ (Al3). Solid horizontal line indicates the approximate transition time between the solar-radiation-dominated phase and the convection-dominated phase of RT instability development. Solid arrows are a guide to the eye which indicate the bounds of the character V shape discussed further in the text.

Figure 15

Intrusion front positions of Al3 and Al6 as a function of the intrusion front position for the main shadowed system (T02). Symbols are front positions extrapolated from SPINS output separated by a time interval of 60 s, connecting lines are a guide to the eye. If a curve is above the diagonal, dashed line, then it has propagated further than T02 up to that time. If it is below the diagonal line T02 has propagated further over the same time interval. Curves parallel to the diagonal are propagating at the same speed.

Figure 16

Averaged and full temperature fields for La2, To2, and La8. (a), (c), (e) Vertically averaged temperature field vs time. Vertical $\lambda$. (b), (d), (f) Is the full temperature field at the time indicated by the horizontal dashed line in (a), (c), (e). (a), (b) $\lambda =0.2$ m (La2), (c), (d) $\lambda =0.4$ m (T02), (e), (f) $\lambda =0.8$ m (La8). Solid horizontal line indicates the approximate transition time between the solar-radiation-dominated phase and the convection-dominated phase of RT instability development. Solid arrows are a guide to the eye which indicate the bounds of the character V shape discussed further in the text.

Figure 17

Intrusion front positions of La2 and La8 as a function of the intrusion front position for the main shadowed system (T02). Symbols are front positions extrapolated from SPINS output separated by a time interval of 60 s, connecting lines are a guide to the eye. If a curve is above the diagonal, dashed line, then it has propagated further than T02 up to that time. If it is below the diagonal line T02 has propagated further over the same time interval. Curves parallel to the diagonal are propagating at the same speed.

Figure 18

Intrusion in the temperature field for case T02 at 2100 s with (a) no noise in temperature field at the onset of the simulation and (b) normally distributed noise in the temperature field at the onset of the simulation.

Figure 19

Position of the intrusion front vs time with and without noise (see legend). Vertical dashed line indicates the time corresponding to Fig. 18. Approximate maximum difference between the intrusion position with and without noise is labeled by the vertical bar at $t=1800$ s.