Topographic Models and Numerical Conditions
In order to ascertain the topographic screening effect of the dead-ice zone on valley winds, three-dimensional numerical simulation of airflow was carried out (Fig. 8) . Topographic figures of actual size for the water surfaces and surrounding moraines of Tsho Rolpa and Imja (Fig. 1A and B) were built up in the calculation domain of x x y x z = 7,000m x 2,000m x 400m. By referring to 1:50,000 scale topographic maps of 1996 and 1997 and data of topographic surveys around Tsho Rolpa and Imja, the topographic figures were assembled by some simple geometrical figures. The roughness length of debris-covered surface and lake surface was assumed to be 0.1 m and 0 m (smooth boundary), respectively. The geometry of the lake surfaces was assumed to be rectangles 3,100m long and 400m wide for Tsho Rolpa and 1,200m long and 400m wide for Imja. The height, He, of end moraine and the height, H of glacier terminus above the lake surface were given at 1 m and 35 m for Tsho Rolpa, respectively. For Imja, H = 25 m, and the height of the end moraine to the dead-ice zone was set as He (= 25 m). The height of the side moraine above the lake surface was commonly given at a constant of 70 m. Considering the air pressure at the high altitude [16, 17], the air density and the standard air pressure were set at 0.75 kg/m3 and 0.6 atm, respectively. The airflow velocity at the inlet (y - z plane at x = 0) was given constant at 1.0 to
5.0 m/s for Tsho Rolpa and at 1.0 to 6.3 m/s for Imja, which correspond to the valley wind speeds observed (Fig. 7).
Figure 8. Topographic figures of Tsho Rolpa and Imja built up in the calculation domain of wind simulation .
A CFD (computational fluid dynamics) program called "PHOENICS ver. 3.5" was used for the simulation. In the program, the completely implicit and hybrid methods were used to resolve discrete types of integral equations of continuity and motion (Navier-Stokes) for incompressible fluids. The eddy viscosity in the Navier-Stokes equation was calculated by using a revised k - e model (MMK model) developed by Murakami et al. , Mochida et al.  and Kondo et al. , where k is the turbulence kinetic energy and e is the turbulence dissipation rate. The basic grid number in the present simulation is x x y x z = 80 x 80 x 80, and the grid size near the ground and lake surface was set to be relatively small. The airflow velocity and pressure fields under steady state were repeatedly calculated until their values were converged into constants.
Fig. 9 shows spatial distributions of horizontal airflow velocity, U2, at 2 m above the lake surface for Tsho Rolpa (He=1 m and Hi =35 m) and Imja (He=Hi =25 m). In order that the same velocity of 5.06 m/s (near the maximum wind velocity observed) is obtained at the points (black circles in Fig. 9) corresponding to the sites of meteorological stations (Fig. 1), the flow velocity at the inlet was then given at 5.0 m/s for Tsho Rolpa and 4.7 m/s for Imja. The flow velocity, Ue, at 2 m above the end moraine was then 6.47 m/s for Tsho Rolpa and 5.06 m/s (black circle) for Imja. The velocity distribution in Tsho Rolpa is affected by both the end moraine and the cliff-shaped glacier terminus, especially by the glacier terminus, showing a decrease of U2 in the downwind direction within ca. 800 m upwind of the glacier terminus. The U2 in Imja decreases by the end moraine (and dead-ice zone) and glacier
Imja terminus within ca. 100 m and 200 m, respectively. The end moraine (and dead-ice zone) and glacier terminus thus can produce a topographic screening effect and a topographic barrier effect, respectively. The flow velocity U 2 averaged over the lake area was then 5.10 m/s for Tsho Rolpa and 2.98 m/s for Imja. The flow velocity for Imja is thus 42 % smaller than that for Tsho Rolpa (depletion rate of 0.42). The average wind shear stress, Tb, at the water surface is given by Tb = paCdU2 2. The average shear stress imposed on the surface of Imja is thus ca. 66 % smaller than for Tsho Rolpa, since the drag coefficient Cd is given at 0.0012 to 0.0015
Figure 9. Spatial distributions of horizontal airflow velocity, U2 (m/s), at 2 m above the lake surface calculated for topographic figures of Tsho Rolpa (upper) and Imja (lower) . The airflow velocity, U , averaged over the lake surface and the air flow velocity, Ue, at 2 m above the end moraine (black circles in Fig. 8) are numerically shown.
for wind velocity of 1 to 6 m/s [27, 28]. The depletion rate in flow velocity was 0.33 for 1.07 m/s calculated at the points corresponding to the weather stations. The decrease in depletion rate is thus small for the flow velocity of 1 to 5 m/s at the points corresponding to the meteorological stations. Hence, the weak mixing or vertical water circulation in the surface layer of Imja probably results from the wind shear stress being not enough to develop wind-driven currents (Fig. 4B).
Fig. 10 shows relations between the velocity ratio, a , the end moraine's height, He, and the glacier terminus' height, Hh above the lake surface. For Imja (Z=1,200 m), the velocity ratio a decreases linearly with increasing He with the correlation of R2 = 0.8885, while, for Tsho Rolpa, a decreases rapidly at He = 0 to 8 m, but declines gradually at He > 8 m (Fig. 10A). Hence, it is suggested that, for glacial lakes dammed up by the relatively long dead-ice zone, the screening effect on the wind velocity over the lake surface is more effective than for glacial lakes dammed up directly by the end moraine. The screening effect of end moraine may be seasonally sensitive in Tsho Rolpa, since the lake level fluctuates at a range of ± 0.6 m in May to October with no ice cover , thus being He < 8 m. The three-dimensional topographic shape of the end moraine (and dead-ice zone), damming up glacial lakes, and a change in the relative height due to the fluctuation in lake level is thus important to know if or not wind-driven water (and thermal) circulation prevails in the lakes. The velocity ratio a in Imja tends to decline gradually at the lengths of 2,200 m and 3,100 m. The basin expansion of Imja may thus weaken the screening effect of the dead-ice zone and end moraine.
The barrier's effect of the glacier terminus is marked for Imja 1,200m long at H = 5 to 25 m, but nearly constant at H = 25 to 55 m (Fig. 10B). This tendency seems to be unchangeable for the lake expansion up to the lengths of 2200 m and 3100m, though the velocity ratio a wholly increases with increasing lake length. The glacier terminus of Tsho Rolpa tends to give nearly constant a at 0.72 to 0.79 for Hic = 0 to 45 m (Fig. 10B). The three-dimensional geometry of the end moraine (and dead-ice zone) could thus affect the efficiency in the barrier's effect of glacier terminus. Any changes in the height of the side moraine and/or the width of the end moraine did not change significantly spatial distributions of the horizontal airflow velocity at 2 m above the lake surface. The existence of the dead-ice zone upwind of glacial lakes and the increase in the height above the lake surface lead to efficiently decrease the wind velocity near the lake surface and control the thermal structure within the lakes.
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