### 2.12.5.9. Convection[¶](#convection "Permalink to this headline")

Convective mixing is based on [Hostetler et al.’s (1993, 1994)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#hostetleretal1993) coupled lake-atmosphere model, adjusting the lake temperature after diffusion and phase change to maintain a stable density profile. Unfrozen lakes overturn when \\(\\rho \_{i} >\\rho \_{i+1}\\), in which case the layer thickness weighted average temperature for layers 1 to \\(i+1\\) is applied to layers 1 to \\(i+1\\) and the densities are updated. This scheme is applied iteratively to layers \\(1\\le i<N\_{levlak} -1\\). Unstable profiles occurring at the bottom of the lake (i.e., between layers \\(i=N\_{levlak} -1\\) and \\(i=N\_{levlak}\\) ) are treated separately ([Subin et al. (2012a)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#subinetal2012a)), as occasionally these can be induced by heat expelled from the sediments (not present in the original [Hostetler et al. (1994)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#hostetleretal1994) model). Mixing proceeds from the bottom upward in this case (i.e., first mixing layers \\(i=N\_{levlak} -1\\) and \\(i=N\_{levlak}\\), then checking \\(i=N\_{levlak} -2\\) and \\(i=N\_{levlak} -1\\) and mixing down to \\(i=N\_{levlak}\\) if needed, and on to the top), so as not to mix in with warmer over-lying layers.

For frozen lakes, this algorithm is generalized to conserve total enthalpy and ice content, and to maintain ice contiguous at the top of the lake. Thus, an additional mixing criterion is added: the presence of ice in a layer that is below a layer which is not completely frozen. When this occurs, these two lake layers and all those above mix. Total enthalpy _Q_ is conserved as

(2.12.57)[¶](#equation-12-52 "Permalink to this equation")\\\[Q=\\sum \_{j=1}^{i+1}\\Delta z\_{j} \\rho \_{liq} \\left(T\_{j} -T\_{f} \\right)\\left\[\\left(1-I\_{j} \\right)c\_{liq} +I\_{j} c\_{ice} \\right\] .\\\]

Once the average ice fraction \\(I\_{av}\\) is calculated from

(2.12.58)[¶](#equation-12-53 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{l} {I\_{av} =\\frac{\\sum \_{j=1}^{i+1}I\_{j} \\Delta z\_{j} }{Z\_{i+1} } ,} \\\\ {Z\_{i+1} =\\sum \_{j=1}^{i+1}\\Delta z\_{j} ,} \\end{array}\\end{split}\\\]

the temperatures are calculated. A separate temperature is calculated for the frozen (\\(T\_{froz}\\) ) and unfrozen (\\(T\_{unfr}\\) ) fractions of the mixed layers. If the total heat content _Q_ is positive (e.g. some layers will be above freezing), then the extra heat is all assigned to the unfrozen layers, while the fully frozen layers are kept at freezing. Conversely, if \\(Q < 0\\), the heat deficit will all be assigned to the ice, and the liquid layers will be kept at freezing. For the layer that contains both ice and liquid (if present), a weighted average temperature will have to be calculated.

If \\(Q > 0\\), then \\(T\_{froz} =T\_{f}\\), and \\(T\_{unfr}\\) is given by

(2.12.59)[¶](#equation-12-54 "Permalink to this equation")\\\[T\_{unfr} =\\frac{Q}{\\rho \_{liq} Z\_{i+1} \\left\[\\left(1-I\_{av} \\right)c\_{liq} \\right\]} +T\_{f} .\\\]

If \\(Q < 0\\), then \\(T\_{unfr} =T\_{f}\\), and \\(T\_{froz}\\) is given by

(2.12.60)[¶](#equation-12-55 "Permalink to this equation")\\\[T\_{froz} =\\frac{Q}{\\rho \_{liq} Z\_{i+1} \\left\[I\_{av} c\_{ice} \\right\]} +T\_{f} .\\\]

The ice is lumped together at the top. For each lake layer _j_ from 1 to _i_ + 1, the ice fraction and temperature are set as follows, where \\(Z\_{j} =\\sum \_{m=1}^{j}\\Delta z\_{m}\\) :

1.  If \\(Z\_{j} \\le Z\_{i+1} I\_{av}\\), then \\(I\_{j} =1\\) and \\(T\_{j} =T\_{froz}\\).
    
2.  Otherwise, if \\(Z\_{j-1} <Z\_{i+1} I\_{av}\\), then the layer will contain both ice and water. The ice fraction is given by \\(I\_{j} =\\frac{Z\_{i+1} I\_{av} -Z\_{j-1} }{\\Delta z\_{j} }\\). The temperature is set to conserve the desired heat content that would be present if the layer could have two temperatures, and then dividing by the heat capacity of the layer to yield
    
    (2.12.61)[¶](#equation-12-56 "Permalink to this equation")\\\[T\_{j} =\\frac{T\_{froz} I\_{j} c\_{ice} +T\_{unfr} \\left(1-I\_{j} \\right)c\_{liq} }{I\_{j} c\_{ice} +\\left(1-I\_{j} \\right)c\_{liq} } .\\\]
    
3.  Otherwise, \\(I\_{j} =0\\) and \\(T\_{j} =T\_{unfr}\\) .
    

