**Appendix A**

Numerical temperature model,

The following equation is discretizised,

$$\frac{\partial}{\partial \mathbf{x}} \text{Kh} \frac{\partial T}{\partial \mathbf{x}} + \frac{\partial}{\partial z} \text{Kv} \frac{\partial T}{\partial z} = \frac{\partial}{\partial t} (cT) \tag{A1}$$

where T is the temperature, Kh is the horizontal conductivity, and Kv is the vertical conductivity. Finite differences and a cell-centered grid are used. In the block with indices (ij) the expression

$$\frac{\partial}{\partial z} \text{Kh} \frac{\partial T}{\partial z} \tag{A2}$$

is evaluated by the following formula [91]:

$$\left[\frac{\partial}{\partial \mathbf{x}} \mathcal{K} \mathbf{h} \frac{\partial T}{\partial \mathbf{x}}\right] \dot{\boldsymbol{v}} = \frac{1}{\delta \mathbf{x}\_i} \left[ \mathcal{K} \mathbf{h}\_{i+\frac{1}{2},j} \left( \frac{2\left(T\_{i+1,j} - T\_{i,j}\right)}{\delta \mathbf{x}\_i + \delta \mathbf{x}\_{i+1}} \right) - \mathcal{K} \mathbf{h}\_{i-\frac{1}{2},j} \left( \frac{2\left(T\_{i,j} - T\_{i-1,j}\right)}{\delta \mathbf{x}\_{i-1} + \delta \mathbf{x}\_i} \right) \right] \tag{A3}$$

*Khi*+ 12 ,*j* is the value of *Kh* at the boundary between the blocks (*i*,*j*) and (*i* + 1,*j*). It is computed as the harmonic mean of *Khi,j* and *Khi* + *1,j*.

The expression ∂∂*zKv* ∂*T*∂*z* is treated analogously.

This gives M · N equations to find the Ti,j, unknowns, where *i* = 1,2, ... ,M, *j* =1,2, ... ,N. Here, M and N are the number of blocks in x-direction and z-direction, respectively.

We use both Dirichlet and Neumann boundary conditions for the temperature model. For Dirichlet boundary conditions the temperature, T, at the boundary is given whereas for Neumann conditions the heat flux, *Kh* ∂*T*∂*x* and *Kv* ∂*T*∂*z* , is given. A Neumann boundary condition with a heat flux of zero is used for the basin edges.

An iterative method is used to solve the linear system. Conjugate gradients are used as an acceleration method [92,93]. The conjugate gradient method is preconditioned by nested factorization [94].
