##### 6.3.1. Axial Region

[31] The thermal structure within the one-dimensional axial intrusion region is calculated using a mass and thermal balance. At each depth mass conservation is used to equate the vertical inflow of melt and solid to the vertical outflow of melt and solid and the horizontal outflow of material to the off-axis region. The horizontally outflowing material leaves the axial region with speed *U*, the half-spreading rate, at all depths. The vertical flow of material at crustal depths can be calculated once the distribution of crystallization, θ(*z*), has been specified, as shown in (1).

[32] A differential equation satisfied by the temperature structure, *T*(*z*), in the one-dimensional axial intrusion region can be obtained from a thermal balance (Figure 6). Heat transport takes place by either conduction or advection (of both liquid and solid), while the crystallization of the crust releases latent heat, and hydrothermal circulation cools the crust by acting as a heat sink. Balancing the heat flows generated by these processes gives the differential equation for the temperature structure of the axial region.

[33] By balancing advection of heat by melt and solid, latent heat release during crystallization, and conduction and using the mass balance relationships in (1) gives the differential equation (5).

This equation must be solved in order to determine the temperature structure *T*(*z*) within the lower axial crust, where it is assumed that no hydrothermal heat removal takes place. The parameters and variables that feature in this equation are listed in Tables 1 and 2 while its derivation is given in Appendix B. The first term on the left hand side corresponds to vertical conduction, while the last on the right hand side is horizontal conduction of heat to the off-axis region. The other terms involving the melt and solid fluxes, ℱ_{l} and ℱ_{s}, and the half-spreading velocity, *U*, account for the advection of latent and specific heat both vertically within the axial region and horizontally to the off-axis region. Evaluation of the function *T*_{i}, the melt input temperature, and the crystal fractions γ and σ is outlined in section 6.4.

[34] In the upper crust, at depths shallower than *z*_{m}, hydrothermal circulation can act to cool the crust, providing an additional term in the heat balance. For all models considered later on in the paper, it is furthermore assumed that in the upper crust the vertical solid flux is zero, and the fluid intrusion temperature is constant (*T*_{i}^{0}), as is expected for emplacement of the upper crust in dykes or lava flows. Under these assumptions, the temperature equation for the upper crust is given by

with the hydrothermal cooling incorporated in the final term on the right hand side, and the crystal fraction τ evaluated as described in section 6.4.

[35] In calculation of the temperature structure of the mantle it is assumed that the input temperature gradient is that which results from melting that produces a crustal thickness of *z*_{c} [*McKenzie and Bickle*, 1988]. This gradient is then matched to the melt input temperature at the onset of crystallization. The temperature equation used for the mantle is then derived by a heat balance which neglects vertical motion of melt and solid, and instead assumes that melt is supplied rapidly to the Moho at the ridge axis, giving

The effect of the melt extraction style on mantle temperatures has been studied by *Asimow* [2002] who showed that varying the steady state melt extraction mechanism can lead to variation in the Moho temperatures of up to 20°C.

[36] In the models which simulate focused crystallization in a shallow melt lens the depth of the lens is set to be *z*_{m}. The crystallization and cooling that takes place at this depth acts as a source of heat and must be accounted for in the models. Heat is bought into the melt lens by liquid at temperature *T*_{i}(*z*_{m}), with flux −ℱ_{l}(*z*_{m}), and is removed upward from the melt lens by liquid with temperature *T*_{i}^{0}, and flux *U z*_{m}. Solid flows downward from the melt lens with flux ℱ_{s}(*z*_{m}), temperature *T*(*z*_{m}) and crystal fraction γ(*T*(*z*_{m})). The rate of heat release in the melt lens is calculated from the change in latent and specific heat carried into and out of the lens, and is thus given by

The heat source can be taken into account by adding a delta function of this size acting at *z*_{m} as a source term to the differential equation for temperature.

[37] Equations (5), (6) and (7) provide a second order differential equation that *T*(*z*) satisfies at all depths. Once the style and distribution of crystallization, ϕ(*z*) and θ(*z*), have been specified, the equations can be solved, using a numerical relaxation scheme described in Appendix C, for a depth range 0 < *z* < *z*_{g} (where *z*_{g} is the depth to the bottom of the mantle melting region), subject to the boundary conditions *T*(*z*_{g}) = *T*_{i}(*z*_{g}) and *T*(0) = 0. This relaxation scheme has the advantages of being rapid and allowing resolution of features of vertical extent of 10 m or less, in contrast to the numerical cost and low resolution of previous finite difference [*Henstock et al.*, 1993] or finite element [*Phipps Morgan and Chen*, 1993] models.

##### 6.3.2. Off-Axis Region

[38] In the off-axis region the temperature field *T*(*x*, *z*) satisfies an advection-diffusion equation, with a constant horizontal advection velocity *U* [*Sleep*, 1975; *Morton and Sleep*, 1985]:

The two source terms on the right hand side of this equation represent heat removed by hydrothermal circulation and latent heat released by crystallization. Equation (9) is solved in the region *x* ≥ *A*, 0 ≤ *z* ≤ *z*_{l}, where *z*_{l} is a depth at which isotherms are approximately horizontal. The boundary conditions that are applied are that: (1) *T*(*A*, *z*) matches the temperature structure found from the one-dimensional calculation in the axial region; (2) *T*(*x*, 0) = 0; (3) *T*(*x*, *z*_{l}) = *T*_{l}, where *T*_{l} is the temperature in the axial region at depth *z*_{l}; (4) *T* *T*_{l}*z*/*z*_{l} as *x* ∞, so that far from the ridge the temperature approaches a conductive profile.

[39] Latent heat is assumed to be released uniformly as crystallization proceeds, so that *Q*(*T*, *z*) is given by

The nature of *Q* means that (9) is nonlinear, and so it is solved using an iterative scheme of the form

At each stage of the iteration the right hand side of this equation becomes a known function (*x*, *z*). In a similar fashion to the treatment of *Morton and Sleep* [1985], each inhomogeneous equation in the iterative scheme is solved by the use of a Fourier sine series expansion

as explained in Appendix D. In order to join the axial and off-axis regions, it is necessary to use another iterative scheme. The temperature in the axial region was calculated and then used as the boundary condition *T*(*A*, *z*) for the off-axis region. The resulting temperature field for the off-axis region was then used to calculate the horizontal heat conduction at *x* = *A*, which is used as an input for the axial calculation. Convergence of this scheme usually takes two or three iterations.