## 1. Introduction

[2] Understanding the heat budget of the Earth's mantle is important for studies of dynamic evolution and chemical composition of the Earth's core and mantle. It is generally agreed that the total surface heat flux of the Earth is about 43 TW. Excluding radiogenic heating of ∼7 TW in continental crust leads to 36 TW heat flux that can be attributed to mantle convection processes. Three main heating sources for this 36 TW heat flux include the heat flux from the core, *Q*_{cmb}, the radiogenic heating in the mantle, *Q*_{rad}, and the heat associated with secular cooling of the mantle, *Q*_{sec}. However, how these three heating sources are partitioned to make up the total surface heat flux remains unresolved [*Davies*, 1999; *Zhong*, 2006].

[3] *Q*_{cmb} provides the basal heating for the mantle and controls the cooling of the core, while *Q*_{rad} and *Q*_{sec} constitute the internal heating for mantle convection. Mantle upwelling plumes, which may form at the core-mantle boundary (CMB) due to thermal boundary layer instability and rise to the surface to generate hot spot volcanisms [*Morgan*, 1971], have been considered as the most important agent to transfer *Q*_{cmb} to the Earth's surface. The plume heat flux estimated from swell topography and plate motion is 2.4–3.5 TW [*Sleep*, 1990; *Davies*, 1988]. This is often taken as *Q*_{cmb}, based on two assumptions: (1) *Q*_{cmb} is entirely transferred by the plumes and (2) the plume heat flux does not change when plumes rise from the CMB to surface.

[4] However, these two assumptions have been questioned recently based on numerical modeling of mantle convection [*Labrosse*, 2002; *Bunge*, 2005; *Mittelstaedt and Tackley*, 2006; *Zhong*, 2006]. First, *Labrosse* [2002] found from 3-D Cartesian isoviscous models with the Boussinesq approximation that the CMB heat flux is controlled by cold downwellings rather than plumes, and he suggested that the plume heat flux is only a fraction of *Q*_{cmb}. *Mittelstaedt and Tackley* [2006] quantified the plume heat flux using 2-D numerical models in three different geometries and also found that the plume heat flux in the upper mantle is a fraction of the CMB heat flux. *Bunge* [2005] suggested that subadiabatic temperature may have significant effects on the plume heat flux. On the basis of his 3-D spherical model calculations that may yield as large as 500 K subadiabatic temperature, *Bunge* [2005] suggested that plume heat flux in the upper mantle may only be 1/3 of *Q*_{cmb}, if plume excess temperature in the upper mantle is ∼250 K. However, Bunge did not quantify the plume heat flux as did by *Mittelstaedt and Tackley* [2006].

[5] *Zhong* [2006] quantified the depth-dependence of plume heat flux in 3-D regional spherical models of mantle convection with the extended-Boussinesq approximation. *Zhong* [2006] found that the plume heat flux represents a large fraction of *Q*_{cmb} as plumes form near the CMB, but the plume heat flux decreases continuously by as much as a factor of 2-3 as plumes rise from the CMB to the upper mantle. *Zhong* [2006] suggested that the reduction in plume heat flux as plumes rise is caused by adiabatic cooling and diffusive cooling of plumes and subadiabaticity. *Zhong* [2006] also showed that *Q*_{cmb} may be required to account for 35% of the surface heat flux from mantle convection, (i.e., internal heating rate for the mantle is 65%), to reproduce plume-related observations including the plume heat flux and plume excess temperature in the upper mantle.

[6] It is important to understand to what extent plume heat flux represents *Q*_{cmb} and what controls the reduction of plume heat flux as plumes rise. In this paper, we formulated a simple analytic model for plume heat flux and plume temperature variations with depth. We performed model calculations similar to those by *Zhong* [2006], but with a larger parameter space and higher resolution. The higher resolution enables us to better quantify the variations of heat flux and temperature with depth. Also, the larger parameter space, particularly larger activation energy, varying lithospheric viscosity and varying dissipation number, ensures the robustness of our results. We quantified the ratio of plume heat flux to *Q*_{cmb} and also the ratio of plume heat flux reduction as plumes rise.

[7] This paper is organized as follows. We first describe our models and a plume detection scheme. Then, the results for several groups of models are shown and analyzed. After a simple analytic model for the plume heat flux variation is derived and compared with numerical model results, the plume-related observations are used to constrain the internal heating rate of the mantle. Finally, the main conclusions are drawn and discussed.