## 1 Introduction

### 1.1 Mantle Convection and Plumes

[2] Slow convection in the Earth's mantle is expressed at the surface by creation of plates at midoceanic ridges and subsequent destruction of the plates upon subduction at convergent margins with related volcanism. Hot spots such as Hawai‘i and Iceland are characterized by midplate volcanism or excess volcanism at ridges. These hot spot regions appear to be supported by the long-term presence of hot material below the lithosphere. The best fluid dynamical explanation for hotspots remains the rise of mantle plumes, which are columnar upwellings with vertical speeds that are significantly higher than they move laterally [e.g., *Morgan*, 1971]. These plumes are ubiquitous features of thermal convection in viscous fluids. Fluid dynamical modeling using laboratory experiments, theoretical arguments, and numerical simulations (for recent reviews, see *Ribe et al*. [2006], *Ito and van Keken* [2007], and *Davaille and Limare* [2007]) demonstrates that the physical characteristics of hotspots in many cases can be explained by plumes rising from the deep mantle, although shallow mantle origins have also been suggested [*King and Ritsema*, 2000]. Seismological evidence for low velocity regions below hotspots have also been used to argue for the existence of plumes [*Montelli et al*., 2004; *Wolfe et al*., 2009], although it is likely that application of standard tomographic techniques is limited due to the effects of wave front healing that may render plumes invisible in the deeper mantle [*Hwang et al*., 2011].

[3] In order to improve our understanding of the formation of hotspots in the convecting mantle, it is essential to develop a strong fluid dynamical basis for studying the development of mantle plumes. Numerous studies have been devoted to that task in the last 50 years. The temperature is hot and uniform over the whole core-mantle boundary and several plumes are expected to develop from the hot thermal boundary layer there. However, in order to better focus on the development of a single plume, a small patch heated at constant temperature can be taken as a proxy. This confined heat source ensures that only one plume is generated, and that it evolves to a quasi steady state. This is the geometry that we shall also adopt in this study.

[4] *Turner* [1962] was the first to study what he called the starting plume, comprising a steady plume conduit capped by a large buoyant head. He further suggested that the buoyancy of the head increases since the head is fed by the stem as a result of the slower upwards motion of the head compared with the steady stem below it. Since then, much effort has been devoted to understand plume dynamics and to provide scalings for plume ascent velocity [*Whitehead and Luther*, 1975; *Shlien*, 1976; *Olson and Singer*, 1985; *Chay and Shlien*, 1986; *Griffiths and Campbell*, 1990; *Moses et al*., 1993; *Couliette and Loper*, 1995; *van Keken*, 1997; *Kaminski and Jaupart*, 2003; *Rogers and Morris*, 2009; *Davaille et al*., 2011] or for steady state plume stem structure [*Batchelor*, 1954; *Fujii*, 1962; *Shlien and Boxman*, 1979; *Tanny and Shlien*, 1985; *Worster*, 1986; *Moses et al*., 1993; *Olson et al*., 1993; *Couliette and Loper*, 1995; *Vasquez et al*., 1996; *Laudenbach and Christensen*, 2001; *Whittaker and Lister*, 2006a, 2006b; *Davaille et al*., 2011]. Additionally, plume growth by entrainment of fluid by thermal diffusion, continuous feeding from the source, laminar entrainment of surrounding material at the rear of a leading vortical head has been studied [*Griffiths and Campbell*, 1990; *Moses et al*., 1993; *Couliette and Loper*, 1995; *Kumagai*, 2002]. It is well recognized that plume morphology and time evolution can be complex due to phase changes, rheological variations, mean ambient shear flow, and compositional effects [e.g., *Bercovici and Mahoney*, 1994; *van Keken*, 1997; *Thompson and Tackley*, 1998; *Davaille*, 1999; *Farnetani and Samuel*, 2005; *Lin and van Keken*, 2005; *Kumagai et al*., 2008].

[5] Although the different studies generally agree on the broad picture of plume dynamics, they often propose quantitatively different scalings. One reason probably resides in the differences in boundary conditions adopted by the authors. Theoretical studies often need to assume an infinite or semi-infinite fluid, while laboratory experiments generally use rigid bottom and side boundaries and deformable or rigid top boundary. On the other hand, numerical studies for a long time preferentially prescribed free-slip nondeformable boundaries. In some cases, rigid boundaries [e.g., *Blankenbach et al*., 1989], deformable boundaries [e.g., *Schmeling et al*., 2008], or open box conditions approximating a semi-infinite fluid [e.g., *Olson et al*., 1993] have been used as well. We nevertheless lack a detailed comparison of the different assumptions on boundary conditions to better understand their role on plume dynamics.

[6] In this paper, we will combine laboratory models with numerical simulations to quantify the influence of the proximity and nature of boundary conditions on the dynamics of a laminar thermal plume that rises from a small constant heater in a nearly constant-viscosity fluid. The experimental results obtained in finite domains will be compared to the theoretical predictions obtained in infinite fluid. This will enable us to discuss the limitations of the different techniques of investigation for mantle plume modeling.

### 1.2 Goal of the Paper

[7] This paper is one in a sequence of three. It follows up on *Vatteville et al*. [2009] and is a companion to *Vatteville* *et al*. (J. Vatteville et al., Development of a laminar thermal plume in a cavity, paper II, in preparation, *Journal of Fluid Mechanics*). This set of papers studies the formation of thermal plumes rising from a small heater in very viscous fluids. We use a combination of laboratory experiments (using silicone oils) and numerical simulations (using finite element methods). In *Vatteville et al*. [2009] we directly compared numerical simulations with the laboratory experiments for a single small box geometry. In paper II we consider the various stages of the plume formation, from the conductive growth of the boundary layer to the formation of the steady state plume.

[8] In the present paper, we consider the establishment of the steady state structure and how the boundaries of the box affect the plume structure. We will first review the theory of the velocity and thermal structure of thermal plumes at high Prandtl number *Pr* (*Pr* = *ν*/*κ*, where *ν* is the kinematic viscosity and *κ* is the thermal conductivity). We will confirm the good comparison between laboratory models and numerical simulations [*Vatteville et al*., 2009]. We then use the numerical models to scale the models to much larger domains that would resemble laboratory experiments in very large tanks, with a viscous fluid filling a cavity that has no-slip (zero velocity) boundary conditions at the base and the sides, and a free-slip (zero normal velocity, zero tangential stress) boundary at the top. We will demonstrate that these boundary conditions play a crucial role in the formation of the plume structure. While the side boundaries become unimportant at a sufficiently large aspect ratio of the domain, the top free-slip boundary condition is important even at very large box sizes. We will finally show that a modification of the boundary conditions will allow for a good reproduction of the predicted structure for a semi-infinite fluid from independent theory [*Whittaker and Lister*, 2006a].

### 1.3 Predictions for the Stem Structure of a Laminar Plume at High Prandtl Number in a Semi-infinite Fluid

[9] *Batchelor* [1954] predicted that, for a steady state laminar plume rising from a point source in an infinite and constant viscosity fluid at infinite Prandtl number, the velocity at the plume axis *V* should be height-independent and scale with a typical velocity *V*_{0}:

where *g* is the gravitational constant, *α* is the thermal expansion coefficient, *Q* is the power of the plume, *ρ* is the density, *C*_{p} is the specific heat and *ν* is the kinematic viscosity. The power of the plume is related to the temperature contrast Δ*T* between plume and ambient fluid and the plume velocity *V*. It can be found at any depth in the fluid from the horizontal integral

where *r* is the distance from the plume axis.

[10] A typical radius of the plume can be defined by a radius *a* outside of which the temperature anomaly is zero and inside which the temperature anomaly is quasi-constant. At high *Pr* the radius grows principally by diffusion and *a* ∼ *κt* where *t* is time. Using the definition of a typical length scale *L*_{0}

we can write *Fujii* [1962] suggested that at the center line of a plume at infinite *Pr* the temperature contrast varies as

where *k* is the thermal conductivity of the fluid and *z* is the height above the source.

[11] Using an asymptotic approach, *Worster* [1986] developed an analytical solution for the velocity and temperature contrast in the plume for fluids that have high, but not infinite *Pr*, which introduced an explicit *Pr* dependence in (1):

where in which *ε* is the solution to the equation *ε*^{4}ln(1/*ε*^{2}) = 1/*Pr* in the interval . In this analysis, the temperature contrast at the plume axis is given by (4).

[12] In all cases above, the fluid is assumed to be infinite and the velocity is independent of the height above the source. *Whittaker and Lister* [2006a] developed a boundary layer theory for a very viscous plume rising from a point source that sits on a plane boundary. This semi-infinite geometry is more relevant to studies of plumes in natural or laboratory settings. *Whittaker and Lister* [2006a] derived that there is a typical height *z*_{0} above the heater at which advection and diffusion are comparable:

and that above this height the plume radius and vertical velocity in the plume slowly increase with height:

and

[13] The temperature in the plume center remains the same as (4).

[14] *Whittaker and Lister* [2006b] provided an extension of this study to that of a finite point source (Figure 1). In this case the temperature at the center of the plume is similar to that of (4) if it is assumed that there is a virtual point source at some depth *z*∗ < 0 below the actual heater (see also *Shlien and Boxman* [1979] and *Shlien and Boxman* [1981]):

and

where we have introduced a proportionality constant *C*_{v}. The prediction for *C* remains equal to . We will refer to this set of predictions (9 + 10) of centerline velocity and temperature as WL06.

[15] The fluid dynamical experiments described in *Vatteville et al*. [2009] and used here are for finite *Pr* (5000–50,000) fluids in a confined tank. *Whittaker and Lister* [2006a] describe that inertial effects introduce secondary corrections to their theory, which become important at a radius of . For our laboratory experiments and related numerical simulations this radius is well in excess of 10 m, which is obviously much larger than the laboratory dimensions. In the mantle, plumes carrying 20–300 GW [*Olson et al*., 1993] would feel inertial effects for radii greater than 10^{25} km. The infinite *Pr* description of [*Whittaker and Lister*, 2006a] should therefore be appropriate for plumes in the laboratory as well as in the mantle. Nevertheless, the confined medium introduces important influences of the side and top boundaries. Due to the high *Pr* nature of the fluid these are felt by the plume in its rise and development of steady state structure. This leads to a centerline velocity profile that is quite distinct from (10). It is our main goal in this paper to evaluate this difference quantitatively and to determine if we can approximate (10) better if (a) we simulate the plume evolution in boxes with very large aspect ratio (to minimize the effect of the side boundaries) and (b) use an open boundary condition at the top of the model (to minimize the effect of the top boundary).