## 1 Introduction

[2] In recent years, geodynamicists have focused increasingly on “dynamic topography” associated with flow in the mantle, or mantle dynamics [e.g., *Braun*, 2010; *Hager*, 1984; *Hager et al*., 1985; *McKenzie*, 1977; *Morgan*, 1965a, 1965b; *Parsons and Daly*, 1983; *Richards and Hager*, 1984]. To 0 order, all topography above sea level is affected by flow and deformation of the mantle; otherwise, erosion would have eventually destroyed it. Flow within the mantle moves crust so as to thicken or thin it, and it moves hot and cold material in response to a balance of stress given by the Navier-Stokes equation. Where isostasy prevails, lateral differences in density associated with lateral variations in both crustal thickness and temperature determine lateral differences in surface height. By definition, isostasy pertains to a static state, however, and consequently the following questions arise: If much of the topography is isostatically compensated, can the signature of mantle flow be detected by topography associated with dynamic processes? and What fraction of topography is supported by dynamic stresses—pressure and deviatoric stress—induced by viscous flow?

[3] Analytic solutions for convective flow in a viscous medium [e.g., *Hager*, 1984; *McKenzie*, 1969, 1977; *Morgan*, 1965a; *Parsons and Daly*, 1983] and countless numerical calculations show that the surface rises over upwelling flow and subsides over downwelling flow. For regions where the lithosphere deforms, however, and in particular where crust thickens or thins, the surface topography might not follow the sublithospheric flow [e.g., *Hoogenboom and Houseman*, 2006; *Neil and Houseman*, 1999]; convergent flow and downwelling may lead to sufficient crustal thickening that the surface will stand high and form a mountain range. Accordingly, we might expect topography over regions where lithospheric deformation occurs to show a different signature from those regions affected by sublithospheric flow.

[4] We consider a simple problem for which *Neil and Houseman* [1999] obtained semi-analytical solutions: Rayleigh-Taylor instability of a stratified lithosphere consisting of a low-density crust over a more dense mantle lithosphere that overlies an inviscid, slightly less dense asthenosphere (Figure 1). Gravity anomalies provide a test of the extent to which isostatic balance accounts for surface topography, and therefore the extent to which topography, both at the surface and on internal density interfaces, is supported by nonlithostatic stresses associated with flow [e.g., *Morgan*, 1965b]. So, we extend *Neil and Houseman*'s [1999] results to describe the impact of Rayleigh-Taylor instability on free-air gravity anomalies at the surface.

Symbols | Parameters |
---|---|

D | Flexural rigidity |

E | Young's modulus |

f_{d} | Flexural factor see equation (4) |

g | Gravity |

G | Newton's gravitational constant |

h | Thickness of lithosphere |

k | Wave number |

k′ = kh | Dimensionless wave number |

LAB | Lithosphere-asthenosphere boundary |

m | Crustal thickness |

m′ = m/h | Dimensionless crustal thickness |

p | Pressure |

q | Growth rate |

q′ = qT | Dimensionless growth rate |

t | Time |

t'=t/T | Dimensionless time |

Time scale | |

T_{e} | Equivalent elastic thickness |

u_{i} = (u,w) | Velocity |

u | Horizontal component of velocity |

U | Horizontal component of velocity eigenfunction |

w | Vertical component of velocity |

W | Vertical component of velocity eigenfunction |

W′ = WT/h | Dimensionless velocity eigenfunction |

x | Horizontal coordinate |

x_{j} | Coordinate (x, z) |

z | Vertical coordinate |

Z | Admittance |

Z′ = Z/2πG(ρ_{m} − ρ_{a}) | Dimensionless admittance |

δ_{ij} | Kronecker delta |

δh | Vertical component of displacement of the LAB |

δh′ = δh/h | Dimensionless vertical component of displacement of the LAB |

δm | Vertical component of displacement of the Moho |

δm′ = δm/h | Dimensionless vertical component of displacement of the Moho |

δs | Vertical component of surface displacement |

δs′ = δs/h | Dimensionless vertical component of surface displacement |

Δg | Gravity anomaly |

Strain rate tensor delta | |

η | Viscosity |

η′ = η_{c}/η_{m} | Dimensionless viscosity (viscosity ratio) |

η_{c} | Viscosity of crust |

η_{m} | Viscosity of mantle lithosphere |

ν | Poisson's ratio |

ρ | Density |

Dimensionless density contrast at the Moho | |

Dimensionless density contrast at surface | |

ρ_{a} | Density of asthenosphere |

ρ′_{a} = ρ_{a}/(ρ_{m} − ρ_{a}) | Dimensionless density of asthenosphere |

ρ_{c} | Density of crust |

ρ_{m} | Density of mantle lithosphere |

σ_{ij} | Stress tensor |

τ_{ij} | Deviatoric stress tensor |

[5] The admittance, *Z*(*k*), the ratio of Fourier transforms of gravity and topography as a function of wave number, provides a scaling of gravity to surface deflection that is useful in assessing the role of nonlithostatic stress in supporting topography:

where and are Fourier transforms of the gravity anomaly, *Δg*(*x*), and surface topography, *δs*(*x*). Obviously, both and should be complex, and hence so should *Z*(*k*), but studies show that, except for large values of *k*, the phase of *Z*(*k*) is essentially 0 for relevant data sets [e.g., *Fielding and McKenzie*, 2012; *McKenzie and Bowin*, 1976; *Watts*, 1978]. In the simple model that we consider, boundary conditions ensure that the phase is 0 (or *π*, which is equivalent to multiplying the admittance by −1).

[6] Where stresses associated with lithospheric deformation affect surface elevations, free-air gravity anomalies can be negative where the surface rises (and crust thickens), as *Hoogenboom and Houseman* [2006] showed for calculations of finite amplitude deformation with radial symmetry. Thus, unlike most cases with sublithospheric flow, negative admittance is possible. The conditions under which it may be negative motivate this study.