Differential heating applied at a single horizontal boundary forces ‘horizontal convection’, even when there is no net heat flux through the boundary. However, almost all studies of horizontal convection have been limited to a special class of problem in which temperature or heat flux differences were applied in only one direction and over the horizontal length of a box (the Rossby problem; Rossby, Deep-Sea Res., vol. 12, 1965, pp. 9–16). These conditions strongly constrain the flow. Here we report laboratory experiments and direct numerical simulations (DNS) extending the results of Griffiths & Gayen (Phys. Rev. Lett., vol. 115, 2015, 204301) for horizontal convection forced by boundary conditions imposed in a two-dimensional periodic array at a horizontal boundary. The experiments use saline and freshwater fluxes at a permeable base with the imposed boundary salinity having a horizontal length scale one quarter of the width of the box. The flow reaches a state in which the net boundary buoyancy flux vanishes and the bulk of the fluid shows an inertial range of turbulence length scales. A regime transition is seen for increasing water depth, from an array of individual coherent plumes on the forcing scale to convection dominated by emergent larger scales of overturning. The DNS explore the analogous thermally forced case with sinusoidal boundary temperature of wavenumber
$n=4$
, and are used to examine the Rayleigh number (
$Ra$
) dependence for shallow- and deep-water cases. For shallow water the flow transitions with increasing
$Ra$
from laminar to turbulent boundary layer regimes that are familiar from the Rossby problem and which have normalised heat transport scaling as
$Nu\sim Ra^{1/5}$
and
$Nu\sim (Ra\,Pr)^{1/5}$
, with
$Nu$
the Nusselt number and
$Pr$
the Prandtl number, in this case maintaining a stable array of coherent turbulent plumes. For deep-water and large
$Ra$
the laminar scaling transitions to
$Nu\sim (Ra\,Pr)^{1/4}$
, with the scales of turbulence extending to the dimensions of the box. The
$1/4$
power law regime is explained in terms of the momentum of symmetric, inviscid large scales of motion in the interior coupled to diffusive loss of heat through stabilised parts of the boundary layer. The turbulence production is predominantly by shear instability rather than convection, with viscous dissipation distributed throughout the bulk of the fluid. These conditions are not seen in the highly asymmetric flow in the Rossby problem even at Rayleigh numbers up to six orders of magnitude greater than the transition found here. The new inertial interior regime has the rate of supply of available potential energy, and its removal by mixing of density, increasing as
$Ra^{5/4}$
, which is faster than
$Ra^{6/5}$
in the Rossby problem. Irreversible mixing is confined close to the forcing boundary and is very much larger than the viscous dissipation, which is proportional to
$Ra$
.