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We present observations of oscillations in the shape of the temperature profile of the large-scale circulation (LSC) of turbulent Rayleigh-B{\'e}nard convection. Temperature measurements are broken down into Fourier moments as a function of $\theta-\theta_0$, where $\theta$ is the azimuthal angle in a horizontal plane at mid-height, and $\theta_0$ is the LSC orientation. The oscillation structure is dominated by a 3rd order sine moment and 3rd order cosine moment in a cubic cell. In contrast, these moments are not found to oscillate in a cylindrical cell. This geometry-dependent behavior can be explained by a model that assumes that the heat transported by the LSC is conducted from the thermal boundary layers, and is proportional to pathlength of the LSC along boundary layers at the top and bottom plates. In a non-circular cross-section cell, oscillations of the LSC orientation $\theta_0$ result in an oscillation in the container shape in the reference frame of the LSC, resulting in an oscillation in the pathlength of the LSC at a given $\theta-\theta_0$. In a square-cross-section cell, this model predicts the dominant 3rd order sine moment and 3rd order cosine moment with magnitudes within 50\% of measured values, when using the amplitude of the oscillation of $\theta_0$ as input. A cylindrical cell is special in that the pathlength is independent of $\theta_0$, and so these oscillating moments are not induced. In a cylindrical cell, the model reproduces the sinusoidal mean temperature profile with a sloshing oscillation dominated by the 2nd order sine moment, consistent with previous observations in that geometry.

Oscillation in the temperature profile of the large-scale circulation of turbulent convection induced by a cubic container