## 3 Results and Discussions
The lateral migration scale increases result in an increase in the amplitude of transverse velocity. Upon reaching a particular threshold of temperature difference, the vortex core will be severely reciprocating. The transverse velocity thus enters the large-amplitude pulsation oscillation phase. The oscillation of the middle monitoring point exhibits more pulsating oscillation characteristics than the transverse velocity oscillation of the upper monitoring point. Figure 9b shows that there is only one main frequency in the spectrum when there is a small temperature  
difference. The frequencies are \(f_{1}\) = 0.242, \(f_{2}\) = 0.244, and \(f_{3}\) = 0.24, respectively. The power spectral density is between 2.8 \(\times\) 10\({}^{-11}\) and 2.8 \(\times\) 10\({}^{-9}\) and slowly increases as the temperature difference increases. Therefore, both small-amplitude pulsation oscillations and regular oscillations under small temperature differences are single-periodic oscillations. Multiple main frequencies appear in the spectrum diagram when the temperature difference is 6. These frequencies are independent of each other, so the large-amplitude pulsati on oscillation under a large temperature difference is the multi-periodic oscillation. Figure 8: (**a**) Oscillation of longitudinal velocity at monitoring point “a” under different temperature difference conditions; (**b**) power spectrum density of longitudinal velocity (\(R\) = 5 mm, \(Ar\) = 1.0, \(Pr\) = 5.54, \(Le\) = 25.8, \(Ca_{C}\) = \(-\)0.021, \(\rho_{l}/\rho_{g}\) = 597.9). Figure 9: (**a**) Oscillation of transverse velocity at monitoring point “b” under different temperature difference conditions; (**b**) power spectrum density of transverse velocity (\(R\) = 5 mm, \(Ar\) = 1.0, \(Pr\) = 5.54, \(Le\) = 25.8, \(Ca_{C}\) = \(-\)0.021, \(\rho_{l}/\rho_{g}\) = 597.9). Figure 10a exhibits the oscillation of the longitudinal velocity of the monitoring point \"b\" under different temperature difference conditions, and Figure 10b is the PSD of longitudinal velocity. As listed in Figure 10a, the magnitude of oscillation in the longitudinal velocity increases as the temperature difference increases. It still exhibits a small-amplitude pulsation oscillation under small temperature differences, while it exhibits a large-amplitude pulsation oscillation under large temperature differences. Similar to transverse velocity, the spectrum of longitudinal velocity has only one fundamental frequency at small temperature differences, and the small amplitude oscillation at this time is a single-periodic oscillation. However, multi-periodic oscillations occur under large temperature differences. Monitoring the velocity at the high and middle points, it can be observed that the oscillation characteristics of thermo-solutocapillary convection are strongly affected by the temperature difference. As the temperature difference increases (\(\Delta T=1-6\)), the oscillation amplitude at the monitoring point gradually increases, and the oscillation transforms into large-amplitude pulsation. This indicates that the flow becomes more unstable at higher temperature differences, consistent with previous studies on thermocapillary convection. The analysis of power spectral density (PSD) also reveals interesting information about the oscillation characteristics of capillary convection. Under small temperature differences (\(\Delta T=1-3\)), the PSD diagram has only one or two main frequencies, indicating the existence of single- or double-periodic oscillations. However, under large temperature differences (\(\Delta T=6\)), multiple main frequencies can be observed, resulting in multi-periodic oscillation. This indicates that the flow becomes more complex at higher temperature differences, and multiple frequencies indicate more complex oscillations. In summary, these results indicate that the oscillation characteristics of thermo-solutocapillary convection are strongly affected by temperature difference and become more complex with the increase in temperature difference.