Hayashi, Itoh

Setup

• Linearize the QHE on an equatorial $\beta$ plane with rayleigh dampiong of velocity (bizarrely, laplacian diffusion is retained)
• No base flow.

Scale analysis:

• Time scale of 10 days.
• Deep convective heating rate of 10K/day. No source given for where this comes from.
• Zonal wind is chosen to be 10 m/s
• Results aren't super relevant for identifying cases.

Linear eigenproblem:

• Heating profile looks like $$Q_a \cos\left(\frac{\pi y}{H_y}\right) \cos\left(\frac{\pi x}{H_x}\right) \sin\left(\frac{\pi z}{z_h}\right) \exp\left(\frac{z}{2H}\right) \exp\left(-i\omega t \right)$$ on $[-H_x/2, H_x/2] \times [-H_y/2,H_y/2] \times [0, z_h]$
• Reaches maximum at approx 9km.
• $Q_a$ is 10 K/day.
• Constant lapse rate atmosphere for base state. No background wind.
• Relative differences of 10-20 percent (higher when $H_x/H_y = 2$).
• Phase speed of 50 days. Results are not sensitive to this.

Explanation of what's happening:

• Strength of heating determines $\pder{w_{\textrm{deep}}}{y}$, inducing meridional tilting of planetary vorticity.
• Temperature structure is not coupled to this tilting.

Takeaways:

• Find places where diabatic heating and $\pder{w_{\textrm{deep}}}{y}$ are tighly coupled vs instances where there are other factors driving synoptic structure of vertical motion