}{h}\right)\left(\frac{h}{U_b} \right)^2 \frac{\left(\frac{\partial \overline{\theta}_n}{\partial z_n} \right)}{S_n^2}.$$

Using the definition of $Ri_b$ (see Sect. 2), we re-write $Ri_g$ as follows: $$Ri_g = Ri_b \frac{\left(\frac{\partial \overline{\theta}_n}{\partial z_n} \right)}{S_n^2}.$$ Similarly, $N^2$ can be written as: $$N^2 = Ri_b \left(\frac{U_b^2}{h^2}\right) \left(\frac{\partial\overline{\theta}_n}{\partial z_n}\right).
    \label{EqN2}$$ The velocity variances and TKE can be normalized as:

$$\sigma_{u_n}^2 = \frac{\sigma_u^2}{U_b^2},$$

$$\sigma_{v_n}^2 = \frac{\sigma_v^2}{U_b^2},$$

$$\sigma_{w_n}^2 = \frac{\sigma_w^2}{U_b^2},
\label{EqSigw}$$

$$\overline{e}_n = \frac{\overline{e}}{U_b^2}. 
\label{Eqe}$$

Following the above normalization approach, we can also derive the following relationship for the energy dissipation rate: $$\overline{\varepsilon} = \nu \left(\frac{U_b}{h}\right)^2 \overline{\varepsilon}_n. 
    \label{EqEDR}$$ In order to expand $\overline{\varepsilon} = \overline{e} N$, we use Eq. \[EqN2\], Eq. \[Eqe\], and Eq. \[EqEDR\] as follows: $$\nu \left(\frac{U_b}{h}\right)^2 \overline{\varepsilon}_n = U_b^2 \overline{e}_n
    Ri_b^{1/2} \left(\frac{U_b}{h}\right) \left(\frac{\partial\overline{\theta}_n}{\partial z_n}\right)^{1/2}.$$ This equation can be simplified to: $$\overline{\varepsilon}_n = Re_b Ri_b^{1/2} \overline{e}_n \left(\frac{\partial\overline{\theta}_n}{\partial z_n}\right)^{1/2}.$$ In a similar manner, $\overline{\varepsilon} = \overline{e} S$ can be re-written as: $$\overline{\varepsilon}_n = Re_b \overline{e}_n S_n.$$


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abstract: 'Using 0.37 megaton$\cdot$years of exposure from the Super-Kamiokande detector, we search for 10 dinucleon and nucleon decay modes that have a two-body final state with no hadrons. These baryon and lepton number violating modes have the potential to probe theories of unification and baryogenesis. For five of these modes the searches are novel, and for the other five modes we improve th