**Authors:**

(1) Scott Conn, California Institute of Technology, Pasadena, California;

(2) Joseph Fitzgerald, California Institute of Technology, Pasadena, California;

(3) Jorn Callies, California Institute of Technology, Pasadena, California.

## Table of Links

## APPENDIX B

**YBJ Upper Boundary Condition**

Beginning from (A3) with the choice C = 0, we can vertically integrate from ๐ง = โ๐ป to ๐ง = 0 and use ๐ = 0 at ๐ง = โ๐ป to arrive at

The no-normal flow condition is imposed by requiring โ๐ = 0 at ๐ง = 0 (Young and Ben Jelloul 1997), which eliminates the advection and dissipation terms. We then horizontally average (denoted by ยท) equation (B1). Because ๐(๐ฅ, ๐ฆ,0,๐ก) has no horizontal structure, it is equal to its horizontal average. On a horizontally periodic domain, all but two terms vanish in the averaged equation:

Because the forcing is horizontally uniform in all of our simulations, this reduces to (13). Note that subtracting (B2) from (B1) yields a condition on the integral of ๐ด:

This paper is available on arxiv under CC 4.0 license.