2.9 Internal Mode 



Internal Mode Equations 



The internal mode of the flow is described by the vertical flow 

 structures and the temperature and salinity distributions. Defining 

 perturbation (not necessarily infinitesimal) velocities as u'=u-U/H and 

 v'=v-V/H, the equations for the internal mode are obtained by subtracting the 

 vertically-averaged momentum equations from the three-dimensional equations: 



1 3Hu' 



H at 



^.-k 



3a 



J. /hu:+u\ 



(2.42) 



1 aHv' 



H at 



ri^ + j_ J. 



H m2 3a 



A ^ 



( Hv'+v \ 



(2.43; 



where B^ and B^, defined in (2.25) and (2.26), represent a1 1 terms in the 

 A y 



transformed three-dimensional momentum equations except the surface slopes and 

 the vertical diffusion terms, and Dj^ and Dy are defined in the 

 vertically-integrated equations (2.37) and (2.38). Notice that the above 

 equations retain the three-dimensionality and hence are different from the 

 model of Nihoul and Ronday (1982), which is actually a superposition of a 

 lateral two-dimensional model and a vertical one-dimensional model. 



The above equations do not contain the surface slope terms and hence a 

 large time step (much larger than the limit imposed by the gravity wave 

 propagation) may be used in the numerical computation. The internal mode also 

 includes equations for temperature, salinity, and vertical velocity as defined 

 by Eqs. (2.27), (2.28), and (2.30). 



40 



