Energy Budget

The energy budget for a density-stratified Boussinesq fluid was first discussed by Winters et al. in 1995. We will largely follow their notation and conceptual framework.

In the following, I will ignore surface fluxes or additional body forces (like tides or rotation). Addition of these terms is not difficult, but since these situations do not arise much (as yet) in the research projects at UW, we will not include them here.

The energy within the system can be broken down into the follow categories

Mechanical Energy (ME)			Internal Energy (Int)
Kinetic (KE)	Potential (PE)
	APE	BPE

Graphically, this looks like:

All the energy components are:

Component	Definition	computed live in SPINS?
KE	${\frac {1}{2}}\rho _{0}\int _{V}|{\vec {u}}|^{2}dV$	Yes
PE	$\int _{V}\rho gzdV$	Yes
BPE	$\int _{V}\rho gz_{*}dV$	Yes
APE	${\text{PE}}-{\text{BPE}}$	No

where $z_{*}$ is the depth of an element of the sorted density field which minimized the potential energy.

The definitions for the energy conversion rates are:

Rate	Definition	computed live in SPINS?
$\epsilon$	$2\mu e_{ij}e_{ij}$	Yes
$\phi _{z}$	$\int _{V}\rho gwdV$	No
$\phi _{d}$	${\frac {d{\text{BPE}}}{dt}}$	No
$\phi _{i}$	$-g\kappa \int _{A}\rho (z=z_{u})-\rho (z=z_{l})dA$	Yes
$\phi _{m}$	$\phi _{d}-\phi _{i}$	No

The energy components and conversion rates which are not computed live in spins are computed post using matlab in the plot_diagnos.m function in SPINS MATLAB tools.