Energy Budget

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

The following 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:

where Num is the energy lost/gained due to the numerical model/scheme/filter.

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
ME	${\text{KE}}+{\text{PE}}$	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 rates of change of each component are

${\begin{aligned}{\frac {d{\text{KE}}}{dt}}&=-\phi _{z}-\epsilon \\{\frac {d{\text{APE}}}{dt}}&=\phi _{z}-\phi _{m}\\{\frac {d{\text{BPE}}}{dt}}&=\phi _{m}+\phi _{i}\\{\frac {d{\text{Int}}}{dt}}&=\epsilon -\phi _{i}\end{aligned}}$

Including the rate of energy added/subtracted due to the numerical model or the filter, $\Gamma$, the evolution equation of the total mechanical energy is

${\frac {dME}{dt}}=\phi _{i}-\epsilon -\Gamma$

The energy components and conversion rates that are not computed live in spins (as well as the energy lost to numerics) are computed in post using Matlab in the plot_diagnos.m function in SPINS MATLAB tools. SPINS returns enough budget information for the uncomputed variables to be evaluated from the above definitions.