Heat and Energy in lakes
I. Heat Budgets – The Balance of heat inputs and exports from a lake
A. Total heat content of a lake
H=M*t*s
Mass * temperature * specific heat - units calories
B.
θ
1. Definition ba The annual heat budget = The total amount of heat that has entered a
lake between the time of its lowest heat content and its highest heat content expressed on a per unit
area basis
2. Calculation
a) direct in an unstratified isothermal lake
volume * density of water = mass * temperature (isothermal so only 1 value) * specific heat of water (1.01
cal*g-1*oC-1)
b) in stratified lakes use
θsz
θwz
summer heat income for depth z
winter heat income for depth z
and areal hypsographic curve
Az (θsz-θwz) * 106
Depth (m)
C. Analytical energy budgets
1. Inputs
Qs
Radiation from the sun
Qh
Radiation from the sky
Qa
Long λ atmospheric radiation
Qm
Long λ radiation from mountains
Qi
Direct conductive heat transfer from the atmosphere
2. Lossess
Qp
Qu
Qw
QE
Qe
Reflection
Scattering
long λ back radiation
Evaporation
Conduction of sensable heat to atmosphere
Net sensible heat transfer
QΘ = Heat storage
QΘ = {Qs + Qh +-QR – Qu} + { Qa + Qm - Qw} + Qs + Qi - QE –Qe
Net advective
transfer
Net radiation flux Qs’
Net radiation surplus QB
II Stability
1. S = Stability the amount of work needed to mix the entire body of water
S = 1/A0 ∫(ρz - ρ̄) * ( z - zρ̄) Az dz
ρ̄ = 1/V ∫ ρz Az dz
Where
Mean density i.e. if the lake were uniform
And
zρ̄
The depth of the lake’s center of gravity at uniformity
Then Birge’s “work of the wind – the amount of work needed to distribute the summer heat income at the
fall overturn is
B = 1/ A0 ∫ z(ρi - ρz) Az dz
for dimictic lakes
A0
Az
z
ρi
ρz
Where
is the lake surface
is the area at depth z
is the depth
Initial density
Observed density at depth z
For monomictic lakes replace density with (ρzw - ρzs)
WATER MOTION IN LAKES
I. HYDRODYNAMICS
A. Turbulent vs. Laminar Flow
Reynolds Number
Re = vl/µ
Where  = density; v = velocity; l = length of contact & µ = viscosity
This dimensionless number is the ratio of:
inertial forces (v ρ - i.e. the force of moving water)
to
viscous forces ( µ/l - i.e. the forces that retard the movement of the water
B. Vortex Formation
Richardson Number
This dimensionless number is the ratio of potential to kinetic energy
In a water body of uniform density:
Ri = gh/v2
Where g = gravitational constant; h represents the length scale and v = velocity
If the water body has density layering:
Ri= g(d/dz)/ (dv/ dz)2
Where g = gravitational constant;  = density; v = velocity and z = depth
C. Eddies
II. APERIODIC MOVEMENTS
A. Currents
1. wind drift
2. density gradients
3. inflow of water from an external source
B. Geostrophic Effects
C. Langmuir Circulation
III. PERIODIC MOVEMENTS
A. Lakes as Mechanical Oscillators
B. Surface Waves (Traveling waves)
1. capillary vs. gravity waves
2. wavelength () and fetch (x)
hw = 0/105 √ x
where hw = wave height and x = fetch
3 Short (deep water)waves
 < z - v ~ ½
4 Long (shallow water) waves
> 20* z - v~ z ½
C. Standing Waves
period t ~ 2*l/ (g z ) ½
where t = period; l = basin length; z = mean depth ; = gravitational constant
D. Seiches
E. Internal Waves
1. progressive waves
2. seiche-caused waves
F. Kelvin and Poncaré Waves