SME
Oceanography
Biological oceanography
Use this for “Light fades with k = 0.1 per m. How deep is the euphotic zone, and what share of the light reaches 20 m?”
Chart
I
I₀(z)
=
100
e
−0.1z
The vertical axis grows downward.
This is an example. Type your own values to start a new problem.
I
%
=
100
×
e
−
×
zₑᵤ
Euphotic depth
worked out (not typed) · m
Formulas
I
=
100
×
e
−
k
×
z
13.5335
=
100
×
e
−0.1 × 20
zₑᵤ
=
ln(100)
k
46.0517
=
ln(100)
0.1
Assumptions
•
Each meter down keeps the same share of the light, so I ÷ I₀ = e−kz, with k the same at every depth.
•
The euphotic zone ends where 1% of the surface light is left, about where photosynthesis just matches respiration.
•
Clear open ocean has k near 0.03 per m; murky coastal water has 0.3 per m or more.
Step-by-step
WE KNOW
k = 0.1 m⁻¹, z = 20 m
FIND
light left (% of surface) (I), euphotic depth (zₑᵤ)
Step 1 · Find light left (% of surface): I
Use I = 100 × e−k × z
Each
meter
keeps
the
same
share,
so
the
light
fades
as
e
−
kz
;
times
100
for
percent.
I
=
100
×
e
−0.1 × 20
I
=
100
×
e
-2
I
=
100
×
0.13534
I
=
13.5335%
Step 2 · Find euphotic depth (zₑᵤ)
Use zₑᵤ = ln(100) ÷ k
The
zone
ends
at
1%
of
the
light:
e
−
kz
=
1
100,
so
kz
=
ln
100.
zₑᵤ
=
ln(100)
0.1
zₑᵤ
=
4.605
0.1
zₑᵤ = 46.0517 m
Check
Put the numbers back in. Both sides should match.
13.5335
=
100
×
e
−0.1 × 20
✓
46.0517
=
ln(100)
0.1
✓
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