(This page is not finished. It will be revised after the lecture.)

Dimension of space, dimension of physical quantities ... different concept.

We live in 3-dimensional space, or 4-dimensional space-time.

3-dimensional space: position of a point can be expressed in 3 real numbers.

- linear orthogonal coordinate (Cartesian [=DesCartes] coordinate): 3 lengths (x, y, z)
- spherical cordinates (r, theta, phi)
- latitude, longitude, height ... variant of spherical coordinates. Note different conventions

- length: metre [m]
- Original concept: 1/10
^{7}of arc of a meridian from equator to pole - Present definition: distance travelled by light in a certain number of seconds

- Original concept: 1/10
- area: m
^{2} - volume: m
^{3}

- SI unit: second [s]
- day = 24 * 60 * 60 s ["*" means multiplication here]
- year = approx. 365.25 days

Primitive concept "weight", "heavy"

differentiated scientific concepts "gravity", "mass", "density"

- Law of motion (Newton; formulated later): force = mass * acceleration
- Law of gravity (Newton): force = constant * mass * mass / distance
^{2} - "Conservation of mass" (typically Lavoisier)

- Unit of mass: kilogram [kg] (= 1000 g)
- Original concept: mass of water of l litre (a 10 cm * 10 cm * 10 cm cube) at 4 deg. C
- Official definition was changed recently. (I will check references if you want.)

- Unit of force: newton [N] = kg m / s
^{2} - Unit of pressure: pascal [Pa] = N / m
^{2} - Unit of work, of energy: joule [J] = N m
- Unit of power, of energy flow per unit time: watt [W] = J / s

"Conservation" in physics is a different concept from the term in "conservation of cultural heritage" or "nature conservation".

No spontaneous generation, no spontaneous decay.

- In a system where there is no exchange of X across the boundary,
X does not change in time.
- dX/dt = 0

- In a system where there are flows of X across the boundary
(in a open system), X may change, but just because of net inflow of X.
- dX/dt = F
_{net in}= F_{in}- F_{out}

- dX/dt = F
- If an open system is in a steady state where X does not change in time,
net inflow of X is zero.
- 0 = F
_{net in}= F_{in}- F_{out}... steady state

- 0 = F

This concept of conservation is useful for "budget analysis", by which we can evaluate quantities which are not directly measurable.

Also, this concept of conservation is used for "prognostic" (i.e. forcasting-type) computations.

- ( X(t+Δt) - X(t) )/Δt = F
_{net in} - X(t+Δt) = X(t) + Δt * F
_{net in}

- air ... a (mixed) substance
- atmosphere ... a physical body (air surrounding a planet)

thermosphere (ionosphere) ... ionized, not well mixed

troposphere, stratosphere, mesosphere ... electrically neutral, well mixed

- well mixed gases: N2, O2, Ar, ... CO2 ...
- H2O in the troposphere
- O3 (ozone) in the stratosphere and other photochemically reactive gases
- aerosols (inhomogeneous)

- fraction by mass [kg/kg (non-dimensional); g/kg etc.]
- fraction by number of molecules, fraction by mol, "fraction by volume" (ideal gas assumption). [ppm, ppb, ppt]

2-component approximation of the atmosphere: "dry air" and water vapor

Unit: Pa = N/m^{2}, hPa = 100 Pa

Sea level pressure : around 1000 hPa

- pressure = g * {mass (per unit area) above}
- g : acceleration of gravity, approx. constant 9.8 m/s
^{2}

- g : acceleration of gravity, approx. constant 9.8 m/s
- 1000 hPa approx. corresponds to 10 t/m
^{2}, where "t" (tonne) is 1000 kg. - layer-by-layer hydrostatic balance
- dp/dz = -ρ g
- p: pressure, ρ(rho) : density = mass/volume

- dp/dz = -ρ g
- equation of state (ideal gas)
- p = ρ R
_{air}T- T: absolute temperature
- R
_{air}(gas constant for air) = R_{univ}/ M_{air} - R
_{univ}... universal gas constant = Boltzmann constant * Avogadro constant - M
_{air}... mass of air per mol [kg/mol] (corresponding to molecular weight, but note the unit!)

- p = ρ R
- If T is approximated as a constant, both p and ρ are proportional to exp(-z/H)
- H = R
_{air}T / g

- H = R

What proportion (in terms of mass) of the atmosphere is below 11 km (approx. in the troposphere)?

Density at 1000 hPa ... approx. 1.2 kg/m^{3} (Note: not g/m^{3})

Range of "specific humidity" (mass fraction of water vapor) at 1000 hPa ... 0 to 50 g/kg

Vertically integrated mass of water vapor per unit mass (global average)... 25 kg/m^{2}

Ocean vs. seawater ... similar to atmosphere vs. air

Ocean covers 70 % of earth's surface

Maximum depth 11 km, average depth 4 km

- Water
- "Salt": ions Cl-, Na+, SO4(2-), Mg2+, ...
- dissolved gases (esp. oxygen)
- dissolved organic matter
- particulate matter

2-component approximation "water and salt"

"Salinity": mass of salt / mass of seawater. Typically 35 g/kg

Density of water 1.0 g/cm^{3}, 1000 kg/m^{3}

At first approximation, density is constant.

For discussion of dynamics, variation of density due to tempearture and salinity is relevant.

Temperature dependence: higher temperature, lower density (thermal expansion).

Note: freshwater (salinity near zero) has abnormal characteristics "higher temperature, higher density" below 4 deg. C. Seawater (salinity near 35 g/kg) does not have this characteristics.

Salinity dependence: higher salinity, higher density.

Inhomogeneity of salinity is mainly caused by input/output of water (rather than input/output of salt).

- rainfall or inflow from rivers -> lower salinity
- evaporation from sea surface -> higher salinity
- formation of sea ice -> higher salinity of ambient water
- melting of sea ice -> lower salinity of ambient water

(To be discussed after the energy balance of the climate system)

- (near-surface) mixed layer (very roughly 100 m) ... 1 year
- "upper ocean" (very roughly 500 m) ... tens of years
- "deep ocean" ... thousand years

- ice on the sea
- sea ice ... frozen seawater
- containing salt, but less than in the water phase
- mostly seasonal (depth ca. 1 m) some multi-year (depth may be 10 m)

- icebergs ... ice cut off from glaciers
- "fresh" (nearly no salt)
- minor in terms of coverage of the earth

- sea ice ... frozen seawater
- ice on land (mostly "fresh")
- snowpack
- mostly seasonal (depth ca. 1 m)

- glaciers
- made from multi-year snowpack; flow (creep) by own weight, typical speed 10 m/year (cf. rivers 1 m/s)
- continental ice sheets
- Antarctica (depth ca. 3 km)
- Greenland
- (e.g. 20000 years ago, also in N America and N Europe)

- mountain glaciers
- Himalayas, Patagonia, Alaska, Alps, New Zealand ...

- snowpack
- ice below ground (including below seafloor)
- frozen ground (containing frozen soil water or frozen groundwater)
- permafrost (ground frozen all around the year)
- seasonally frozen ground

- frozen ground (containing frozen soil water or frozen groundwater)

- Taikan
**Oki**[沖 大幹], 1999: The global water cycle.*Global Energy and Water Cycles*(K. A. Browning & R. J. Gurney, R.J. eds., Cambridge University Press) Section 1.2.

2019-Apr-25 (before lecture)

MASUDA Kooiti