Lecture Guidelines for GEOF110 Chapter 2 + Thermodynamics (2 hours) Chapters 3+ 4 (2 hours) Ilker Fer Guiding for blackboard presentation. Following Pond & Pickard, Introductory Dynamical Oceanography

GEOF110 Guidelines / 2 2 Pond & Pickard, Ch.2 Density: Fresh water  (P,T) Seawater  (P,T, S) Air  (specific humidity, T, P) Salinity  measure of mass of dissolved salt in 1 kg of seawater (about 35 gr/kg; titration  not practical to measure  modern: practical salinity scale) Extreme properties of seawater: - high specific heat (Def.: heat energy required to increase T of a unit quantity of substance by a certain T interval, unit: J/(kgK))  ocean currents carry much thermal energy - high latent heat of fusion (Def.: amount of thermal energy absorbed or released to change state from solid  liquid, vice-versa)  T is maintained close to freezing point (T f  0.054xS) in polar regions - high latent heat of evaporation  heat transfer sea to air - high molecular heat conductivity

GEOF110 Guidelines / 2 3 Density  as T  or S   t =  kg/m 3  (P,T,S) is non-linear ; more so for T rule-of-thumb:  t  1 by  T = -5K  S = 1  P = 2000 kPa (= 200 dbar  200 m) (note kPa is SI; dbar not SI) Narrow range of ocean T-S properties (show volumetric T/S plot). How do we measure in the ocean  CTD (in situ C,T, P; then work out S,  t and   Practical salinity scale– make use of electrical conductivity  easier to measure ( K15 = ratio of cond of seawater at 15degC, 1 atm to cond of KCl with same T and P and a certain mass fraction. Salinity = f(K15). K15=1  S=35 )

GEOF110 Guidelines / 2 4 Thermodynamics 1st law: dq – p d  = de (1) dq : change of heat per unit mass d  : change in specific volume (  = 1/  ) p d  : work done by pressure (expansion) de : change in internal energy If the fluid particle receives heat (dq > 0) and work done by p is positive (i.e., compressing particles, d  0. Note e is proportional to T abs  hence T abs increases. (Tabs = K) 2nd law: dq = Td  (2) d  : entropy per unit mass Entropy is a measure of how close a thermodynamic system is to equilibrium. It is unavailability of a system's energy to do work! From (1) & (2): Td  – p d  = de (3)

GEOF110 Guidelines / 2 5 For a gas or fluid, eq. of state gives the relation between the state variables: f(p,T,S,  )=0, where S is composition (e.g. salinity for seawater, specific humidity for air). For simplicity assume composition does not change, dS = 0. Density then changes with p and T only. Specific entropy change: Change in g i : dg i = de + pd  +  dp - Td  -  dT Using (3): dg i = Td  - pd  + pd  +  dp - Td  -  dT dg i =  dp -  dT (4) Define  T : Tendency to change in volume in response to change in T Define g i : Max. amount of non- expansion work which can be extracted from a closed system, or the max can be attained in a completely reversible process. It is the thermodynamic potential obtainable form an isothermal, isobaric thermodynamic system.

GEOF110 Guidelines / 2 6 1st law of thermodynamics

GEOF110 Guidelines / 2 7 Adiabatic Lapse Rate Adiabatic means to alter the state of gas or fluid without adding/removing heat. So, in Eq.(1), dq = 0. Following from Eq. (2), entropy must be constant. From Eq. 7, using d  = 0 and  =1/  : From hydrostatic pressure (fluid at rest), we have dp = -  gdz, so Adiabatic Lapse Rate, or Adiabatic Temperature Gradient For fluids with simple Eq. of state (NOT SEAWATER), easy to compute . For dry, ideal gas  = p/RT (R is universal gas constant). So, This is, DRY ad. lapse rate, for dry air about 1K / 100 m. Smaller if there’s moist K / 1000 m in Ocean

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9 Potential Temperature  is the T a parcel of fluid would acquire if moved adiabatically to a reference pressure level. Derivation: Start with enthalpy form of 1st law of thermodynamics: For seawater, final eq. can be integrated using Eq. of state or using tables. For the atmosphere ideal gas behavior is typically assumed, i.e.  = p/RT and  T =1/T For dry air; R = J/kgK C P = 1004 J/kgK

GEOF110 Guidelines / 2 10 Mindanao Trench

GEOF110 Guidelines / 2 11 Density:  (for water ) Relative density: d Specific volume:  = 1/   t =  (S,T,P=0)-1000 kg/m 3   =  (S, ,P=0)-1000 kg/m 3 Also  1 ;  2 ;  4 ; Sigma introduced simply because the variability is 10XX. Pressure for density calculations refers to hydrostatic pressure. Atmospheric pressure  P = 0 (show plots of typical S, T, theta, sigma_t, sigma_theta)

GEOF110 Guidelines / % of Ocean

GEOF110 Guidelines / 2 13 Left in situ and potential temperature and Right sigma-t and sigma-theta in the Kermadec Trench in the Pacific at °E and °S. Warren (1973).

GEOF110 Guidelines / 2 14 Specific volume:  = 1/  For perfect gas  Eq. of State:  = RT a /P (R = Gas constant; T a = Absolute temperature) Seawater   =  (S, T, P) : complicated Atmosphere   =  (water vapor, T, P) IES80: International Eq. of state for Seawater better than Knudsen-Ekman tables. Systematic difference between tables and IES80 IES80 slightly denser than tables: about 0.01  t at P=0 about 0.03  t at z=5000 m about 0.09  t at z=10,000 m

GEOF110 Guidelines / 2 15 Upper: σθ, showing an apparent density inversion below 3,000 m. Lower: σ4 showing continuous increase in density with depth. From Lynn and Reid (1968).

GEOF110 Guidelines / 2 16 Chapter 3: Basic Laws 1. Conservation of Mass– leads to continuity equation 2. Conservation of Energy (split: heat (leads to heat budgets) and mechanical energy (leads to wave eq.)) 3. Newton #1 law of motion (F net = 0  no change of motion) 4. Newton #2 law of motion (a = rate of change of motion  F net ) 5. Newton #3 law of motion (for any force acting on a body, there’s equal and opposite force acting on some other body) 6. Conservation of angular momentum (leads to conservation of vorticity) 7. Newton’s law for gravitation Laws 3-5 are aspects of conservation of linear momentum (leads to Navier-Stokes). Law 7 relates to astronomical tides; pressure distribution; gravitational instability

GEOF110 Guidelines / 2 17 Forces Primary Forces (cause motion)  Gravitation (  pressure gradient, buoyancy, tides)  Wind stress (parallel to surface: friction force)  Atmospheric pressure (inverted barometer effect: adjustment of sea- level to changes in barometric pressure: +1mbar  -1 cm in sea level)  Seismic (driven by earthquakes, e.g. Tsunami) Secondary Forces (result from motion)  Coriolis force (apparent force on a moving body observed relative to the rotating Earth)  Friction (turbulence) [Force is a vector  magnitude & direction] Solid Fluid From J. Price.

GEOF110 Guidelines / 2 18 Motion Thermohaline Wind driven (major upper circulation, surface waves, upwelling) Tidal currents (driven by tidal potential) Tsunami (note: surface wave) Turbulent motions Waves in the interior  Internal gravity waves  Planetary waves (e.g. Rossby, Kelvin, Equatorial waves)

GEOF110 Guidelines / 2 19 Chapter 4: Eq. of Continuity (of Volume) A1A1 A2A2 u1u1 u2u2 In a pipe there is no loss of flow. Incoming flow at cross-section A 1 equals outgoing flow at cross-section A 2. Volume flow (flux) is mean speed x cross-section area: u 1 A 1 = u 2 A 2 If A 2 = 2A 1 ; then u 2 =u 1 /2 to satisfy continuity.

GEOF110 Guidelines / 2 20 Example Constant channel width; no rainfall; no evaporation; layer 1 thickness h 1 is constant; Cross-section area at A and B is equal. But u B >u A in layer 1. Inlet is NOT emptying  conservation of volume Outflow in layer 1 must be balanced by inflow in layer 2. A A u A = A B u B is not valid (A A = A B but u A < u B )  vertical motion A A u A + A A-B w= A B u B RIVER INLET SEA h1h1 LAYER 1 LAYER 2 AB

GEOF110 Guidelines / 2 21 Derivation of Eq. of Continuity x y z xx zz yy u,  u+  u  +  Control volume,  V=  x  y  z

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GEOF110 Guidelines / 2 23 Alternative: Consider the rate of change of volume of a fluid element x y z xx yy u

GEOF110 Guidelines / 2 24 Infer vertical velocity from horizontal currents Ship Course steered Position fix t=0 Ship Position fix t=24h Actual Course Position using dead reckoning (from constant speed and course) Mean surface current during t=24 hours Mean surface current from ship drift u = 0.3 m/s v = 0.03 m/s  x = 500 km  y= 500 km B u = 0.25 m/s v = 0.05 m/s u =-0.25 m/s v =-0.01 m/s u = m/s v = 0.0 m/s D C A E

GEOF110 Guidelines / 2 25 w << u Take horizontal scale for whole ocean: L = 4000 km (Earth’s radius 6371 km) Vertical scale H = 4 km (mean ocean depth) Aspect ratio: H/L = O(10 -3 ) : Like a sheet of paper Scaling: w  Hu/L  u/1000 !!!

GEOF110 Guidelines / 2 26 Example: Extension to conservation of mass and salt Observations give the salinity of the inflow and outflow across the Gibraltar Strait, and the outflow volume flux in Sv ( 1 Sv = 10 6 m 3 /s ) What is R+P-E ? What is flushing time? (Assume in/outflow density is equal.) Volume=4x10 6 km 3 Conservation of mass: Source: Robert Stewart Conservation of salt: