Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Simple and basic dynamical ideas…..- Newton’s Laws- Pressure and hydrostatic balance- The Coriolis effect- Geostrophic balance- Lagrangian-Eulerian coordinate framesOCN/ATM 587Newton’s Laws….F ma mx= =�&&%% %[a 2nd order vector ODE or PDE; actually 3 scalar ODE/PDEs ]xyzF mxF myF mz===���&&&&&&wherethe 3 ODEs are:( , , ) ; ( , , )x y zx x y z F F F F= =%%F F=r%note notation variants:Properties of an atmosphere and ocean at rest:Note that gases and liquids are fluids. Fluids differ from solids in that fluids at rest cannot support a shearing stress….the distinctive property of fluids. A fluid at rest cannot sustain a tangential force…the fluid would simply glide over itself in layers in such a case. At rest, fluids can only sustain forces normal to their shape (a container, for example). Pressure is the magnitude of such a normal force, normalized by the area of the fluid normal to the force. pressure = normal force / unit area [pressure is a scalar]/p F n A= g%%The pressure cube…Let Fp be the vector force per unit volume, in order to take the size of the cube out of the problem:1 2 1 21 2{ }/ ( )( ) /px px pxpxF F F p y z p y z x y zpF p p xxd d d d d d dddd= - = -= - =-��lim 0xp px xddd����Note:The force/cube equations for all of the faces of the cube can be combined to yield the vector equation{ , , }pp p pF px y z� � �= - - - =- �� � �rSuppose the force is redefined on a per unit mass basis; thenThis equation says, simply, that the net pressure force acting on a fluid element is given by the pressure gradient.1 1 1 1{ , , }mp p pF px y zr r r r� � �= - - - =- �� � �rPressure, continued…..If only pressure forces are acting on an element of fluid (air or seawater), then due to Newton’s Laws the element must be accelerating. If the pressure forces are balanced by some other force, then the fluid element can be in equilibrium (ie, at rest).What other forces are there? Consider gravity…ˆ ˆk ; kmg gF mg F g=- =-r rIf the pressure force is balanced by the force of gravity, then 0 0m gF F F= � + =�r r r1ˆk = 0p gr- � -or,1 10p px yr r� �- =- =� �1 p pg gz zrr� �- = � =-� �the hydrostatic relation[horizontal equations][vertical equation],pgzr�=-�which can be easily integrated.( ) (0)p z g z pr=- +assuming  = constantIf we define the pressure at the sea surface to be zero, then( )p z g zr=-pressure at any depth can be calculatedOceanic case:Note: SI units of pressure are pascals, but oceanographers typically use decibars; numerically the depth in meters is the same as the pressure in decibars.The hydrostatic relation….Check the units (oceanic case)….2 3 2Mg M Mgg L OKL L L=:( )p z g zr=-weight per unit areaNotes…- Depth (z) is negative – downwards.- The hydrostatic pressure is just the weight of the water column above (per unit area).,pgzr�=-�Atmospheric case:p = RT[ideal gas law]For an isothermal atmosphere (T constant) the hydrostatic relation can be integrated to find thatp = ps e-z/H H = RT/g [the scale height]Here ps is the sea level pressure and H  7.6 km. Pressure decreases exponentially with altitude.Variation of pressure with altitude in the atmosphere….Eq90ps  1000 mbarWhen can the hydrostatic relation be used? (ie, what has been assumed here?)We assumed that pressure forces balance gravity…..when will this be true?0 0m gF F F= � + =�r r rBut suppose instead that22m gzF z F F zt�= = � + =��r r r&& &&22m gzF z F F zt�= = � + =��r r r&& &&As long as1gzF<<&&the flow will still be hydrostatic.2/1L T LTg g<< � >>for L = 1 meter,this will be true as long as T  1 sec (ocean)for L = 1 km, this will be true as long as T  30 sec (atmosphere)pressure (continued)……p high p low  p p Fm [thus, a force equal to  Fm is needed to balance the pressure force]The Coriolis effect….(i) Motion on a nonrotating Earth The motion is determined by Newton’s Laws with gravity, ˆF mx gr= =-�&&%%Example: projectile motionNote: Newton’s Laws as normally used are true only in an inertial coordinate frame (one fixed with respect to distant, fixed stars. An Earth-based coordinate system (longitude, latitude, altitude; east, north, up) is not an inertial coordinate system.Coriolis effect, continued(ii) Stationary motion on a rotating EarthThe motion is determined by Newton’s Laws with gravity; the motion is only stationary in an Earth-based coordinate system [ east, north, up; longitude, latitude, altitude].Motion in an Earth-based coordinate system leads to a new effect: the centrifugal force.[inertial frame: centripetal acceleration][rotating frame: centrifugal force]Coriolis effects….continuedOn a rotating Earth the effective gravity is the sum of the force of attraction and the centrifugal force.Coriolis effect, continued(iii) Moving particles on a rotating Earth The motion is determined by Newton’s Laws with gravity; the motion is not stationary in an inertial frame or the rotating frame. Motion in the rotating frame leads to a new effect: the Coriolis force.Coriolis effect (continued)….A point on latitude  is rotating at a tangential velocity VT given by VT = V0 cos , where V0 is the equatorial value.Moving a distance L between A and B at speed Uo requires a time L/Uo .Earth rotation rate = Earth rotation time= 1/ = (Earth time)/(AB time) = Uo/(L ) << 1 (slow AB motion, strong rotation)  >> 1 (fast AB motion, weak rotation)Coriolis effect (continued)….Coriolis acceleration = 2  u u = (u, v, w)Coriolis force =  2  u i, j, k = east, north, up unit vectorsˆ ˆ ˆi j k2 u = 2 0 cos sinˆ ˆ ˆ2 ( ( cos sin )i + sin j- cos k)u v ww v u ul ll l l l� �� �W� W� �� �� �= W -Coriolis effect (continued)….Initial velocity Coriolis forcenorth (0,v,0), v>0 eastsouth (0,v,0), v<0 westeast (u,0,0), u>0 south, upwest (u,0,0), u<0 north, downup (0,0,w), w>0 westdown (0,0,w), w<0 eastCoriolis effect, continued….ˆ ˆ ˆ2 u 2 ( ( cos sin )i + sin j- cos k)w v u ul l l lW� = W -The Coriolis effect is due to rotation +