Inviscid 2D flow (periodic boundaries)

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2 Inviscid 2D flow (periodic t q + r (vq) =@ t q + J(,q)= S + R advection of potential @x d dxdyf (q) =0 dt q = r 2 + h, h(x, y) =f H D H D

3 Quadratic Invariants Potential Enstrophy Q = 1 2 q 2 t q + r (vq) =0 Kinetic Energy E = 1 2 (r ) 2 dxdy d dt and many more invariants dxdyf (q) =0

4 Section 2: +ẑ (r rq) =0 r 1 ± q 1 ± q rq locally = F 0 (q) non-trivial relationship is µ = q = r 2 + h q 1, 1 s = h µ + 2

5 Quadratic Invariants (again) non-zero modes of topography µ 0 s multiple for a given energy (not all necessarily stable) Potential Enstrophy Q s (µ) = 1 2 X µ 2 h 2 (µ + 2 ) 2 Kinetic Energy E s (µ) = 1 2 X 2 h 2 (µ + 2 ) < 2 < 2 1 h 2 = 0 for 2 + < 2 < 2

6 Nonlinear Stability Q + µ E z} { Q Q s + µ(e E s )= 1 2 = 1 2 = 1 2 = 1 2 = 1 2 = 1 2 = s + dxdy q 2 (q s ) 2 + µ(r ) 2 µ(r s ) 2 dxdy (r 2 ( s + )) 2 +2hr 2 (r 2 s ) 2 + µ(r( s + )) 2 µ(r s ) 2 dxdy 2q s r 2 + µ2r r s +(r 2 ) 2 + µ(r ) 2 dxdy (r 2 ) 2 + µ(r ) I 1 s (r ˆn)dS 2 dxdy (r 2 C X 2 ( 2 + µ) 2 ) 2 + µ(r ) 2 Perturb the state to determine stability dxdy 2q s r 2 2q s r 2 for µ> 2 0 µ< 2 1 the perturbation doesn t grow or decay

7 Nonlinear Stability Q + µ E z} { Q Q s + µ(e E s )= 1 2 = 1 2 = 1 2 = 1 2 = 1 2 = 1 2 = s + dxdy q 2 (q s ) 2 + µ(r ) 2 µ(r s ) 2 dxdy (r 2 ( s + )) 2 +2hr 2 (r 2 s ) 2 + µ(r( s + )) 2 µ(r s ) 2 dxdy 2q s r 2 + µ2r r s +(r 2 ) 2 + µ(r ) 2 dxdy (r 2 ) 2 + µ(r ) I 1 s (r ˆn)dS 2 = 1 2 dxdy (r 2 C X 2 ( 2 + µ) 2 a miracle occurs X ) 2 + µ(r ) 2 single-signed Perturb the state to determine stability dxdy 2q s r 2 2q s r 2 2 ( 2 + µ) 2 for µ> 2 0 µ< 2 1 the perturbation doesn t grow or decay in time we don t now anything about the rest of µ though

8 Nonlinear Stability (Q + µe) = X 2 (q µ ) 2 (Q + µe) = X 2 ( 2 + µ) 2 positive for µ > 2 0 minimum enstrophy branch Recall that since flow is inviscid any perturbation does not decay, regardless of stability

9 Moving to Section 3 Inviscid statistical equilibrium

10 a bit of machinery is involved probability density depends on invariants of system P ( ) / e ae let s ignore the rest of these E = 1 2 Q = 1 2 F (q)dxdy X X bq+... h i = h (a/b)+ 2 h( h i)( p h p i)i = 2 a + b 2, p 1 a + b h 2 ((a/b)+ 2 ) 2 2 a + b 2 + (a/b)2 h 2 ((a/b)+ 2 ) 2 a + b 2 > 0 otherwise negative energy -p is to denote complex conjugate what are a, b? for µ eq (a/b), µ eq > 2 0 if b>0 µ eq < 2 1 if b<0 Recall from nonlinear stability s = h µ + 2

11 what does this mean?

12 Infinite Resolution E = 1 2 Q = 1 2 Reminder 1!1 X X 1 a + b h 2 ((a/b)+ 2 ) 2 2 a + b 2 + (a/b)2 h 2 ((a/b)+ 2 ) 2

13 Infinite Resolution (No topography) E = 1 2 Q = 1 2 Reminder 1!1 X X 1 a + b h 2 ((a/b)+ 2 ) 2 2 a + b 2 + (a/b)2 h 2 ((a/b)+ 2 ) 2 maximum energy of minimum enstrophy state

14 Infinite Resolution (E,Q) (E, 2 0E) h =0 remainder of enstrophy 0 1!1 Q E apple E s ( 2 0) E E s ( 2 0) Q h 6= 0 no eddy energy h 6= ! !1

15 Beta plane t + J(, + h + y) =0 Periodic on,h Energy is conserved d dt 1 2 (r ) 2 dxdy =0 Oh No! Small and large scale potential enstrophy are not conserved d dt d dt ( + h) 2 6=0 ( + h + y) 2 6=0

16 Beta plane t + J(, + h + y) =0 Periodic on,h Introduce mean eastward flow = Uy d 1 (r ) 2 dxdy = U dt dxdy Conserved Quantity dxdy apple (r ) 2 U ( + h) 2 E = 1 2 U 2 + dxdy(r ) 2, for du dt = Q = U + 1 dxdy( + h) 2 dxdy

17 let s do the same thing again without a +ẑ (r rq) =0 r with a beta plane rq possible solution µ s = r 2 s + h! = Uy q! q = + h + y possible solution µ( s U s y)=r 2 s + h + y

18 we have parts that are periodic and large scale parts µ( s U s y)=r 2 s + h + y µ = U s µ s = r 2 s + h E s (µ) = 1 2 Q s (µ) = 2 µ µ X X 2 h 2 (µ + 2 ) 2 µ 2 h 2 (µ + 2 ) 2

19 E s (µ) = 1 2 µ Q s (µ) = 2 2 µ X X 2 h 2 (µ + 2 ) 2 µ 2 h 2 (µ + 2 ) 2

20 Nonlinear Stability (again) Q + µ E z} { > 0, for µ>0 < 0, for µ< 2 1 Physically relevant regime

21 Canonical equilibrium (again) mean and variance of is same a>0 a + b 2 > 0

22 Canonical equilibrium (again) positive eddy energy a>0 a + b 2 > 0 for µ eq > for µ eq < 0, hui 2 1, hui is large amplitude and westward is small amplitude and eastward

23 Infinite Resolution (beta plane) For E = E,Q= Q (E,Q) (E,Q Q res ) All energy goes to mean flow 0 1!1 For the same energy canonical equilibrium is equivalent to nonlinearly stable state

24 Section 6

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26 Discussion Is there some scheme that could be used to determine when to include the higher order invariants?

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28 What does conservation of potential vorticity mean? f>0,h>0,q = q 0 q = r 2 + h, h(x, y) =f H D Salmon, R., Lectures on Geophysical Fluid Dynamics

29 Canonical Equilibrium

30 Liouville s Theorem 12 = 1 2 log 12 = log 1 + log 2 Higher order moments of potential vorticity log a = a + E a (,q)+ Q a = a ae bq +... a = P ( ) / e ae bq h i = h (a/b)+ 2 h( h i)( p h p i)i = 2 a + b 2, p but what are a and b?

31 Find a and b from the mean energy and potential enstrophy hfi = hf i a + b 2 > 0 otherwise negative energy

32 Kinetic + J(,q)=@ tr 2 + t r 2 + t r 2 + J(,q) dxdy =0 c r t r ) t r t r 2 I d 1 t r ) ˆndl r r dxdy =0 c dt 2 I d 1 c (@ t r ) ˆndl r r dxdy =0 dt dxdy d dt 1 2 r r dxdy =0 J(,q)dxdy =0