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Figure from:Baroclinic Vortices in Rotating Stratified Shearing Flows: Cyclones, Anticyclones, and Zombie Vortices

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Figure 1.1: Vortices at the surface of the Agulhas Current off the African coast. See http://svs.gsfc.nasa.gov/vis/a000000/a003800/a003827/ for details and a movie of the Earth’s surface currents (Credit: NASA/Goddard Space Flight Center Scientific Visual- ization Studio).
Figure 1.1: Vortices at the surface of the Agulhas Current off the African coast. See http://svs.gsfc.nasa.gov/vis/a000000/a003800/a003827/ for details and a movie of the Earth’s surface currents (Credit: NASA/Goddard Space Flight Center Scientific Visual- ization Studio).
Figure 1.2: Anticyclonic Meddies, form 1 km deep in the Atlantic Ocean, visualized using satellite sea-surface height measurements. Surface signature of anticyclones is elevation in the sea-surface height, which is shown in red (Credit: University of Delaware).
Figure 1.2: Anticyclonic Meddies, form 1 km deep in the Atlantic Ocean, visualized using satellite sea-surface height measurements. Surface signature of anticyclones is elevation in the sea-surface height, which is shown in red (Credit: University of Delaware).
Figure 1.3: The Great Red Spot (right) and Oval BA (left) are two gigantic anticyclones in the Southern Hemisphere of Jupiter. The picture is taken by the Hubble Space Telescope (Credit: NASA).
Figure 1.3: The Great Red Spot (right) and Oval BA (left) are two gigantic anticyclones in the Southern Hemisphere of Jupiter. The picture is taken by the Hubble Space Telescope (Credit: NASA).
Table 2.1: Parameters of the background flows and of the vortices at the “initial” time. For Cases A, B, and D the “initial” time is t = 0, and for Case C the “initial” time is t = tory. All values are in CGS units. For all cases, f = 5 rad/s, g = 980 m/s’, and p, = 1 g/cm”. Ekman number is defined as Ex = v/( fL?). See text for the difference between Cases B1 and B2. For Case Cl, torr = 60 s and Q = —64 cm?/s, and for Case C2, tors = 30s and Q = —128 cm?3/s.
Table 2.1: Parameters of the background flows and of the vortices at the “initial” time. For Cases A, B, and D the “initial” time is t = 0, and for Case C the “initial” time is t = tory. All values are in CGS units. For all cases, f = 5 rad/s, g = 980 m/s’, and p, = 1 g/cm”. Ekman number is defined as Ex = v/( fL?). See text for the difference between Cases B1 and B2. For Case Cl, torr = 60 s and Q = —64 cm?/s, and for Case C2, tors = 30s and Q = —128 cm?3/s.
Figure 2.2: Comparison of ayym with apyR (see text for definitions). The circles show the value of |Ro(1 + Ro)|(anum/@rur)* and the straight lines show the value of this expression if anuM = Qrur. All 4122 data points (circles) collapse on the straight lines (and densely cover them), validating our equation (2.13). Data points are recorded one inertial period (= 47/f) after the initial time (as defined in table 2.1) in Cases A-C, and 50 inertial periods after t = 0 in Case D. Note that all of our simulated vortices have L = R,. The horizontal axis in the inset is the same as in the main Figure; the inset’s vertical axis is the relative difference |1 — (anum/a'rHrR)?| (which is < 0.07). (n.b., the left-most plotted point has Ro(1+ Ro) ~ —0.25 due to the mathematical tautology that Ro(1 + Ro) > —0.25 for all values of Ro.)
Figure 2.2: Comparison of ayym with apyR (see text for definitions). The circles show the value of |Ro(1 + Ro)|(anum/@rur)* and the straight lines show the value of this expression if anuM = Qrur. All 4122 data points (circles) collapse on the straight lines (and densely cover them), validating our equation (2.13). Data points are recorded one inertial period (= 47/f) after the initial time (as defined in table 2.1) in Cases A-C, and 50 inertial periods after t = 0 in Case D. Note that all of our simulated vortices have L = R,. The horizontal axis in the inset is the same as in the main Figure; the inset’s vertical axis is the relative difference |1 — (anum/a'rHrR)?| (which is < 0.07). (n.b., the left-most plotted point has Ro(1+ Ro) ~ —0.25 due to the mathematical tautology that Ro(1 + Ro) > —0.25 for all values of Ro.)
Figure 3.1: Schematic of the experimental setup: the 50 x 50 x 70 cm tank is fixed on a rotating table and filled with salt-water, linearly stratified in the 30 cm middle layer. A pipe linked to a pump and a conductivity probe are mounted atop the tank and can translate vertically along the axis of rotation of the tank using separate motors. A horizontal laser sheet in the midplane (shown as a dotted line) and a top—view camera allows for recording and PIV measurements. Not shown are a side—view camera and a random dots pattern (on the opposite side of the tank facing the camera) for synthetic schlieren visualization.
Figure 3.1: Schematic of the experimental setup: the 50 x 50 x 70 cm tank is fixed on a rotating table and filled with salt-water, linearly stratified in the 30 cm middle layer. A pipe linked to a pump and a conductivity probe are mounted atop the tank and can translate vertically along the axis of rotation of the tank using separate motors. A horizontal laser sheet in the midplane (shown as a dotted line) and a top—view camera allows for recording and PIV measurements. Not shown are a side—view camera and a random dots pattern (on the opposite side of the tank facing the camera) for synthetic schlieren visualization.
Table 3.1: Cases in laboratory experiments (1 F;) and numerical simulations (S). f [rad/s] and N [rad/s] are the Coriolis parameter and the Brunt—Vaisala frequency of the background density stratification, respectively. Q, [cm3/s] is the imposed volumetric suction rate. Suc- tion starts at t = 0 and ends at ft, |s]. The working fluid in the experiments is salt-water with p, = 1.02 g/cm? and v = 0.01 cm?/s. In the simulations, p, = 1 g/cm? and v = 0.01 cm?/s, unless otherwise stated. The three col umns on the right show the Rossby number Ro (defined in (3.20)), normalized Brunt—Vaisala frequency at the center of the vortex N,/N, and normalized horizontal divergence evaluated the center of the cyclone, all at the end of suction t = tg.
Table 3.1: Cases in laboratory experiments (1 F;) and numerical simulations (S). f [rad/s] and N [rad/s] are the Coriolis parameter and the Brunt—Vaisala frequency of the background density stratification, respectively. Q, [cm3/s] is the imposed volumetric suction rate. Suc- tion starts at t = 0 and ends at ft, |s]. The working fluid in the experiments is salt-water with p, = 1.02 g/cm? and v = 0.01 cm?/s. In the simulations, p, = 1 g/cm? and v = 0.01 cm?/s, unless otherwise stated. The three col umns on the right show the Rossby number Ro (defined in (3.20)), normalized Brunt—Vaisala frequency at the center of the vortex N,/N, and normalized horizontal divergence evaluated the center of the cyclone, all at the end of suction t = tg.
Figure 3.2: Effect of rotation and stratification on the divergence of the velocity field during suction. The solid lines show the normalized horizontal divergence ((Vi- v)/q)c and the broken lines show the normalized vertical divergence ((Ow/0z)/q)-. (a) The flow in Case S1 (blue, no symbol) is non-rotating (f = 0) and constant—density and has ((V_-v)/q). = 2/% and ((Ow/0z)/q)- = 1/3. The vertical divergence Ow /0z dominates in Cases $2 (red e) whick is rotating and constant—density. The flow in Case $3 (black Ml) is non-rotating (f = 0) anc stratified, and has a dominant horizontal divergence. (b) Case $4 has f/N = 2.5 (blue, nc symbol), Case $5 has f/N = 1 (red e), and Case $6 has f/N = 5 (black Ml).
Figure 3.2: Effect of rotation and stratification on the divergence of the velocity field during suction. The solid lines show the normalized horizontal divergence ((Vi- v)/q)c and the broken lines show the normalized vertical divergence ((Ow/0z)/q)-. (a) The flow in Case S1 (blue, no symbol) is non-rotating (f = 0) and constant—density and has ((V_-v)/q). = 2/% and ((Ow/0z)/q)- = 1/3. The vertical divergence Ow /0z dominates in Cases $2 (red e) whick is rotating and constant—density. The flow in Case $3 (black Ml) is non-rotating (f = 0) anc stratified, and has a dominant horizontal divergence. (b) Case $4 has f/N = 2.5 (blue, nc symbol), Case $5 has f/N = 1 (red e), and Case $6 has f/N = 5 (black Ml).
Figure 3.3: Azimuthal velocity profile along y = z = 0 obtained from the PIV measurements for Cases E1 (blue e) at t/t, = 0.15 and E2 (red Ml) at t/t, = 0.8. Slope of the approximate solid—body rotation in the core Q, gives Ro=Q,/f 1 (E1) and 0.5 (E2).
Figure 3.3: Azimuthal velocity profile along y = z = 0 obtained from the PIV measurements for Cases E1 (blue e) at t/t, = 0.15 and E2 (red Ml) at t/t, = 0.8. Slope of the approximate solid—body rotation in the core Q, gives Ro=Q,/f 1 (E1) and 0.5 (E2).
Figure 3.8: (a) Evolution of the Rossby number Ro = w,/(2f) for Cases S4 (blue solid line), S5 (red broken line), and S7 (black dot-dashed line), where w, is the the vertical vorticity at the center of the vortex. (b) Case S54, evolution of different terms of equation (3.21) at x = 0: (Ow,/0t) (blue marked line), —f(Vi-v)- (red broken line), —w.(Vi-vi)e (black dot-dashed line), and v(V?w,), (light blue solid line).
Figure 3.8: (a) Evolution of the Rossby number Ro = w,/(2f) for Cases S4 (blue solid line), S5 (red broken line), and S7 (black dot-dashed line), where w, is the the vertical vorticity at the center of the vortex. (b) Case S54, evolution of different terms of equation (3.21) at x = 0: (Ow,/0t) (blue marked line), —f(Vi-v)- (red broken line), —w.(Vi-vi)e (black dot-dashed line), and v(V?w,), (light blue solid line).
Figure 3.9: Evolution of Ro during suction in the experiments for Cases E2 (black a), Et E2, ble 3.1). (blue Ml) and E7 (red e). Cases (0.875, —0.24), respectively (see Ta K6, E7 have (f/N,Q,) = D (0.875, —0.6), (1, —1.2)
Figure 3.9: Evolution of Ro during suction in the experiments for Cases E2 (black a), Et E2, ble 3.1). (blue Ml) and E7 (red e). Cases (0.875, —0.24), respectively (see Ta K6, E7 have (f/N,Q,) = D (0.875, —0.6), (1, —1.2)
Figure 3.11: Azimuthal velocity profile obtained from the PIV measurements for Cases E1 (blue e) at (t —t,)/T = 3.5 and E2 (red M) at (t —t,)/T = 12. Slope of the approximate solid—body rotation in the core gives Ro + 0.9 (E1) and Ro & 0.24 (E2).
Figure 3.11: Azimuthal velocity profile obtained from the PIV measurements for Cases E1 (blue e) at (t —t,)/T = 3.5 and E2 (red M) at (t —t,)/T = 12. Slope of the approximate solid—body rotation in the core gives Ro + 0.9 (E1) and Ro & 0.24 (E2).
Figure 3.15: Evolution of N. in experiments and simulations after suction stops for Cases (a) S4 (blue solid line), 55 (red broken line), and $7 (black dot-dashed line); (b) El (blue M), E8 (black A) and E10 (red e). See Table 3.1 for the parameters of each case. T = 41/f is the inertial period.
Figure 3.15: Evolution of N. in experiments and simulations after suction stops for Cases (a) S4 (blue solid line), 55 (red broken line), and $7 (black dot-dashed line); (b) El (blue M), E8 (black A) and E10 (red e). See Table 3.1 for the parameters of each case. T = 41/f is the inertial period.
Figure 3.16: Evolution of aspect ratio calculated from (3.24) (lines) and (3.23) (symbols) for Cases S4 (blue MI), S5 (red e), S6 (black A), and S7 (green HM with broken line).
Figure 3.16: Evolution of aspect ratio calculated from (3.24) (lines) and (3.23) (symbols) for Cases S4 (blue MI), S5 (red e), S6 (black A), and S7 (green HM with broken line).
Figure 4.1: Zombie vortices start near the origin in the x—y plane with subsequent generations sweeping outward in x (the horizontal axis is x, the vertical axis is y). The vorticity w, is red for cyclones; blue for anticyclones; and green for w, = 0. This Couette flow has f/N = 1 and g/N = —3/4. The perturbing vortex at the origin cannot be seen because the plane shown here is at z = —0.404. The x-y computational domain is |x| < 4.7124; |y| < 2.3562, and is larger than shown. (a) t = 64/N. (b) t = 256/N. (c) t = 576/N. (d) t = 2240/N. See text for details.
Figure 4.1: Zombie vortices start near the origin in the x—y plane with subsequent generations sweeping outward in x (the horizontal axis is x, the vertical axis is y). The vorticity w, is red for cyclones; blue for anticyclones; and green for w, = 0. This Couette flow has f/N = 1 and g/N = —3/4. The perturbing vortex at the origin cannot be seen because the plane shown here is at z = —0.404. The x-y computational domain is |x| < 4.7124; |y| < 2.3562, and is larger than shown. (a) t = 64/N. (b) t = 256/N. (c) t = 576/N. (d) t = 2240/N. See text for details.
Figure 4.2: Zombie vortices sweep outward from the perturbing vortex at the origin in the x—z plane shown at y = 0 (the horizontal axis is x, the vertical axis is z). Anticyclonic w, is black and cyclonic is white. This is the same flow as in Figure 4.1. The computational domair has |z| < 4.7124 and is larger than shown. The Rossby number of the initial perturbing anticyclone at the origin is Ro = w,/f = —0.31. (a) t = 128/N. Only the vortex at the origin is present, but critical layers with s = 0 and |m| = 1, 2, and 3 are visible. The faint diagonal lines correspond to internal inertia-gravity waves with shear, not critical layers. (b) t = 480/N. 1*'-generation vortices near |x| = 1 and 1/2 have rolled-up from critical layers with s = 0 and |m| = 1 and 2, respectively. (c) t = 1632/N. 2"¢-generation |m| = 1 vortices near |x| = 0 and 2 were spawned from the 1* generation vortices near |x| = 1. Another 2”¢_veneration of |m| = 1 vortices is near |x| ~ 1/2 and 3/2, which were spawned by the 1% generation near |x| = 1/2. All 2"¢-generation vortices rolled up from critical layers with |m| = 1. (d) t = 3072/N. 1%, 2”¢ and 3"¢ generation vortices.
Figure 4.2: Zombie vortices sweep outward from the perturbing vortex at the origin in the x—z plane shown at y = 0 (the horizontal axis is x, the vertical axis is z). Anticyclonic w, is black and cyclonic is white. This is the same flow as in Figure 4.1. The computational domair has |z| < 4.7124 and is larger than shown. The Rossby number of the initial perturbing anticyclone at the origin is Ro = w,/f = —0.31. (a) t = 128/N. Only the vortex at the origin is present, but critical layers with s = 0 and |m| = 1, 2, and 3 are visible. The faint diagonal lines correspond to internal inertia-gravity waves with shear, not critical layers. (b) t = 480/N. 1*'-generation vortices near |x| = 1 and 1/2 have rolled-up from critical layers with s = 0 and |m| = 1 and 2, respectively. (c) t = 1632/N. 2"¢-generation |m| = 1 vortices near |x| = 0 and 2 were spawned from the 1* generation vortices near |x| = 1. Another 2”¢_veneration of |m| = 1 vortices is near |x| ~ 1/2 and 3/2, which were spawned by the 1% generation near |x| = 1/2. All 2"¢-generation vortices rolled up from critical layers with |m| = 1. (d) t = 3072/N. 1%, 2”¢ and 3"¢ generation vortices.