2.1. Spatial structure of $w_d$ and $w_a$

Here we revisit the spatial structure of the pumping terms in the presence of a uniform wind to better understand their influence on cyclones and anticyclones. The surface stress $\boldsymbol {\tau }$ (1.4) and its curl are calculated using a uniform large-scale northerly wind

(2.2a,b)\begin{equation} \boldsymbol{U}=(0,-V_w),\quad |\boldsymbol{U}| \equiv V_w. \end{equation}

The vortex model is an isolated circular flow whose vorticity and azimuthal velocity profiles in polar coordinates are given by

(2.3)$$\begin{gather} \omega(r)=\omega_0 \left[1-\frac{\alpha}{2} \left(\frac{r}{R} \right)^{\alpha} \right] \exp\left[-\left(\frac{r}{R}\right)^{\alpha}\right], \end{gather}$$

(2.4)$$\begin{gather}v(r)=\frac{\omega_0 r}{2} \exp\left[-\left(\frac{r}{R}\right)^{\alpha}\right], \end{gather}$$

where $\omega _0$ is the peak vorticity, $R$ a horizontal length scale, $r$ the radial distance to the centre of the vortex and $\alpha$ a shape parameter that determines the steepness of the profiles (Carton & McWilliams Reference Carton and McWilliams1989). The structure is isolated because it has zero total circulation: the cyclonic or anticyclonic core is surrounded by an annulus of opposite-sign vorticity. This vortex is often used to model experimental vortices in the laboratory (Kloosterziel & van Heijst Reference Kloosterziel and van Heijst1992; van Heijst & Clercx Reference van Heijst and Clercx2009; Zavala Sansón & van Heijst Reference Zavala Sansón and van Heijst2014), and similar structures are also helpful for representing mesoscale oceanic vortices (e.g. Gaube et al. Reference Gaube, Chelton, Samelson, Schlax and O'Neill2015).

Figure 1 shows the spatial structure of $w_d$ and $w_a$ for three magnitudes of a uniform northerly wind (average, moderately intense and strong) for an intense anticyclonic vortex. The current-induced Ekman pumping (figure 1a–c) has a dominant pattern with an opposite sign of the vortex core (upwelling). In contrast, the vorticity-advection vertical velocity (figure 1d–f) has a dipolar structure with upwelling (downwelling) to the right (left)-hand side of the wind. In both cases, the lower intensity regions surrounding the vortex are of secondary importance and are associated with the chosen vortex model.

Figure 1.

Spatial structure of the pumping terms (a–c) $w_d$ and (d–f) $w_a$ generated by a uniform northerly wind (2.2a,b) with three different magnitudes $V_w$ and blowing over an anticyclonic vortex with vorticity profile (2.3). Red (blue) colours indicate upwelling (downwelling). The vortex parameters are $R=100$ km, $\omega _0/f_0=-0.25$, $\alpha =2$. The vortex is centred at the origin of the coordinate system.

To better appreciate the magnitude and distribution of the vertical velocities, figure 2 presents the profiles $w_d(x,0)$ and $w_a(x,0)$ for cyclones and anticyclones subject to different wind strength. The following are observed.

1. (i) The Ekman velocities in all cases are enhanced for stronger winds. For instance, the current-induced velocity within the anticyclone in figure 2(a) goes from less than 1 m d$^{-1}$ for $V_w=7.5$ m s$^{-1}$ to nearly 3 m d$^{-1}$ for the intense wind $V_w=22.5$ m s$^{-1}$. The upwelling and downwelling generated by the vorticity-advection term are even larger: approximately 6 and $-$6 m d$^{-1}$, respectively, see figure 2(c).

2. (ii) As expected, the profiles are mirror-shaped for cyclones and anticyclones, but the absolute magnitudes are not the same. The positive $w_d$ velocity in the anticyclone is stronger than the negative velocity in the cyclone (figure 2a,b). This difference is an indication that anticyclones decay faster than cyclones.

3. (iii) Regarding the magnitude of $w_a$, note that the downward velocity is larger than the upward velocity in both anticyclones (figure 2c) and cyclones (figure 2d). The asymmetry is caused by the relative wind stress, which is enhanced when the northerly wind and the current flow are in opposite directions.

4. (iv) Both the upwelling and downwelling $w_a$ velocities are more significant in the anticyclone than in the cyclone (compare figures 2c and 2d).

Figure 2.

Profiles along the $x$-direction of the pumping terms $w_d$ and $w_a$ for (a,c) anticyclones and (b,d) cyclones. The colours indicate the wind magnitude: $V_w=7.5$ m s$^{-1}$ (black), $V_w=15$ m s$^{-1}$ (blue), $V_w=22.5$ m s$^{-1}$ (red). Vortex parameters as in figure 1.

The pumping fields lift up or push down water through the mixed layer, but the fluid parcels or any passive tracer is also subject to the swirling flow around the vortex core. Thus, an appropriate way to evaluate the net vertical velocity is by taking azimuthal averages (McGillicuddy et al. Reference McGillicuddy, Ledwell and Anderson2008). Using the azimuthal coordinate $\theta$, we calculate such averages as

(2.5)\begin{equation} \left.\begin{gathered} \bar{w}_d(r)=\int_0^{2{\rm \pi}} w_d \,{\rm d}\theta, \\ \bar{w}_a(r)=\int_0^{2{\rm \pi}} w_a \,{\rm d}\theta. \end{gathered}\right\} \end{equation}

Figure 3(a,b) shows the radial profiles of the azimuthally averaged $w_d$ pumping terms. The velocity scale is kept the same as in figure 2(a,b), so we can appreciate that the integrated magnitude of the pumping terms at the vortex centre is similar to that found in the zonal profiles. In contrast, the integrated $w_a$ profiles are strongly reduced, as shown in figure 3(c,d), because the upwelling side is nearly cancelled out by the downwelling region in both vortices. However, the residual pumping is not zero but slightly negative and off the vortex centre for cyclones and anticyclones. This net downward flow suggests that $\bar {w}_a$ produces a weak damping effect on the cyclone and slightly intensifies the anticyclone, as we shall verify below.

Figure 3.

Radial profiles of the azimuthally averaged pumping terms $w_d$ and $w_a$ for (a,c) anticyclones and (b,d) cyclones. The colours and vortex parameters are as in figure 2.

2.2. Effects of $w_d$: vortex decay

Let us now examine the nonlinear decay of circular vortices due to the Ekman pumping term $w_d$. The change of relative vorticity on the $f$-plane is approximated as

(2.6)\begin{equation} \frac{\partial \omega}{\partial t} = \frac{f_0}{H} w_d. \end{equation}

Using $\omega < f_0$, the Ekman pumping is expanded in Taylor series as

(2.7)\begin{equation} w_d=\frac{1}{\rho_0 (f_0+\omega)}\boldsymbol{\nabla} \times \boldsymbol{\tau} \approx \frac{1}{\rho_0\, f_0} \left(1-\frac{\omega}{f_0} \right) \boldsymbol{\nabla} \times \boldsymbol{\tau}. \end{equation}

Without loss of generality, we concentrate on the large-scale northerly wind (2.2a,b). Then, the surface stress $\boldsymbol {\tau }$ (1.4) can be simplified by using the fact that the wind speed is much faster than the ocean currents, $u,v<< V_w$. After some algebra to simplify the product $|\boldsymbol {U}-\boldsymbol {u}| (\boldsymbol {U}-\boldsymbol {u})$, the stress is approximated as

(2.8)\begin{equation} \boldsymbol{\tau} = \rho_a C_D V_w ({-}u,-V_w-2v). \end{equation}

The curl is

(2.9)\begin{equation} \boldsymbol{\nabla} \times \boldsymbol{\tau} = \rho_a C_D V_w \left({-}2 \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\right). \end{equation}

An equivalent expression for $\boldsymbol {\nabla } \times \boldsymbol {\tau }$ was found by Dewar & Flierl (Reference Dewar and Flierl1987). Assuming that the vortex remains circular, then $\partial v/\partial x = -\partial u/ \partial y$ at the centre, so the stress curl at the origin can be written as

(2.10)\begin{equation} (\boldsymbol{\nabla} \times \boldsymbol{\tau})_c ={-}\tfrac32 \rho_a C_D V_w \omega_c, \end{equation}

where $\omega _c$ is the peak vorticity. Plugging (2.10) in (2.7) yields

(2.11)\begin{equation} w_d ={-}\frac32 \frac{\rho_a C_D V_w}{\rho_0\, f_0} \left(1-\frac{\omega_c}{f_0} \right) \omega_c. \end{equation}

Substituting in (2.6) gives an equation for the peak vorticity decay:

(2.12)\begin{equation} \frac{{\rm d}\omega_c}{{\rm d} t} ={-}\frac32 \frac{\rho_a C_D V_w}{\rho_0 H } \left(1-\frac{\omega_c}{f_0} \right) \omega_c \equiv{-}\frac{1}{Tf_0} (f_0-\omega_c) \omega_c, \end{equation}

where the time scale $T$ is defined as

(2.13)\begin{equation} T=\frac23 \frac{\rho_0 H}{\rho_a C_D V_w}. \end{equation}

The solutions of (2.12) are of the form

(2.14)\begin{equation} \omega_c(t)= \frac{\omega_0 \exp({-t/T})}{1-(\omega_0/f_0)(1-\exp({-t/T}))}, \end{equation}

where $\omega _0$ is the peak vorticity at $t=0$. These are decaying solutions at the characteristic time $T$. The most important feature is that (2.14) distinguishes the vortex polarity in the denominator, implying that anticyclones ($\omega _0<0$) decay faster than cyclones ($\omega _0>0$). The solutions can be called nonlinear because of the form of the evolution equation (2.12). Solutions (2.14) are equivalent – but not the same – to those applied for representing the decay of vortices due to bottom Ekman friction in rotating tank experiments (Zavala Sansón & van Heijst Reference Zavala Sansón and van Heijst2000). The comparison of top and bottom friction effects is further discussed in the last section.

The more conventional (exponential) decay is recovered in the limit of weak vortices $\omega /f_0<<1$. In that case, the vertical velocity simplifies as

(2.15)\begin{equation} w_{lin} = \frac{1}{\rho_0\, f_0} \boldsymbol{\nabla} \times \boldsymbol{\tau}. \end{equation}

Thus, the vorticity equation (2.6) becomes linear:

(2.16)\begin{equation} \frac{{\rm d}\omega_c}{{\rm d} t} ={-}\frac32 \frac{\rho_a C_D V_w}{\rho_0 H} \omega_c \equiv{-}\frac{1}{T} \omega_c, \end{equation}

with exponentially decaying solutions

(2.17)\begin{equation} \omega_c(t)=\omega_0 \exp({-t/T}). \end{equation}

In this approach, cyclones and anticyclones decay at the same rate. The linear decay was discussed by Gaube et al. (Reference Gaube, Chelton, Samelson, Schlax and O'Neill2015), who found the same eddy attenuation time scale $T$ but followed a different approach. They considered a randomly directed wind $\boldsymbol {U}=V_w(\cos \theta,\sin \theta ) > > \boldsymbol {u}$, and then averaged over a uniform $\theta$-distribution to obtain a mean surface stress $\bar {\boldsymbol {\tau }}$ proportional to the surface currents. Using the curl $\boldsymbol {\nabla } \times \bar {\boldsymbol {\tau }}$ in the linear Ekman condition (2.15) yields the same stress curl (2.9) and therefore the exponential decay (2.17) (see also Renault et al. Reference Renault, Molemaker, McWilliams, Shchepetkin, Lemarié, Chelton, Illig and Hall2016).

2.3. Effects of $w_a$: vortex drift

From the azimuthally averaged profiles in figure 3(c,d), we observed that $\bar {w}_a<0$, so this downward velocity slightly intensifies anticyclones and weakens cyclones. However, the effect is relatively weak because the net pumping is small, as we shall corroborate in numerical simulations.

Another effect associated with $w_a$ is the vortex translation in the direction of the Ekman transport, a process correctly predicted by Stern (Reference Stern1965) and, to the best of our knowledge, not discussed in previous studies. From Ekman theory, and ignoring the contribution of a solenoidal vector field (Wenegrat & Thomas Reference Wenegrat and Thomas2017), the transport components are

(2.18)\begin{equation} \left.\begin{gathered} HU_{E}=\frac{\tau^y}{\rho_0 (f_0+\omega)}, \\ HV_{E}={-}\frac{\tau^x}{\rho_0 (f_0+\omega)}, \end{gathered}\right\} \end{equation}

where $(U_{E},V_{E})$ is the integrated velocity in the Ekman layer that the vortex eventually acquires. Since the northerly wind is such that $V_w > > v$, the only relevant stress component from (2.8) is $\tau ^y \approx -\rho _a C_D V_w^2$, which implies that the vortex drifts westward with speed $U_{E}$. Assuming small $\omega /f_0$ again, we approximate the speed using the peak vorticity:

(2.19)\begin{align} U_{E} &\approx \frac{\tau^y}{\rho_0 H f_0^2} (f_0-\omega_c), \nonumber\\ &={-}\frac{\rho_a C_D V_w^2}{\rho_0 H f_0^2} (f_0-\omega_c) \equiv{-}\frac23 \frac{V_w}{f_0 T} \left(1-\frac{\omega_c}{f_0}\right), \end{align}

where we introduced the characteristic time $T$ defined in (2.13). Note that anticyclones move faster than cyclones due to factor $1-\omega _c/f_0$. The vortex speed is approximately constant because $\omega _c$ does not change much under the influence of $w_a$ (as we shall see later). Thus, the vortex displacement after time $T_a$ is

(2.20)\begin{equation} {\rm \Delta} x ={-}\frac23 \frac{V_w T_a}{f_0 T} \left(1-\frac{\omega_c}{f_0}\right). \end{equation}

A conventional (or linear) displacement that would not distinguish between cyclones and anticyclones is obtained by assuming $\omega _c<< f_0$, which yields

(2.21)\begin{equation} {\rm \Delta} x_{lin}={-}\frac23 \frac{V_w T_a}{f_0 T}. \end{equation}

2.4. Relative importance of $w_d$ and $w_a$

There has been some discussion on the relative importance of $w_d$ and $w_a$ for typical oceanic vortices due to their differences in magnitude and spatial structure (McGillicuddy et al. Reference McGillicuddy2007; Mahadevan et al. Reference Mahadevan, Thomas and Tandon2008). Here we propose an alternative view based on our previous results in § 2.2. The relative magnitude of the pumping terms is estimated as

(2.22)\begin{align} \left|\frac{w_a}{w_d}\right| \approx \left|\frac{-\tau^y \partial \omega/\partial x}{(f_0+\omega)\boldsymbol{\nabla} \times \boldsymbol{\tau} }\right| &\approx \left|\frac{(\rho_a C_D V_w^2) \quad (\omega_0/R)}{(f_0+\omega_0) \quad ({-}3/2) \rho_aC_DV_w \omega_0} \right| \nonumber\\ &= \frac{2/3}{(1+Ro)}\frac{|V_w|}{f_0R }, \end{align}

where we considered $\tau ^y \approx -\rho _aC_DV_w^2$ from (2.8), the wind stress curl $\boldsymbol {\nabla } \times \boldsymbol {\tau }$ from (2.10), $f\approx f_0$ and the dynamical Rossby number $Ro=\omega _0/f_0$. Note that the denominator depends on the vortex sign contained in $Ro$, so it is verified that

(2.23)\begin{equation} \left|\frac{w_a}{w_d}\right|_{cyclones} < \left|\frac{w_a}{w_d}\right|_{anticyclones}. \end{equation}

This fact can be observed in figure 4(a), which presents the $|w_a/w_d|$ versus $Ro$ curves for the three wind regimes. For the average wind, the current-induced velocity $w_d$ dominates (black curve). For the moderately intense forcing, $|w_a/w_d|>1$ for anticyclones and $<1$ for cyclones (blue curve). The vorticity-advection velocity $w_a$ is dominant in the strong wind regime (red line).

Figure 4.

Ratio of the pumping terms as a function of the dynamical Rossby number. (a) Ratio based on the peak values of $w_a$ and $w_d$ estimated by (2.22) for three wind speeds. (b) Ratio of the azimuthally integrated pumping terms (2.24). This ratio does not depend on the wind speed. The stars in both panels indicate the anticyclonic and cyclonic cases in figures 2 and 3.

The previous estimate (2.22) only considers the characteristic magnitude of the pumping terms but not their spatial structure. A different comparison consists of using the azimuthally integrated pumping components. For the current-induced velocity $\bar {w}_d$, we can use again $\boldsymbol {\nabla } \times \boldsymbol {\tau }$ from (2.10) because the integrated pumping is approximately the same with or without integration (see the peak values in figures 2a,b and 3a,b). For $\bar {w}_a$, we have to consider that the upwelling at a flank of the vortex cancels most of the downwelling at the other side, and the remaining part is the negative velocity found in figure 3(c,d). Thus, using (2.8), we estimate $\bar {w}_a \propto \overline {\tau ^y \partial \omega /\partial x} \approx (-\rho _aC_D V_w 2v) (\omega _0/R)$. The vortex azimuthal velocity is well approximated by its value at $R/2$, where the negative pumping takes place, so $v\approx \omega _0 R/6$ (to check for the factor 1/6, evaluate, for instance, the azimuthal velocity (2.4) at $r=R/2$). Thus,

(2.24)\begin{align} \left|\frac{\bar{w}_a }{\bar{w}_d }\right|\approx \left|\frac{\overline{-\tau^y \partial \omega/\partial x}}{(f_0+\omega)\boldsymbol{\nabla} \times \boldsymbol{\tau}}\right| &\approx \left|\frac{(\rho_a C_D V_w \omega_0R/3) \quad (\omega_0/R)}{(f_0+\omega_0) \quad ({-}3/2) \rho_aC_DV_w \omega_0} \right| \nonumber\\ &= \frac{2/9}{(1+Ro)}|Ro|. \end{align}

Now the comparison of the Ekman terms does not depend on the wind magnitude as its effect was cancelled out when estimating $\bar {w}_a$. Figure 4(b) shows the resulting curve, which reveals that the top drag velocity dominates, $|\bar {w}_a/\bar {w}_d|<1$, for any (small) Rossby number.

3.1. Dynamical model and description of experiments

We consider a two-dimensional (2-D) homogeneous flow governed by the vorticity equation (2.1). The vorticity-stream function formulation of this model is

(2.1)\begin{equation} \frac{\partial \omega}{\partial t} + J\left(\omega+\beta y,\psi \right) = \frac{f_0}{H} w, \end{equation}

where $J(a,b)=\partial _x a \partial _y b-\partial _y a \partial _x b$ is the Jacobian operator, $H$ a depth scale and $w$ the surface Ekman pumping. The horizontal velocity components are defined from the stream function $\psi$ as

(3.2a,b)\begin{equation} u = \frac{\partial \psi}{\partial y},\quad v ={-} \frac{\partial \psi}{\partial x}. \end{equation}

The relative vorticity is the 2-D Laplacian of the stream function:

(3.3)\begin{equation} \omega ={-}\nabla^2 \psi. \end{equation}

Table 1 shows the parameters that are kept constant in all simulations. The flow domain is a square, closed box with lateral walls at $-L\le x,y\le L$. The fluid depth is constant, and there is no bottom friction. Some simulations are performed on the $f$-plane, in which the flow dynamics are equivalent to the purely 2-D case, while some others are performed on the $\beta$ plane. The simulations are initialised with an intense isolated vortex (2.3) with peak vorticity $\omega _0=0.25f_0$ and radial scale $R=100$ km. Note that we consider a rather strong vortex with maximum velocities of approximately $u_{max}\sim 1$ m s$^{-1}$. The vortex initial position is at the origin $(0,0)$, so the structure is sufficiently far away from the walls ($R/L<1$) at all times. We use no-slip conditions (by setting a small viscous parameter) to confine the wall effects within a thin boundary layer that does not influence the vortex.

Table 1.

Fixed flow parameters in all simulations.

The Ekman pumping $w$ may be either $w_d$ (1.2), $w_a$ (1.3) or the combination $w_d+w_a$, as outlined in table 2. The wind stress is parametrised with the atmosphere-ocean relative velocity (1.4) using a constant wind with magnitude $V_w$ in a fixed, arbitrary direction $\theta$ with respect to the positive $x$-direction:

(3.4)\begin{equation} \boldsymbol{U}=V_w(\cos \theta,\sin \theta),\quad -{\rm \pi} \le \theta \le {\rm \pi}. \end{equation}

For all $f$-plane simulations, we use the ‘northerly’ wind $\theta =-{\rm \pi} /2$, but we also examine other directions in the $\beta$-plane simulations. To simulate strong atmospheric fronts, we focus on intense winds of $V_w=20$ ms$^{-1}$, for which $C_D=2.5 \times 10^{-3}$ (e.g. García-Nava, Ocampo-Torres & Hwang Reference García-Nava, Ocampo-Torres and Hwang2012). The duration of the simulations is 30 days to appreciate significant changes in the vortex strength and position.

Table 2.

Outline of the simulations. The pumping term in experiments A4 and C4 is given by (2.15).

The numerical model is a finite differences code used in previous works to simulate experimental rotating flows over topography (Zavala Sansón & van Heijst Reference Zavala Sansón and van Heijst2014) and turbulent wind-driven flows (Zavala Sansón Reference Zavala Sansón2022). Further details can be consulted in those studies. The nonlinear terms are discretised in a rectangular grid and treated with an Arakawa scheme to avoid the spurious production of energy or enstrophy in the inviscid limit. Time advancement is performed with an explicit, third-order Runge–Kutta method. In all cases shown below, there are $513 \times 513$ grid points ($\textrm {d}x=\textrm {d}y\approx 2.3$ km) and the time step is $\textrm {d} t=3600$ s. Several preliminary runs were performed with a lower resolution of $257 \times 257$ points, and the results were essentially the same.

3.2. Evolution of vortices

To illustrate the effects of the Ekman pumping terms separately, we consider an anticyclonic vortex subject to large-scale northerly wind, $\theta =-{\rm \pi} /2$ in (3.4), equivalent to (2.2a,b) (experiments A1, A2 and A3 in table 2). Although these simulations are performed on the $f$-plane, we will refer to the uniform wind as ‘northerly’ because it is directed downward in the planar plots, and for a better comparison with the $\beta$-plane simulations presented in § 3.4.

Figure 5 presents the vorticity fields at the initial time and 30 days after in the three experiments. When the Ekman pumping is only $w_d$ (figure 5a), the vorticity field decays (as appreciated from the colour scale); however, the vortex stays at its initial position and does not change its shape appreciably. When the pumping is only $w_a$ (figure 5b), the vorticity field does not change appreciably but the vortex drifts westward with a slight southward component. Recall that the vortex drift takes place in the direction of the Ekman transport (westward, for the northerly wind). When the two Ekman pumping components are considered (figure 5c), we observe that the vortex experiences both effects: it decays and moves westward.

Figure 5.

Numerically calculated relative vorticity surfaces in: (a) experiment A1 using only $w_d$; (b) experiment A2 using $w_a$; (c) experiment A3 using $w_d+w_a$ for (top) the initial time and (bottom) 30 days. The flow parameters are shown in table 1. The vertical and horizontal dashed lines are drawn to identify the initial vortex position and displacement.

The vertical velocity fields induced by the pumping terms are shown in figure 6. For $w_d$ (figure 6a), we observe the characteristic upwelling at the central part (flanked by two elongated downwelling regions), which is weakened after 30 d. The up-and-downwelling pattern generated by the vertical velocity $w_a$ (figure 6b) drifts westward with the vortex and does not decay in time. Note that the magnitude of $w_a$ is greater than that of $w_d$ at the initial and final times. When the two effects are considered (figure 6c), we observe an upwelling/downwelling pattern that differs from the current-induced upwelling $w_d$ (in figure 6a) and is more similar to the $w_a$ field (figure 6b). Again, the structure weakens in time and drifts to the west.

Figure 6.

Ekman pumping surfaces in experiments A1, A2 and A3 shown in figure 5: (a) $w_d$; (b) $w_a$; (c) $w_d+w_a$.

To compare the difference between cyclones and anticyclones, figure 7 presents the vorticity profiles along the $x$-direction at times 0 and 30 d. In figure 7(a), we observe that the anticyclone decays more than the cyclone due to $w_d$; figure 7(b) shows that $w_a$ promotes the westward drift of both vortices while the peak vorticity remains constant; finally, figure 7(c) shows the combined effects in the two vortices (decay and displacement). In what follows, we evaluate these effects quantitatively and test the theoretical predictions derived in the previous section.

Figure 7.

Vorticity $x$-profiles (passing over the vortex centre) for cyclones (red) and anticyclones (blue) at $t=30$ d in experiments: (a) A1, C1; (b) A2, C2; (c) A3, C3. The anticyclone profiles are multiplied by $-1$ to facilitate the comparison. The profiles at $t=0$ d are shown with dashed curves.

3.3. Decay of cyclones and anticyclones

Let us consider the influence of the Ekman pumping terms on the vortex strength. Figure 8(a) presents the time evolution of the normalised peak vorticity $\omega _c(t)$ in four simulations: two of them for an anticyclone using $w_d$, $w_a$ (A1 and A2) and the corresponding cases for a cyclone (C1 and C2). The curves show that the peak vorticity in the anticyclone decays faster than in the cyclone when the only pumping term is $w_d$. The analytical expression (2.14) for the nonlinear decay is also shown for both vortices (magenta curves), which agree very well with the numerical results. The decay time scale (2.13) is $T\approx 84$ days. The figure also shows the exponential decay given by (2.17) (dashed line), which lies between the cyclone and anticyclone curves. Regarding the $w_a$ effects, the peak vorticity remains constant because $w_a$ is zero at the centre of a circular vortex, as can be observed in the vorticity profiles in figure 2(c,d). This is also evident by noticing from the $w_a$ definition (1.3) that $\partial \omega /\partial x=\partial \omega /\partial y = 0$ at the centre of a circular structure. However, $w_a$ plays a role in the kinetic energy budget, as discussed below.

Figure 8.

Temporal evolution of the peak vorticity in anticyclonic (blue) and cyclonic (red) vortices subject to different pumping effects. (a) Experiments A1, C1 (using $w_d$) and A2, C2 (using $w_a$). The analytical nonlinear decay (2.14) for cases A1, C1 is indicated with solid magenta lines. The linear decay (2.17) is represented by the dashed curve. (b) Simulations of the same vortices but now using $w_d+w_a$ (experiments A3, C3). The black curve corresponds to simulations with linear decay A4, C4.

In figure 8(b), we compare the peak vorticity decay of the cyclone and the anticyclone in simulations using the two Ekman terms (experiments A3 and C3). The curves are virtually the same as the previous simulations with $w_d$ because the contribution of $w_a$ to this parameter is nearly zero. We also plotted the exponential decay from simulations using $w_{lin}$ (experiments A4 and C4), which collapse into one single curve (black line). Overall, the results prove that the peak vorticity value at the eddy's centre decays faster in the anticyclone than in the cyclone and that the mechanism is the $w_d$ pumping term.

Now we examine the evolution of the kinetic energy integrated over a circular area $S$ surrounding the vortex

(3.5)\begin{equation} E=\frac12 \int_{S} (u^2 + v^2)\,{\rm d} S. \end{equation}

We use $S={\rm \pi} R_s^2$ with $R_s=3R$. Figure 9 shows the energy curves for the same experiments shown in previous plots. In figure 9(a), it is found again that, when using only $w_d$, the kinetic energy in the anticyclone decays faster than in the cyclone. Between the two decreasing curves, we plotted the expected energy curve corresponding to the exponential decay:

(3.6)\begin{equation} E(t)=E(0)\exp({-2t/T}). \end{equation}

The e-folding time $T/2$ is due to the quadratic form of the kinetic energy (see e.g. Zavala Sansón et al. Reference Zavala Sansón, van Heijst and Backx2001; Renault et al. Reference Renault, Molemaker, McWilliams, Shchepetkin, Lemarié, Chelton, Illig and Hall2016). What is different now is the effect of $w_a$: the energy increases in the anticyclone and decays in the cyclone. These changes due to $w_a$ are smaller than the decay due to $w_d$, but they are evident. The cause for these behaviours is the net downwelling flow that results from the azimuthal integration of the $w_a$ dipolar field (described in figure 3c,d). Indeed, the average downward velocity $\bar {w}_a$ is faster in the anticyclone, so its intensification (approximately 12 % in kinetic energy after 30 days) is greater than the weakening of the cyclone (approximately 9 %).

Figure 9.

Temporal evolution of the vortex kinetic energy defined in (3.5) for cyclones (red) and anticyclones (blue). (a) Experiments A1, C1 (using $w_d$) and A2, C2 (using $w_a$). The standard decay (3.6) is represented by the dashed curve. (b) Simulations of the same vortices but now using $w_d+w_a$ (experiments A3, C3).

Figure 9(b) presents the kinetic energy decay of the cyclone and anticyclone in the simulations using the two Ekman terms. The curves almost collapse, with the anticyclonic decay being slightly faster. For the anticyclone, this means that the energy decay due to $w_d$ is compensated by the energy increase due to $w_a$. For the cyclone, both effects produce energy decay but with a smaller magnitude, so the final curve is almost identical to that of the anticyclone.

How is the anticyclone/cyclone asymmetric decay in terms of the wind speed and vortex strength? To explore this question, we performed new experiments using $w_d+w_a$ and measured the time evolution of the anticyclone/cyclone peak vorticity and the kinetic energy ratios, defined as $|\omega ^-|/\omega ^+$ and $E^-/E^+$, respectively. The results are presented in figure 10(a,b) for different combinations of the initial peak vorticity $\omega _0$ and wind speed $V_m$. At $t=0$, the curves start at 1 and, since anticyclones decay faster, the curves decay in time. The resulting ratios for the vortices presented in figures 8(b) and 9(b) are plotted with black lines. Both ratios decay less for weaker vortices and wind speeds (see e.g. the green curves), indicating a more symmetrical behaviour of cyclones and anticyclones.

Figure 10.

Temporal evolution of (a) the peak vorticity ratio $|\omega ^-|/\omega ^+$ between anticyclones and cyclones for different vortex amplitudes $\omega _0$ and wind speeds $V_m$, and (b) the corresponding kinetic energy ratio $E^-/E^+$.

3.4. Vortex drift

We showed in figures 5–7 that the Ekman pumping $w_a$ induces the vortex drift (2.20) to the right of the wind direction. Figure 11 presents the kinetic energy field and the numerically calculated trajectory of anticyclone A2 and cyclone C2 (that is, under the influence of $w_a$ alone) at day 30. The westward displacements are 56.3 km for the anticyclone and 42.2 km for the cyclone. These values agree with the displacements calculated with (2.20), which gives ${\rm \Delta} x=59.6$ and 35.8 km, respectively. Note also that the numerical trajectory is not purely westward, but there is a small southward shift (approximately 9 km in both simulations). Overall, the experiments confirm that the anticyclone drifts to the right of the wind at the predicted speed and faster than the cyclone. The kinetic energy fields demonstrate that the anticyclone is reinforced compared with the cyclone, as shown in figure 9(a).

Figure 11.

Kinetic energy surfaces and velocity vectors in simulations using $w_a$: (a) anticyclone A2; (b) cyclone C2. The vortex displacement is shown with green curves. The maximum velocity is 0.55 m s$^{-1}$. Vertical and horizontal dashed lines are drawn to identify the vortex initial position at the origin.

Now we will examine the vortex trajectory on the $\beta$-plane. Recall that nonlinear barotropic anticyclones (cyclones) on a $\beta$-plane drift southwestward (northwestward) (Adem Reference Adem1956). In particular, intense shielded vortices as those defined by (2.3) follow a trochoid trajectory with a strong meridional component (Zavala Sansón, van Heijst & Doorschot Reference Zavala Sansón, van Heijst and Doorschot1999; Flór & Eames Reference Flór and Eames2002). Then we may ask, what is the effect of the Ekman term $w_a$ on the vortex drift? To answer this question, we performed several simulations for cyclones and anticyclones (see table 2). Note that we only consider $w_a$ because it is the pumping term responsible for the vortex drift, so we omit the decaying effects associated with $w_d$. It must be taken into account that now the wind direction is of fundamental importance. Thus, we analysed the vortex trajectories in five simulations: a reference experiment with no Ekman terms (cases A0 and C0, dashed lines), and four simulations using $w_a$ and having the wind provenance from the cardinal directions (figure 12).

Figure 12.

Vortex trajectories calculated in $\beta$-plane simulations with different wind provenance: (a) northerly wind (simulations AN, CN); (b) southerly (AS, CS); (c) easterly (AE, CE); (d) westerly (AW, CW). Cyclone (anticyclone) trajectories are denoted with red (blue) curves. In all panels, the trajectories in the absence of wind are shown with dashed lines (experiments A0, C0). In each panel, the black arrow indicates the wind direction and the magenta arrow the direction of the Ekman transport.

Let us focus first on the anticyclones (drifting southwestward – blue curves). In all cases, we can observe that the trajectory is deflected from the reference run according to the wind direction. For instance, for the northerly wind (figure 12a), the vortex trajectory is deflected to the west. In contrast, for southerly wind (figure 12b), the Ekman drift is eastward, and the trajectory is deflected more to the south. Note that the $\beta$-drift may be arrested (figure 12c) or augmented (figure 12d). For cyclones (drifting northwestward), we observe that the trajectories are also deflected from the reference experiment depending on the wind direction. An exception is the case shown in figure 12(d) (westerly wind). In general, the nonlinear character of the problem rules out the possibility of explaining the details of the trajectories because of the complex convolution of the vorticity contours as the eddies drift, and the radiation of Rossby waves (McWilliams & Flierl Reference McWilliams and Flierl1979; Flór & Eames Reference Flór and Eames2002).

3.5. Extended vorticity equation

Let us examine additional simulations based on the vorticity equation that retains the absolute vorticity in the forcing term

(3.7)\begin{equation} \frac{\partial \omega}{\partial t} + J\left(\omega+\beta y,\psi \right) = \frac{(f+\omega)}{H} w_s. \end{equation}

Although we show in Appendix A that the $\beta$ and $\omega$ terms on the right-hand side can be neglected because of magnitude grounds (so the dominant forcing term is $f_0 w/H$), here we explore the consequences of keeping them.

When using the Ekman pumping terms $w_d$ and $w_a$ in (3.7), the absolute vorticity $f+\omega$ cancels out, such that:

(3.8)\begin{equation} \frac{(f+\omega)}{H}(w_d+w_a)= \frac{1}{H\rho_0}\boldsymbol{\nabla} \times \boldsymbol{\tau} + \frac{1}{H\rho_0 (f+\omega)} \left(\tau^x \frac{\partial \omega}{\partial y} - \tau^y \frac{\partial \omega}{\partial x}\right). \end{equation}

Thus, the forcing term $(f+\omega )w_d/H$ becomes identical to the linear case $f_0w_{lin}/H$, with $w_{lin}$ given by (2.15). A notorious consequence is that cyclones and anticyclones now decay at the same rate: $w_d$ no longer distinguishes the vortex polarity. In contrast, the effects due to $w_a$ remain nonlinear, so its influence on vortices is similar as before: $w_a$ weakens (intensifies) cyclones (anticyclones) and promotes their drift to the right of the wind direction.

To show these assertions, we reproduced all the simulations in table 2 but now using (3.7). In general, the results are qualitatively similar, though there are important differences to mention. We focus on the global energy evolution. Figure 13 presents the vortex energy for the same cases shown in figure 9. First, note that in figure 13(a), when using $w_d$, the cyclone and the anticyclone decay in the same fashion: the curves are overlapped and are given by (3.6). Second, when using $w_a$, the energy evolves in a similar way as in figure 9: the anticyclone is intensified and the cyclone weakened. Interestingly, when using $w_d+w_a$ (figure 13b), the net energy decay of the cyclone is notoriously faster than that of the anticyclone. Of course, the difference is due to $w_a$, since $w_d$ produces the same decay.

Figure 13.

Temporal evolution of the vortex kinetic energy of cyclones (red) and anticyclones (blue) in experiments using the extended vorticity equation (3.7). Compare with the curves shown in figure 9: (a) A1, C1 (using $w_d$) and A2, C2 (using $w_a$). A1, C1 decay at the same rate (the curves are overlapped); (b) A3, C3 ($w_d+w_a$).